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UBC Theses and Dissertations

Die and mould making using the polyhedral concept Lau, Charles Yu-Kit 1984

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DIE AND MOULD MAKING USING THE POLYHEDRAL CONCEPT by CHARLES YU-KIT LAU B a c h e l o r Of E n g i n e e r i n g , McMaster U n i v e r s i t y , 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF ' THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y 1984 © C h a r l e s Y u - K i t L a u , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y of B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : J a n u a r y 31, 1984 i i A b s t r a c t The u l t i m a t e g o a l of many e n g i n e e r i n g p u r s u i t s i s t h e a p p l i c a t i o n o f s c i e n c e and m a t h e m a t i c s t o t h e p r o d u c t i o n of m a n u f a c t u r e d p r o d u c t s . M a n u f a c t u r i n g i s t h e t r a n s f o r m a t i o n of a d e s i g n e r s ' s i d e a s i n t o t h r e e - d i m e n s i o n a l o b j e c t s w i t h p r a c t i c a l a p p l i c a t i o n i n t h e r e a l w o r l d . M a n u f a c t u r e of p r o d u c t s r e q u i r e t o o l s ( d i e s , m o u l d s , p u n c h e s , e t c . ) i n p r o c e s s e s r a n g i n g f r o m c a s t i n g and i n j e c t i o n -m o u l d i n g t o f o r g i n g , p u n c h i n g and c o i n i n g . T h e s e t o o l s , as a r e a l l t h r e e - d i m e n s i o n a l . s o l i d s , a r e bounded by s u r f a c e s . D i f f e r e n t m a n u f a c t u r i n g p r o c e s s e s p r e s e n t d i f f e r e n t p r o b l e m s t o d e s i g n e r s ; f o r example, s h r i n k a g e and f l a s h i n c a s t i n g and s p r i n g - b a c k i n f o r g i n g or deep d r a w i n g . The t r a d i t i o n a l a p p r o a c h i n t o o l and d i e - m a k i n g i s b a s e d on e x p e r i e n c e d p a t t e r n -makers or s c u l p t o r s making t h e r e q u i r e d o b j e c t b a s e d on e n g i n e e r i n g b l u e - p r i n t s a s w e l l as t h e i r own i n t u i t i o n and judgement. W i t h t h e a d v e n t o f h i g h s p e e d c o m p u t e r s and n u m e r i c a l l y c o n t r o l l e d m a c h i n e s , t h e s e t r a d i t i o n a l p r o c e d u r e s can be i n c o r p o r a t e d i n t o an i n t e g r a t e d a p p r o a c h by a p p l y i n g CAD/CAM t e c h n i q u e s . The p u r p o s e of t h i s r e s e a r c h i s t o d e v e l o p s u c h g e n e r a l methods f o r t h e m o d e l l i n g and making of d i e s and m o u l d s . C a v i t y d i e s c o n s i s t o f b o u n d i n g s u r f a c e s t h a t a r e e i t h e r a n a l y t i c a l o r n o n - a n a l y t i c a l . A n a l y t i c a l s h a p e s a r e u s u a l l y d e s i g n e d s u r f a c e s w hich a r e c o m b i n a t i o n s of s u r f a c e - e l e m e n t s r e p r e s e n t e d by w e l l known m a t h e m a t i c a l e q u a t i o n s . Non-a n a l y t i c a l s h a p e s a r e o f t e n n a t u r a l s u r f a c e s d e f i n e d by r a n d o m l y measured d a t a . These r e q u i r e s o r t i n g and o r d e r i n g . In a d d i t i o n , s h a p e s s u c h as d u c t s , s h e l l s and b o t t l e s l e n d t h e m s e l v e s t o s p e c i a l t r e a t m e n t s r e q u i r i n g t h e i n p u t of p a r t i c u l a r p a r a m e t e r s f o r p r o d u c t i o n of s i m i l a r i t e m s o v e r a l o n g p r o d u c t i o n r u n . In t h e work which f o l l o w s , a l l o f t h e s e t y p e s of d i e -c a v i t i e s have been e x a m i n e d . Examples a r e g i v e n t o show how v a r i o u s r e q u i r e m e n t s may be h a n d l e d by an i n t e g r a t e d CAD/CAM a p p r o a c h . Computer r o u t i n e s have been d e v e l o p e d i n s u c h a way t h a t no s p e c i a l s k i l l s i n m a t h e m a t i c s and programming a r e r e q u i r e d on t h e p a r t o f t h e u s e r o f t h e p r o g r a m s w h i c h can be i n c o r p o r a t e d i n t o a low c o s t , f u l l y a u t o m a t e d t u r n - k e y s y s t e m . T a b l e of C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s v i i L i s t o f F i g u r e s v i i i A cknowledgement x C h a p t e r I INTRODUCTION .1 1. THE INCIDENCE OF SURFACES IN ENGINEERING 1 2. RELATIONS BETWEEN SURFACE DESIGN AND MANUFACTURING 1 3. THE ROLES OF MOULDS AND DIES IN SURFACE-FORMING 6 4. THE MAKING OF DIES AND MOULDS 7 5. OBJECTIVES OF RESEARCH . 10 C h a p t e r I I THE TECHNICAL/MATHEMATICAL FEATURES OF SURFACE AS AN ENTITY 11 1. PHYSICAL SURFACES DEFINED BY ANALYTICAL FUNCTIONS 11 2..PHYSICAL SURFACES AS A MANIFOLD OF POINTS 12 3. METHODS OF SURFACE DEFINITION 14 4. SURFACE INTERPOLATION . 15 5. THE POLYHEDRAL CONCEPT 16 C h a p t e r I I I SCULPTURED DIE-SURFACES 21 1. GENERAL FEATURES OF DIES . 21 1.1 C h a r a c t e r i s t i c s Of D i e C a v i t i e s 21 2. DESIGN AND MACHINING OF DIES USING THE CAD/CAM APPROACH 23 2.1 M a c h i n i n g Of D i e s By The POLYHEDRAL NC System 23 2.2 C o m p u t a t i o n Of S u r f a c e - r e l a t e d P r o p e r t i e s U s i n g The P o l y h e d r a l C o n c e p t 25 C h a p t e r IV THE ANALYTICAL DIE 27 1. PIECEWISE ANALYTICAL AND COMPOUND SURFACES 27 2. MODELLING OF COMPOUND SURFACES USING THE METHOD OF HIGHEST POINT 27 V 2.1 Subdomains W i t h i n The G l o b a l Domain 31 2.2 M u l t i v a l u e d S u r f a c e s And N a t u r a l L i m i t s 31 2.3 G e n e r a l P r o c e d u r e F o r E x e c u t i n g The Method Of H i g h e s t P o i n t 34 3. GEN7 : A GENERAL PROGRAM FOR EXECUTING THE METHOD OF HIGHEST POINT 35 3.1 G e n e r a l E q u a t i o n Of A Q u a d r i c S u r f a c e 37 3.2 G e n e r a l T r a n s f o r m a t i o n Of Axes 38 3.3 S t r u c t u r e Of GEN7 43 3.4 Sample Runs Of GEN7 49 3.4.1 P i p e - T e e P a t t e r n 49 3.4.2 A u t o m o b i l e Rear Lamp Punch Model 51 3.4.3 Vacuum C l e a n e r H o u s i n g Punch Model 53 3.4.4 O t h e r Examples 57 4. COMPOUND SURFACES WITH NON-ANALYTICAL'SURFACE-PIECES ..59 5. MACHINING OF A DIE 60 C h a p t e r V THE NON-ANALYTICAL DIE 61 1. ARBITRARY ( FREE FORM ) SURFACES 61 1.1 Measurement Of A r b i t r a r y S u r f a c e s 61 2. MACHINING OF CAVITY MOULDS FOR MEASURED SURFACES .64 3. EXAMPLES ON REPLICATING MEASURED SURFACES 65 3.1 R a d i u s Bone 65 3.2 F a c i a l M o u l d 71 3.3 Ox T i b i a Bone 78 4. GENERAL PROCEDURE FOR REPLICATING A MEASURED SURFACE ..90 5. OTHER CONSIDERATIONS 90 C h a p t e r VI SPECIAL DIE CAVITY SURFACES 91 1. SPECIALIZED DIES 91 2. SPECIALIZED MOULD FOR A SHELL 94 2.1 I n t r o d u c t i o n 94 2.2 O r g a n i z a t i o n Of M a c h i n i n g P r o c e s s 94 2.2.1 C a l c u l a t i o n Of C u t t e r L o c a t i o n D a t a 99 2.2.2 M a c h i n i n g Of D i e s 103 2.3 E x t e n s i o n Of Method 108 C h a p t e r V I I DISCUSSIONS AND CONCLUSIONS 111 1. CONSIDERATIONS IN DIE AND MOULD MAKING 111 v i 2. CASTING AND MOULDING OF MODELS 112 3. DIE DESIGN AND MACHINING SYSTEM BASED ON POLYHEDRAL NC SYSTEM 114 3.1 Work A c h i e v e d In T h i s R e s e a r c h 115 3.2 Scheme F o r P r o p o s e d D i e D e s i g n And M a c h i n i n g S y s t e m 116 4. PROPOSED FURTHER WORK 121 5. CONCLUSIONS 121 APPENDIX A - GENERAL TRANSFORMATION OF QUADRIC SURFACES .123 APPENDIX B - USER MANUAL FOR PROGRAM GEN7 128 1 . HOW TO RUN 128 2. USER-INPUTS 128 3 . PROGRAM OUTPUTS 1 28 4. SAMPLE INPUTS 129 5. PROGRAM LISTING FOR GEN7 131 APPENDIX C - USER MANUAL FOR PROGRAM CAVITY6 140 1 . HOW TO RUN 14 0 2 . USER INPUTS 140 3. PROGRAM OUTPUT 140 4. SAMPLE INPUT 141 5. PROGRAM LISTING FOR CAVITY6 142 BIBLIOGRAPHY 146 V I I L i s t of T a b l e s I n p u t F a r a m e t e r s f o r GEN7 45 Summary of R o u t i n e s t o be u s e d f o r P r o p o s e d D i e D e s i g n and M a c h i n i n g System 120 v i i i L i s t o f F i g u r e s 1.1 Model o f t h e p u n c h i n g d i e of an a u t o m o b i l e r e a r lamp h o u s i n g 2 1.2 Model o f a human f a c e 2 1.3 S k e t c h e s o f i n j e c t i o n m o u l d i n g s y s t e m s 4 1.4 Some common m e t a l s t r e t c h i n g and s q u e e z i n g o p e r a t i o n s . 5 1.5 T o o l - p a t h f o r a c u t t e r f o r m i l l i n g and d r i l l i n g 9 2.1 P h y s i c a l s u r f a c e d e f i n e d by c l o s e l y s p a c e d d a t a p o i n t s • 13 2.2 S u r f a c e a p p r o x i m a t i o n a s an i r r e g u l a r p o l y h e d r o n 17 2.3 C a l c u l a t i o n o f c u t t e r l o c a t i o n d a t a by p o l y h e d r a l c o n c e p t 20 3.1 T y p i c a l f e a t u r e s of a c a v i t y - d i e • 22 3.2 C l o s e r c o n f o r m i t y of s p h e r i c a l l y ended m i l l i n g c u t t e r t o f e m a l e c a v i t y 24 3.3 C o m p u t a t i o n of s u r f a c e - p r o p e r t i e s u s i n g t h e p o l y h e d r a l c o n c e p t 26 4.1 T y p i c a l e n g i n e e r i n g component c o n t a i n i n g s i m p l e a n a l y t i c e l e m e n t s of p r i s m s and c y l i n d e r s 28 4.2 M o d e l l i n g o f compound s u r f a c e s u s i n g t h e Method o f H i g h e s t P o i n t 30 4.3 Subdomain w i t h i n g l o b a l domain 32 4.4 N a t u r a l l i m i t s f o r s u r f a c e - e l e m e n t s 33 4.5 S u r f a c e - e l e m e n t s f o r p r o g r a m GEN7 36 4.6 T r a n s f o r m a t i o n of a x e s f o r a q u a d r i c s u r f a c e - p i e c e .... 38 4.7 T y p i c a l p r o m p t i n g s e q u e n c e o f GEN7 48 4.8 S k e t c h o f a p i p e - t e e p a t t e r n 49 4.9 P i p e j u n c t i o n t e e m o d e l l e d by GEN7 50 4.10 P r i n c i p a l s e c t i o n s of an a u t o m o b i l e r e a r lamp punch ... 51 4.11 T a i l lamp punch m o d e l l e d by GEN7 52 4.12 I n i t i a l p r o p o s e d s k e t c h o f vacuum c l e a n e r h o u s i n g mould 54 4.13 P r i n c i p a l s e c t i o n s a d o p t e d f o r vacuum c l e a n e r h o u s i n g mould 54 4.14 T u b u l a r s u r f a c e w i t h p a r a b o l i c p r o f i l e 55 4.15 Vacuum c l e a n e r h o u s i n g mould m o d e l l e d by GEN7 56 4.16 I n c l i n e d c y l i n d e r w i t h e l l i p s o i d 57 4.17 Compound s u r f a c e m o d e l l e d by GEN7 58 4.18 M a r r i a g e o f a n a l y t i c a l and n o n - a n a l y t i c a l p i e c e s by Method of H i g h e s t P o i n t 59 5.1 S c h e m a t i c c o n f i g u r a t i o n o f a CAT s c a n n e r 62 5.2 G o v e r n i n g b o u n d a r y - c u r v e s o f a v i o l i n 63 5.3 Shadow M o i r e f r i n g e s o f human r a d i u s bones 65 5.4 Computer r e p l o t o f r e g i o n o f i n t e r e s t 66 5.5 P e r s p e c t i v e p l o t of b o n e - s u r f a c e f i t t e d by TRUEPERS ... 67 5.6 P l o t o f c a v i t y - m o u l d ( f e m a l e s u r f a c e ) f o r r a d i u s bone . 68 5.7 M a c h i n e d models o f male and f e m a l e s u r f a c e s 69 5.8 Computer r e p l o t of c o n t o u r map d e f i n i n g human f a c e .... 72 5.9 C o n t o u r map w i t h added d a t a f o r b a s e - p l a n e 73 5.10 F i t t e d s u r f a c e of human f a c e by TRUEPERS 75 i x 5.11 P e r s p e c t i v e p l o t of f e m a l e mould 76 5.12 M a c h i n e d c a v i t y - m o u l d and p l a s t e r - m o d e l o f a human f a c e 77 5.13 R e p l o t s o f s l i c e s o f c r o s s - s e c t i o n s o f ox t i b i a bone .. 80 5.14 T h r e e v i e w s of p r o j e c t e d shape of ox t i b i a bone by s u p e r i m p o s i t i o n of s l i c e s (a) f r o n t view 81 (b) p r o f i l e view 82 (c ) t o p vie w 83 5.15 T h r e e v i e w s o f s u p e r i m p o s e d s l i c e s a f t e r t r a n s f o r m a t i o n (a) f r o n t v i e w 84 (b) p r o f i l e view 85 (c) t o p v i e w 86 5.16 D a t a f o r ' h o l e s ' a r e b l a n k e d o u t 87 5.17 Data f o r b a s e - p l a n e 88 5.18 F i t t e d s u r f a c e f o r bone and c o r r e s p o n d i n g f e m a l e mould 89 6.1 T o o l - p a t h f o r m a c h i n i n g a c i r c u l a r c y l i n d e r 92 6.2 T o o l - p a t h f o r m a c h i n i n g a d u c t 93 6.3 G o v e r n i n g b o u n d a r y - c u r v e s of a shoe-mould 93 6.4 C a v i t y - d i e o f a s h e l l - m o u l d 95 6.5 Upper d i e - b l o c k made by c a s t i n g i n t o m a c h i n e d c a v i t y .. 96 6.6 C u t t i n g p a t h f o r upper c a v i t y 98 6.7 E l l i p t i c a l f l a s h - l i n e of d i e 101 6.8 T o o l - m o t i o n f o r c u t t i n g e d g e - w a l l 102 6.9 Lower c a v i t y and c o r r e s p o n d i n g t o o l - p a t h 104 6.10 Upper c a v i t y and c o r r e s p o n d i n g t o o l - p a t h 105 6.11 M a c h i n e d d i e - b l o c k f o r l o w e r c a v i t y 106 6.12 M a c h i n e d s u r f a c e f o r u p p e r c a v i t y 106 6.13 M a c h i n i n g upper c a v i t y w i t h l a r g e t o o l g i v e s "more f l a s h 107 6.14 S h e l l mould w i t h a r b i t r a r y b o u n d i n g s u r f a c e s 109 6.15 A r b i t r a r y f l a s h - l i n e g i v e s n o n - s y m m e t r i c , c y l i n d r i c a l l y c u r v e d p a r t i n g s u r f a c e s 110 7.1 S h e l l - m o u l d of f a c i a l mask from s l i p c a s t i n g 113 7.2 S c h e m a t i c a p p r o a c h t o a g e n e r a l d i e d e s i g n and m a c h i n i n g s y s t e m 117 X Ac knowledgement The a u t h o r w i s h e s t o e x p r e s s h i s most s i n c e r e g r a t i t u d e t o h i s s u p e r v i s o r , P r o f e s s o r James P. Duncan, f o r h i s g u i d a n c e t h r o u g h o u t t h e p r e p a r a t i o n of h i s t h e s i s , p a r t i c u l a r l y f o r b e i n g so c o n s i d e r a t e , e n c o u r a g i n g and e n l i g h t e n i n g . The a u t h o r a l s o w i s h e s t o t h a n k Mr. A l a n S t e e v e s f o r h i s h e l p i n d e v e l o p i n g many of t h e s o f t w a r e r o u t i n e s u s e d f o r t h i s r e s e a r c h , t o Mr. K e n n e t h Law and Mr. P a u l L o u i e f o r t h e i r h e l p i n t h e measurement of d a t a , and t o e v e r y b o d y i n t h e D epartment of M e c h a n i c a l E n g i n e e r i n g f o r t h e i r warmth, c o o p e r a t i o n and e s p e c i a l l y t h e i r s e n s e of humour, w i t h o u t w h i c h t h e months s p e n t a t t h e D e p a r t m e n t would be a l o t l e s s e n j o y a b l e . S u p p o r t f o r t h i s r e s e a r c h was p r o v i d e d by t h e N a t i o n a l S c i e n c e and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada. 1 I . INTRODUCTION 1. THE INCIDENCE OF SURFACES IN ENGINEERING A l l o b j e c t s t h a t e x i s t i n p h y s i c a l s p a ce a r e bounded o r c o n t a i n e d by s u r f a c e s . A s u r f a c e can be c o n s i d e r e d as t h e i n t e r f a c e between p a r t s of s p a c e h a v i n g d i f f e r e n t p h y s i c a l a t t r i b u t e s . In most e n g i n e e r i n g a p p l i c a t i o n s , a s u r f a c e i s v i e w e d a s t h e i n t e r f a c e between a s o l i d o b j e c t and i t s a t m o s p h e r i c s u r r o u n d i n g s . E n g i n e e r s have a l w a y s been c o n c e r n e d w i t h t h e d e s i g n of t h r e e - d i m e n s i o n a l o b j e c t s , t h e c h a r a c t e r i s t i c s of w h i c h a r e t h e i r b o u n d i n g s u r f a c e s . An example i s shown i n F i g u r e 1.1 w h i c h shows t h e model o f a punch f o r f o r m i n g an a u t o m o b i l e r e a r lamp h o u s i n g . T h i s i t e m has a s u r f a c e w h i c h i s a c o m b i n a t i o n of s i m p l e a n a l y t i c a l s u r f a c e e l e m e n t s . O t h e r s u r f a c e s , s u c h as human a n a t o m i c a l p a r t s , may not be a n a l y t i c a l i n n a t u r e . F i g u r e 1.2 shows a model o f a human f a c e u s e d f o r b i o m e d i c a l e n g i n e e r i n g r e s e a r c h a p p l i c a t i o n s . 2. RELATIONS BETWEEN SURFACE DESIGN AND MANUFACTURING N e a r l y a l l e n g i n e e r i n g p u r s u i t s l e a d t o t h e d e s i g n and m a n u f a c t u r e o f t h r e e - d i m e n s i o n a l components. The u l t i m a t e g o a l of an e n g i n e e r i s t h e a p p l i c a t i o n o f s c i e n c e and m a t h e m a t i c s t o t h e p r o d u c t i o n of m a n u f a c t u r e d p r o d u c t s w i t h p r a c t i c a l a p p l i c a t i o n s i n t h e r e a l w o r l d . 2 Figure 1.2 Model of a human face 3 The c h o i c e of s u r f a c e - f o r m f o r any e n g i n e e r i n g component i s o f t e n t h e r e s u l t of compromises between low m a n u f a c t u r i n g c o s t and f u n c t i o n a l r e q u i r e m e n t s . F o r example, p l a n e s and c y l i n d e r s c an e a s i l y be g e n e r a t e d and t u r n e d on s i m p l e machine t o o l s , and a r e a d o p t e d as t h e b u i l d i n g b l o c k s o f most d e s i g n s . However, r e q u i r e m e n t s i n s o l i d m e c h a n i c s , f l u i d d y n a m i c s , a c o u s t i c s , o p t i c s , e t c . , may n e c e s s i t a t e complex s u r f a c e - s h a p e s and o v e r r u l e t h e c o n s i d e r a t i o n s o f e a s y m a n u f a c t u r e . M a n u f a c t u r i n g i s t h e t r a n s f o r m a t i o n of a d e s i g n e r ' s i d e a s i n t o t h r e e - d i m e n s i o n a l o b j e c t s . More s p e c i f i c a l l y , i t can be c o n s i d e r e d as t h e f o r m i n g o f t h e b o u n d i n g s u r f a c e s o f a p a r t i c u l a r component by m a n i p u l a t i o n o f v a r i o u s raw m a t e r i a l s . Most m a n u f a c t u r i n g p r o c e s s e s c a n be c a t e g o r i s e d i n t o one of t h e f o l l o w i n g b a s i c p r o c e s s e s : i C a s t i n g / M o u l d i n g : T h i s i n c l u d e s p r o c e s s e s s u c h as s a n d and d i e c a s t i n g , i n j e c t i o n m o u l d i n g , e t c . , and g e n e r a l l y i n v o l v e s f i l l i n g c a v i t y - m o u l d s or d i e s w i t h l i q u i d o r p l a s t i c m a t e r i a l s . Some examples a r e shown i n F i g u r e 1.3. 4 ( i ) S ingle-stage Plunger type Heotingcyl. Injector-screw Screw drive ( i i ) S ingle-stage R e c i p r o c a t i n g Scew Type chamber Firs'-stage injection \\\*\M from screw ^vJ or plunger ( i i i ) Two-stage Plunger or Scr e w - P l a s t i c s o r types Second -stoge injection F i g u r e 1.3 S k e t c h e s o f i n j e c t i o n - m o u l d i n g s y s t e m s 5 i i M e c h a n i c a l W o r k i n g : Many shapes and forms a r e p r o d u c e d by m e c h a n i c a l w o r k i n g o f m e t a l s i n p r o c e s s e s r a n g i n g f r o m s h e e t m e t a l r o l l i n g , f o r g i n g , d r a w i n g t o p u n c h i n g , h o b b i n g and c o i n i n g . E xamples a r e shown i n F i g u r e 1.4 Stretch Forming Sheet m e t a l — . Jows Jaw Hydraulic o  jack Stretch Draw Forming Ram Hole Stretching or Burring A . Pre-punched hole " Tj -! Punch Necking Workpiece Stop b locks^ s \ ^ \ \ \ \ \ \ \ \ \ \ ^ \ \ V / T Workpiece P u n c n Workpiece ) c Reduction on si Retoining /Impression Blank 'VBolster ploteV/ Hobbing EES ESfl rC(M<C<<lv Gear Stud Staking Ironing Riveting F i g u r e 1.4 Some common m e t a l s t r e t c h i n g and s q u e e z i n g o p e r a t i o n s 6 i i i J o i n i n g Complex s t r u c t u r e s a r e o f t e n f a b r i c a t e d by j o i n i n g s i m p l e r e l e m e n t s u s i n g p r o c e s s e s s u c h as w e l d i n g , b r a z i n g , s o l d e r i n g o r a d h e s i v e b o n d i n g . i v C u t t i n g / E r o s i o n Many components a r e c u t i n t o t h e i r f i n a l s h a p e s by means s u c h as m a c h i n i n g o r f l a m e c u t t i n g . O t h e r s a r e formed by c h e m i c a l o r e l e c t r i c a l e r o s i o n i n p r o c e s s e s s u c h a s ECM ( e l e c t r o c h e m i c a l m a c h i n i n g ) o r EDM ( e l e c t r i c d i s c h a r g e m a c h i n i n g ), t o name o n l y a few. 3. THE ROLES OF MOULDS AND DIES IN SURFACE-FORMING In many of t h e p r o c e s s e s d e s c r i b e d above ( and shown i n F i g u r e s 1.3 and 1.4 ), n o t a b l y i n c a s t i n g , m o u l d i n g , f o r g i n g , p u n c h i n g , c o i n i n g and h o b b i n g , t o o l s i n t h e form of d i e s , moulds o r punches a r e r e q u i r e d . The d e s i g n and making o f d i e s and moulds a r e t h e r e f o r e v e r y i m p o r t a n t f o r m a n u f a c t u r i n g . These t o o l s , as a r e a l l t h r e e - d i m e n s i o n a l o b j e c t s , a r e bounded by s u r f a c e s . In d e s i g n i n g t h e i r b o u n d i n g s u r f a c e s , a d e s i g n e r i s f a c e d w i t h a d d i t i o n a l p r o b l e m s p r e s e n t e d by d i f f e r e n t m a n u f a c t u r i n g p r o c e s s e s . F o r i n s t a n c e , s h r i n k a g e and f l a s h i n c a s t i n g p r o c e s s e s , ' s p r i n g - b a c k ' i n f o r g i n g and deep-d r a w i n g , e t c . , must be t a k e n i n t o c o n s i d e r a t i o n d u r i n g t h e d e s i g n s t a g e . 7 4. THE MAKING OF DIES AND MOULDS A l t h o u g h a l l e n g i n e e r i n g d e s i g n s e x i s t i n t h r e e - d i m e n s i o n a l s p a c e , t h e t r a d i t i o n a l a p p r o a c h has a l w a y s been f o r t h e d e s i g n e r t o c o n v e y h i s i d e a s i n t w o - d i m e n s i o n a l d r a w i n g s . In t o o l and d i e - m a k i n g , t h e g e o m e t r i c a l s p e c i f i c a t i o n s have been p r e s e n t e d i n t h e form o f b l u e p r i n t s . An e x p e r i e n c e d p a t t e r n - m a k e r , f o l l o w i n g t h e i n s t r u c t i o n s on t h e d r a w i n g s as w e l l as h i s own i n t u i t i o n a nd judgement, h a s , i n t h e p a s t , d e v i s e d p r o c e d u r e of e x e c u t i o n and t h e d i r e c t i o n o f machine t o o l s t o make t h e r e q u i r e d p r o d u c t . When s u r f a c e - g e o m e t r y i s c o m p l i c a t e d , t h e o b j e c t must be s c u l p t u r e d . T r a d i t i o n a l l y , t h i s t y p e o f s c u l p t u r e d s u r f a c e has been hand-made by e x p e r i e n c e d s c u l p t o r s . A p h y s i c a l model i s f i r s t s c u l p t u r e d on s o f t m a t e r i a l s s u c h as wax, p l a s t e r , c l a y or wood. Then t h e r e q u i r e d mould i s made u s i n g one of many r e v e r s a l p r o c e s s e s . I n v a r i b l y , t h e form of t h e f i n a l mould depends l a r g e l y on t h e e x p e r i e n c e and s k i l l of t h e s c u l p t o r and a c e r t a i n d e g r e e of a r t i s t i c l i c e n s e i s a l w a y s p r e s e n t . T h i s may n o t n e c e s s a r i l y be d e s i r a b l e i n many s c i e n t i f i c and t e c h n i c a l a p p l i c a t i o n s where a c c u r a c y and r e p e a t a b i l i t y i s of c r i t i c a l i m p o r t a n c e . W i t h t h e d e v e l o p m e n t of n u m e r i c a l l y c o n t r o l l e d m a c h i n e s i n t h e p a s t two d e c a d e s , a more e f f i c i e n t and c o h e r e n t a p p r o a c h b a s e d on t h e i n t e g r a t i o n of d i g i t a l c o m p u t e r s and a u t o m a t e d m a n u f a c t u r i n g s y s t e m s c a n be a d o p t e d . Numerous CAD/CAM sy s t e m s a r e a v a i l a b l e f o r d i f f e r e n t a p p l i c a t i o n s , but many o f them a r e 8 s t i l l f o l l o w i n g t h e t r a d i t i o n a l a p p r o a c h b a s e d on two-d i m e n s i o n a l d r a w i n g s . F i g u r e 1.5 shows t h e d r a w i n g of a component t o be made by an NC m a c h i n e . The d e s i g n e r s p e c i f i e s t h e geometry and t o o l - p a t h s a r e t h e n d e d u c e d from t h e d r a w i n g . More a d v a n c e d s y s t e m s c a n a u t o m a t i c a l l y c a l c u l a t e t h e t o o l - p a t h s and d e v i s e t h e m a c h i n i n g s e q u e n c e but g e n e r a l l y t h e y employ a ' t w o - a n d - a - h a l f - D ' a p p r o a c h . T h i s i s s a t i s f a c t o r y f o r most e n g i n e e r i n g a p p l i c a t i o n s f o r w h i c h o n l y s i m p l e a n a l y t i c s u r f a c e -e l e m e n t s s u c h as p l a n e s and c y l i n d e r s a r e p r e s e n t . D i f f i c u l t y a r i s e s , however, when more c o m p l i c a t e d s c u l p t u r e d s u r f a c e s a r e r e q u i r e d . I t i s t h e p u r p o s e of t h i s r e s e a r c h t o d e v e l o p a g e n e r a l and i n t e g r a t e d a p p r o a c h t o t h e m o d e l l i n g and making of d i e s and moulds u s i n g CAD/CAM t e c h n i q u e s . F i g u r e 1.5 T o o l - p a t h f o r a c u t t e r f o r m i l l i n g and d r i l l i n g 10 5. OBJECTIVES OF RESEARCH C a v i t y moulds c o n s i s t of b o u n d i n g s u r f a c e s t h a t a r e e i t h e r a n a l y t i c a l o r a r b i t r a r y . A n a l y t i c a l s u r f a c e s a r e u s u a l l y d e s i g n e d s h a p e s c o n t a i n i n g s u r f a c e - e l e m e n t s r e p r e s e n t e d by m a t h e m a t i c a l e q u a t i o n s . T h e s e i n c l u d e most e n g i n e e r i n g components. A r b i t r a r y s u r f a c e s a r e u s u a l l y n a t u r a l s u r f a c e s d e f i n e d by measured d a t a . T h e s e r a n g e s from n a t u r a l l a n d s c a p e s t o a n a t o m i c a l p a r t s . S p e c i a l s u r f a c e s , s u c h as d u c t s , b o t t l e s and s h e l l s may r e q u i r e s p e c i a l t r e a t m e n t s . Examples of a l l t h r e e c l a s s e s of s u r f a c e d e s c r i b e d above have been examined i n t h e work f o l l o w i n g . An i n t e g r a t e d CAD/CAM a p p r o a c h f o r m o d e l l i n g and m a c h i n i n g of t h e s e c a v i t y - s u r f a c e s h as been d e v e l o p e d i n t h i s r e s e a r c h . The main o b j e c t i v e s a r e : 1. M o d e l l i n g of C a v i t y - S u r f a c e s : t o d e v e l o p g e n e r a l computer r o u t i n e s t o g e n e r a t e a n a l y t i c a l s u r f a c e s as e n c o u n t e r e d i n many e n g i n e e r i n g a p p l i c a t i o n s ; t o d e v e l o p g e n e r a l p r o c e d u r e s f o r t h e m o d e l l i n g of non-a n a l y t i c s u r f a c e s . 2. O r g a n i z a t i o n o f t h e M a c h i n i n g P r o c e s s : t o g e n e r a t e c u t t e r l o c a t i o n d a t a (CLD) t o machine t h e g e n e r a t e d s u r f a c e . 3. M a c h i n i n g and M o u l d i n g : t o machine d i e s and moulds f r o m t h e c u t t e r l o c a t i o n d a t a u s i n g n u m e r i c a l l y c o n t r o l l e d m a c h i n e s , and t o t e s t t e c h n i q u e s f o r f o r m i n g p h y s i c a l components from s u c h d i e s and m o u l d s . 1 1 I I . THE TECHNICAL/MATHEMATICAL FEATURES OF SURFACE AS AN ENTITY 1. PHYSICAL SURFACES DEFINED BY ANALYTICAL FUNCTIONS A g e n e r a l s u r f a c e c an be c o n s i d e r e d a s a c o n t i n u o u s m a n i f o l d of an i n f i n i t e number of p o i n t s i n s p a c e d e t e r m i n e d by a s p a c e f u n c t i o n . I f t h i s s u r f a c e i s i m a g i n e d t o e x i s t i n a C a r t e s i a n c o - o r d i n a t e frame, t h e p o i n t s r e p r e s e n t i n g t h e s u r f a c e c an be r e l a t e d by a f u n c t i o n a l r e l a t i o n s h i p between t h e c o o r d i n a t e s F(x,y,z) = 0 . • Any p o i n t P on t h e s u r f a c e may be r e p r e s e n t e d by i t s c o o r d i n a t e s ( x ,y ,z ) or by t h e p o i n t p o s i t i o n v e c t o r R = P P P x i + y j + z k . Thus t h e m a n i f o l d of p o i n t s may be m o d e l l e d p— p — p — m a t h e m a t i c a l l y by some f u n c t i o n of two i n d e p e n d a n t v a r i a b l e s ( x , y ) o r p a r a m e t e r s ( u , v ) . T h r e e f u n d a m e n t a l g e n e r a l forms a r e shown below : C l a s s i c a l Form : F( x , y , z ) = 0 ( 2.1 ) Monge's E q u a t i o n : z = F( x,y ) ( 2.2 ) Gauss' Form : x = F , ( u,v ) y = F 2 ( u,v ) ( 2.3 ) z = F 3 ( u,v ) I f t h e f u n c t i o n F( x,y ) i n z = F( x,y ) i s g i v e n o r d e t e r m i n e d i n some way as a m a t h e m a t i c a l e q u a t i o n , t h e v a l u e z w i t h r e s p e c t t o ( x,y ) c a n be d e t e r m i n e d by a n a l y s i s . I f z e x i s t s w i t h i n s p e c i f i e d r a n g e s of ( x,y ), t h e r e s u l t i n g s u r f a c e i s c o n s i d e r e d as an a n a l y t i c a l s u r f a c e . 12 2. PHYSICAL SURFACES AS A MANIFOLD OF POINTS Most n a t u r a l s u r f a c e s , s u c h as a n a t o m i c a l s u r f a c e s c a n n o t be r e p r e s e n t e d by s i m p l e a n a l y t i c a l f u n c t i o n s . A s i n g l e p o i n t P on one of t h e s e s u r f a c e s may be measured and i t s c o o r d i n a t e s t h o u g h t of a s a v e r t i c a l d i s t a n c e Zp above an a r b i t r a r y l o c a t i o n whose h o r i z o n t a l c o o r d i n a t e s a r e ( x p » Y p )• I n t h i s c a s e , t h e measured s u r f a c e may s t i l l be a c o n t i n u o u s m a n i f o l d of p o i n t s i n s p a c e , b ut o n l y a l i m i t e d number of p o i n t s on t h e s u r f a c e a r e measured or d e f i n e d ( F i g u r e 2.1 ). A l a r g e number of c l o s e l y s p a c e d measured p o i n t s c a n r e a d i l y be o b t a i n e d and g i v e an a p p r o x i m a t i o n t o t h e s u r f a c e ; i f more p o i n t s a t p a r t i c u l a r l o c a t i o n s n o t i n t h e g i v e n m e a s u r e d s e t a r e s u b s e q u e n t l y r e q u i r e d , i n t e r p o l a t i o n s must be p e r f o r m e d . The c l o s e n e s s o f t h e a p p r o x i m a t i o n depends on t h e number o f d a t a p o i n t s measured, and i n t e r p o l a t i o n s a r e b a s e d on t h e p o s t u l a t i o n t h a t t h e s u r f a c e has c o n t i n u i t y of p o s i t i o n , s l o p e , and i n some c a s e s , c u r v a t u r e . 1 3 Figure 2 .1 Phys i c a l s u r f a c e d e f i n e d by c l o s e l y spaced data p o i n t s 14 3. METHODS OF SURFACE DEFINITION In g e n e r a l , s u r f a c e s can be d e f i n e d i n the f o l l o w i n g ways : a ) A n a l y t i c a l S u r f a c e s Most s u r f a c e s i n v o l v e d i n e n g i n e e r i n g d e s i g n c o n s i . s t of an a s s e m b l y o f s i m p l e s u r f a c e - p i e c e s ( eg. c y l i n d e r s , s p h e r e s ) whose c h a r a c t e r i s t i c e q u a t i o n s a r e w e l l known. Such s u r f a c e s c a n t h u s be g e n e r a t e d from a n a l y t i c a l e q u a t i o n s , b ) P h y s i c a l M o d e l s F r e q u e n t l y , s u r f a c e s a r e d e f i n e d i n t h e form of p h y s i c a l m o d e l s . T h e s e r e q u i r e measurements by v a r i o u s means : m e c h a n i c a l , o p t i c a l , o r a c o u s t i c a l . In t h i s c a s e , t h e s u r f a c e s a r e r e p r e s e n t e d by c l o s e l y s p a c e d random s u r f a c e - p o i n t s , c ) S u r f a c e s D e v e l o p e d from S p a t i a l B o u n d a r i e s In many e n g i n e e r i n g d e s i g n s , a s u r f a c e i s d e t e r m i n e d by d r a w i n g s of p r o j e c t i o n s o f i t s b o u n d a r i e s . To span a t h r e e d i m e n s i o n a l s u r f a c e f r o m t h e s e two d i m e n s i o n a l p r o j e c t i o n s , v a r i o u s a l g o r i t h m s have been d e v e l o p e d e m p l o y i n g t h e i d e a s o f p r o p o r t i o n a l d e v e l o p m e n t o r v e c t o r e q u a t i o n s . [ D u n c a n & F o r s y t h , 1977] d ) Computed A x i a l Tomography ( CAT s c a n ), P o s i t r o n E m i s s i o n Tomography ( PET s c a n ), o r  N u c l e a r M a g n e t i c R e s o n a n c e ( NMR s c a n ) The t e c h n i q u e o f CAT s c a n n i n g p r o d u c e s c l o s e l y s p a c e d s l i c e s o f s e c t i o n s o f bones and i n t e r n a l o r g a n s of t h e human body. By s u p e r i m p o s i n g t h e s e s l i c e s , t h e s h a p e s of i n t e r n a l o r g a n s can be o b t a i n e d . [ P o r t u g u a l , 1982] 15 4. SURFACE INTERPOLATION U n l i k e a n a l y t i c s u r f a c e s , s c u l p t u r e d s u r f a c e s c a n n o t be d e s c r i b e d by s i m p l e m a t h e m a t i c a l r e l a t i o n s h i p s . A l t h o u g h c o n t o u r - m e a s u r i n g equipment has been h i g h l y d e v e l o p e d and NC c o n t o u r i n g m a c h i n e s can f o l l o w a l m o s t any s u r f a c e , e f f i c i e n t means f o r m a t h e m a t i c a l d e s c r i p t i o n Of a r b i t r a r y s u r f a c e s a r e r e q u i r e d f o r d e v e l o p m e n t of 'smart' CAD/CAM s y s t e m s . Numerous methods t o h a n d l e f r e e - f o r m s c u l p t u r e d s u r f a c e s been s u g g e s t e d , among wh i c h a r e F e r g u s o n ' s M u l t i v a r i b l e C u r v e I n t e r p o l a t i o n [ F e r g u s i o n , 1964], Coon's B i - C u b i c S u r f a c e P a t c h [Coons, 1967] and B e z i e r ' s UNSURF s y s t e m [ B e z i e r , 1972]. T h e s e i n t e r p o l a t i o n t e c h n i q u e s g e n e r a l l y employ h i g h d e g r e e v e c t o r e q u a t i o n s . t o b u i l d s e t s of e l e m e n t a r y s u r f a c e - p a t c h e s i n t e r c o n n e c t i n g one a n o t h e r o v e r a g l o b a l f i e l d ; and t h e s o l u t i o n s of t h e s e e q u a t i o n s a r e f o u n d by s p e c i f y i n g d i s p l a c e m e n t and s l o p e c o n t i n u i t i e s a t t h e b o u n d a r y of e a c h s u r f a c e - e l e m e n t s . To machine t h e s c u l p t u r e d s u r f a c e u s i n g NC e q u i p m e n t , t h e c u t t e r l o c a t i o n d a t a must be g e n e r a t e d . When a s p h e r i c a l l y -ended m i l l i n g c u t t e r i s u s e d , t h e c u t t e r l o c a t i o n d a t a i s r e p r e s e n t e d by an o f f s e t s u r f a c e w h i c h i s t h e l o c u s o f t o o l -c e n t r e p o i n t s . T o o l - p o s i t i o n s c a n be d e t e r m i n e d by c a l c u l a t i n g t h e c o - o r d i n a t e s o f a p o i n t w h i c h h a s an o f f s e t d i s t a n c e e q u a l t o t h e t o o l - r a d i u s a l o n g t h e n o r m a l v e c t o r a t a s u r f a c e - p o i n t . The p o l y h e d r a l c o n c e p t , an a p p r o a c h t a k e n f o r t h i s r e s e a r c h , a p p r o x i m a t e s t h e s u r f a c e a s an i r r e g u l a r p o l y h e d r o n by c o n n e c t i n g n e i g h b o r i n g d a t a p o i n t s w i t h f a c e t s . No a t t e m p t i s 1 6 made t o a v o i d s l o p e d i s c o n t i n u i t i e s , s i n c e t h e c h a r a c t e r i s t i c s o f a l l NC m a c h i n e s a r e s u c h t h a t t h e y move l i n e a r l y from one d a t a p o i n t t o t h e n e x t ; t h e end r e s u l t i s t h a t a l l m a c h i n e d s u r f a c e s a r e a c t u a l l y p o l y h e d r o n s , and no s l o p e c o n t i n u i t y i s e n s u r e d . A more d e t a i l e d d e s c r i p t i o n i s p r o v i d e d i n t h e n e x t s e c t i o n . 5. THE POLYHEDRAL CONCEPT POLYHEDRAL NC i s a computer s o f t w a r e p a c k a g e d e v e l o p e d a t t h e U n i v e r s i t y of B r i t i s h C o l u m b i a i n t h e y e a r s 1969 t o 1976. I t c o n s i s t s of a s y s t e m of programs f o r t h e d e f i n t i o n and m a c h i n i n g of s c u l p t u r e d s u r f a c e s u s i n g n u m e r i c a l l y c o n t r o l l e d m a c h i n e s . The b a s i s of t h e p o l y h e d r a l a p p r o a c h i s t o d e f i n e t h e s u r f a c e by a network of c l o s e l y s p a c e d d i s c r e t e p o i n t s i n C a r t e s i a n c o o r d i n a t e s and t h e n a p p r o x i m a t e t h e s u r f a c e by an i r r e g u l a r p o l y h e d r o n w i t h v e r t i c e s b e i n g t h e s u r f a c e - p o i n t s . By j o i n i n g a d j a c e n t p o i n t s i n s e t s of 3, t r i a n g u l a r p l a n e f a c e t s o f t h e p o l y h e d r o n a r e f o r m e d . The r e s u l t r e s e m b l e s a c u t gem s t o n e . ( F i g u r e 2.2) 17 F i g u r e 2 . 2 S u r f a c e a p p r o x i m a t i o n a s a n i r r e g u l a r p o l y h e d r o n j o i n i n g a d j a c e n t p o i n t s i n s e t s o f t h r e e t o f o r m t r i a n g u l a r p l a n e f a c e t s b y 18 Once t h e c o o r d i n a t e s o f t h e v e r t i c e s a r e p r o c e s s e d and t h e f a c e t s a r r a n g e d i n a l o g i c a l o r d e r , a s p h e r i c a l l y - e n d e d c u t t i n g t o o l c an be d i r e c t e d t o t o u c h e v e r y f a c e t of t h e p o l y h e d r o n , one a t a t i m e . The p o s i t i o n of t h e c u t t i n g t o o l , d e f i n e d by c o o r d i n a t e s known a s t h e c u t t e r l o c a t i o n d a t a ( CLD ), i s f o u n d as f o l l o w s : L e t P , ( x 1 r y , ) , P 2 ( x 2 , y 2 ) , P 3 ( x 3 , y 3 ) r e p r e s e n t s t h e v e r t i c e s of one f a c e t . S i n c e 3 p o i n t s d e f i n e one p l a n e , a p l a n e can be r e p r e s e n t e d by t h e e q u a t i o n : x l v l z l x 2 y 2 z 2 1 x y „ z „ 1 3 2 3 3 ( 2.4 ) o r : Ax + By + Cz + D ( 2.5 ) D i v i d i n g e q u a t i o n 2.5 by JA2 + B 2 + C 2 , t h e e q u a t i o n becomes : ax + /3y + 7 Z + p = 0 where a, /3 and 7 a r e t h e d i r e c t i o n c o s i n e s of t h e n o r m a l t h e f a c e t . ( F i g u r e 2.3 ) 19 L e t C be t h e c e n t r o i d of t h e f a c e t whose c o o r d i n a t e s ( x c ' Y c » z c ) a r e f o u n d f r o m : x c = ( x i + X2 + X3 ) / 3 Yc = ( Y l + Y2 + Y3 ) / 3 ( 2.7 ) z c = ( Z i + z 2 + z 3 ) / 3 L e t T be a p o i n t o f d i s t a n c e R from C a l o n g t h e n o r m a l t o t h e p l a n e t h r o u g h C, t h e n : x t = x c + aR y t = y c + 0R ( 2.8 ) z t = z c + 7 R Now, i f R i s t h e r a d i u s o f t h e s p h e r i c a l l y - e n d e d c u t t i n g t o o l , T w i l l be t h e t o o l - c e n t r e p o s i t i o n a t w h i c h t h e t o o l j u s t t o u c h e s ( i e . i s t a n g e n t i a l ) t o t h e f a c e t a t i t s c e n t r o i d . By r e p e a t i n g t h e above c a l c u l a t i o n s f o r e a c h f a c e t , a s e r i e s o f p o i n t s r e p r e s e n t i n g t h e CLD p a t h c a n be o b t a i n e d . ( F i g u r e 2.3) Pro g r a m SUMAIR i n t h e POLYHEDRAL NC s y s t e m employs t h e l o g i c d e s c r i b e d above t o c a l c u l a t e t h e t o o l - p a t h f o r m a c h i n i n g . E x t e n s i v e m a t h e m a t i c a l a n a l y s i s i s p e r f o r m e d t o g u i d e t h e t o o l i n s u c h a way t o a v o i d u n d e r c u t t i n g o f n e i g h b o u r i n g f a c e t s when ' v i s i t i n g ' e a c h f a c e t . [ D u n c a n & M a i r , 1976] 20 Programs of t h e POLYHEDRAL NC s y s t e m have been e x t e n s i v e l y u s e d and t e s t e d i n numerous p r o j e c t s t h r o u g h o u t t h e y e a r s . I t can be c l a i m e d t h a t , as l o n g as a s i n g l e v a l u e d s u r f a c e i s r e p r e s e n t e d by a t a b l e o f p o i n t s , t h e s y s t e m i s c a p a b l e o f r e p l i c a t i n g t o a s p e c i f i e d a c c u r a c y any p h y s i c a l s u r f a c e . D o c u m e n t a t i o n o f system i s f o u n d i n M a i r and Duncan, 1978. I F i g u r e 2.3 C a l c u l a t i o n of c u t t e r l o c a t i o n d a t a by p o l y h e d r a l c o n c e p t 21 I I I . SCULPTURED DIE-SURFACES 1. GENERAL FEATURES OF DIES As d e s c r i b e d i n t h e p r e v i o u s c h a p t e r s , a l l e n g i n e e r i n g d e s i g n s l e a d u l t i m a t e l y t o some f o r m o f p r o d u c t s w i t h c h a r a c t e r i s t i c b o u n d i n g s u r f a c e s t h a t a r e e i t h e r a n a l y t i c a l or a r b i t r a r y ( s c u l p t u r e d ) . In i n d u s t r i e s i n w h i c h m e t a l s , p l a s t i c s , c e r a m i c s and o t h e r m a t e r i a l s a r e s h a p e d by c a s t i n g , m o u l d i n g , m e c h a n i c a l w o r k i n g and o t h e r p r o c e s s e s , many r e p l i c a t i o n s of. t h e s e p r o d u c t s a r e u s u a l l y r e q u i r e d . To a i d m a n u f a c t u r i n g e i t h e r m a s t e r f o r m s , c l o s e l y r e s e m b l i n g t h e f i n a l p r o d u c t , or c a v i t y - d i e s s haped t o e n c l o s e i t , a r e u s e d . The d e s i g n o f s u c h f o r m s and c a v i t i e s i s b a s e d on t h e geometry o f t h e r e q u i r e d i t e m as w e l l as t h e p r o b l e m s imposed by d i f f e r e n t m a n u f a c t u r i n g p r o c e s s e s . F o r example, a g e n e r a l d i l a t i o n of t h e volume of a c a v i t y - d i e u s e d i n h o t c a s t i n g i s needed t o a c c o u n t f o r t h e c o n t r a c t i o n of m e t a l upon c o o l i n g . In t h i s c a s e , t h e d i e must d i f f e r i n shape from t h e f i n i s h e d c o l d i t e m . 1.1 C h a r a c t e r i s t i c s Of D i e C a v i t i e s C a v i t y - d i e s a r e d e s i g n e d t o e n c l o s e or l i m i t t h e f l o w of l i q u i d or p l a s t i c m a t e r i a l . When s u c h m a t e r i a l has s o l i d i f i e d , t h e moulded p r o d u c t has t o be e x t r a c t e d . In most common m a n u f a c t u r i n g p r o c e s s e s , t h i s r e q u i r e s t h e c a v i t y t o be s p l i t i n t o two h a l f c a v i t i e s . F i g u r e 3.1 shows t h e t y p i c a l f e a t u r e s of a d i e c a v i t y . The 22 two h a l f d i e - b l o c k s a r e b r o u g h t t o g e t h e r a l o n g a common p a r t i n g s u r f a c e w h i c h i s u s u a l l y , but n o t n e c e s s a r i l y , a p l a n e . The c a v i t y i t s e l f i s e n c l o s e d by t h e ' c e i l i n g ' s u r f a c e of t h e upper b l o c k , t h e ' f l o o r ' s u r f a c e o f t h e l o w e r , and t h e s i d e w a l l s s p a n n i n g t h e d e p t h between t h e c e i l i n g and t h e f l o o r . The s i d e w a l l s u s u a l l y s l o p e , o r ' d r a f t ' , t o w a r d s t h e p a r t i n g s u r f a c e t o f a c i l i t a t e t h e r e m o v a l of t h e s o l i d i f i e d p r o d u c t . The ' p a r t i n g l i n e ' i s t h e i n t e r s e c t i o n o f t h e p a r t i n g s u r f a c e and t h e c a v i t y -s u r f a c e . S i n c e m a t e r i a l s t e n d t o e s c a p e a l o n g t h e p a r t i n g s u r f a c e t o f o r m a ' f l a s h ' , t h i s l i n e i s a l s o known as t h e ' f l a s h - l i n e ' . — upper die pattern cei1 inn lower die pattern-F i g u r e 3 . 1 T y p i c a l f e a t u r e s o f a c a v i t y - d i e 23 2. DESIGN AND MACHINING OF DIES USING THE CAD/CAM APPROACH The t r a d i t i o n a l c r a f t - b a s e d a p p r o a c h i n t h e d e s i g n and s h a p i n g o f d i e s and f o r m s can now be i n c o p o r a t e d i n t o a u n i f i e d and i n t e g r a t e d a p p r o a c h t h r o u g h t h e use of c o m p u t e r s . G e o m e t r i c a l s p e c i f i c a t i o n s o f d i e - s u r f a c e s c a n be d e f i n e d by e i t h e r m a t h e m a t i c a l e q u a t i o n s or measured d a t a and s t o r e d i n computer memories. S u r f a c e - p r o p e r t i e s c a n be computed and d e s i g n a d j u s t m e n t s may be a p p l i e d v i r t u a l l y i n s t a n t a n e o u s l y u s i n g h i g h s p e e d c o m p u t e r s and i n t e r a c t i v e g r a p h i c s . M a c h i n e i n s t r u c t i o n s a r e t h e n g e n e r a t e d t o g u i d e t h e c u t t i n g t o o l o f a n u m e r i c a l l y c o n t r o l l e d m a c h i n e t o c r e a t e t h e s u r f a c e . 2.1 M a c h i n i n g Of D i e s By The POLYHEDRAL NC System Many c a v i t y - d i e s c o n t a i n p l a n e s and r i g h t p r i s m s o r c y l i n d e r s of g e n e r a l c r o s s - s e c t i o n . These a r e w e l l d e f i n e d a n a l y t i c a l l y and can be e a s i l y m a c h i n e d by a t w o - a n d - a - h a l f - D (2-1/2 D) a p p r o a c h u s i n g one of t h e many a v a i l a b l e CAD/CAM s y s t e m s . O t h e r s , however, i n c o r p o r a t e d i f f i c u l t - t o - d e f i n e s u r f a c e s , u s u a l l y compound i n n a t u r e ( i e . an a s s e m b l y o f many i n d i v i d u a l c o n t i g u o u s p i e c e s ), and c a n n o t be g e n e r a t e d i n a 2-1/2 D manner. S c u l p t u r e d s u r f a c e s c a n be m a c h i n e d e a s i l y w i t h t h e POLYHEDRAL NC s y s t e m . W i t h t h i s a p p r o a c h , i t i s more s a t i s f a c t o r y i n many r e s p e c t s t o machine t h e c a v i t y d i r e c t l y . B e s i d e s t h e o b v i o u s a d v a n t a g e o f s a v i n g m a n u f a c t u r i n g t i m e , d i r e c t m a c h i n i n g of t h e f e m a l e mould g e n e r a l l y g i v e s a b e t t e r s u r f a c e - f i n i s h t h a n m a c h i n i n g t h e male model whenever a 24 s p h e r i c a l l y - e n d e d m i l l i n g c u t t e r i s u s e d . As can be seen from F i g u r e 3.2, a s p e r i t i e s or c u s p s a r e l e f t between t o u c h e s as t h e t o o l moves from one f a c e t t o t h e o t h e r . The h e i g h t s of t h e s e c u s p s a r e d e p e n d a n t on t h e l e n g t h o f i n c r e m e n t s between t o u c h e s a s w e l l as t h e l o c a l r a d i i o f c u r v a t u r e o f t h e s u r f a c e a t t h e p o i n t s w h i c h a r e t o u c h e d by t h e t o o l . Conformity een tool and ace spherical tool Figure C l o s e r conformity of s p h e r i c a l l y - e n d e d to d i e - s u r f a c e curvatures gives b e t t e r when machining the female c a v i t y m i l l i n g s u r f a c e -c u t t e r f i n i s h 25 F o r a c o n c a v e upwards s u r f a c e , t h e h e i g h t o f t h e c u s p a r o u n d a s u r f a c e - p o i n t i s t h e f u n c t i o n o f t h e d i f f e r e n c e between t h e m a g n i t u d e s o f t h e t o o l r a d i u s and t h e r a d i u s o f c u r v a t u r e of t h e s u r f a c e a t t h a t p o i n t . Whereas f o r a c o n c a v e downwards ( convex ) s u r f a c e , t h e c u s p h e i g h t i s a f u n c t i o n of t h e sum of t h e two. O b v i o u s l y , t h e f e m a l e mould, w h i c h i s u s u a l l y c o n c a v e upwards, w i l l have b e t t e r s u r f a c e - f i n i s h when mach i n e d by a s p h e r i c a l l y -e n d e d c u t t e r . M o r e o v e r , t h e s u r f a c e - n o r m a l s on a conve x s u r f a c e d i v e r g e whereas t h e y c o n v e r g e f o r a c o n c a v e s u r f a c e . The t o o l p o s i t i o n s f o r d i f f e r e n t f a c e t s a r e c l o s e r t o g e t h e r when m a c h i n i n g t h e f e m a l e mould,, w h i c h i n t u r n g i v e s a b e t t e r s u r f a c e - f i n i s h i n terms of a s p e r i t i e s . 2.2 C o m p u t a t i o n Of S u r f a c e - r e l a t e d P r o p e r t i e s U s i n g The  P o l y h e d r a l C o n c e p t In many i n s t a n c e s , i t i s d e s i r a b l e t o have c o n t r o l o v e r s u c h p r o p e r t i e s a s t h e e n c l o s e d volume o r t h e s u r f a c e - a r e a of a c a v i t y d i e . T h i s i s i m p o r t a n t , f o r example, when t h e v o l u m e t r i c c o n t e n t of a b o t t l e has a p r e s c r i b e d v a l u e ; o r when t h e h e a t t r a n s f e r c h a r a c t e r i s t i c s of a c a s t i n g a r e t o be c o n t r o l l e d . By a p p r o x i m a t i n g t h e s u r f a c e a s a m u l t i - f a c e t e d p o l y h e d r o n , t h e c a l c u l a t i o n s f o r many s u r f a c e - p r o p e r t i e s c a n be e a s i l y a c h i e v e d . F o r example, t h e volume, s u r f a c e - a r e a and c e n t r e of mass of an o b j e c t c an be computed as shown i n F i g u r e 3.3 Surface Area dA. - surface area of facet i 1 ( x c . Y c > z ) - centroid coordinates of facet i i i C i Volume Centre of Mass CM 'CM "CM i = l 1 £ . s i ( s i - a i ) ( s i - b i ) ( s r b i ) where : s = . 1 " 7 ( a. + b. + c. ) * i i I V " g W e i where : T. = cos n T, ( T-dA.z . * x CI V ( 7.dA.Z . * y . ) y ; (-r .dA.z . * z . ) i=i 1 1 C 1 C 1 Other parameters, such as moment of i n e r t i a , can a l s o be found. F i g u r e 3.3 C o m p u t a t i o n of s u r f a c e - r e l a t e d p r o p e r t i e s u s i n g t h e p o l y h e d r a l c o n c e p t 27 IV. THE ANALYTICAL DIE 1. PIECEWISE ANALYTICAL AND COMPOUND SURFACES Many d i e - c a v i t i e s and p u n c h e s a r e d e f i n e d g e o m e t r i c a l l y a s compound i n t e r p e n e t r a t i o n o f s e v e r a l s u r f a c e - e l e m e n t s b l e n d e d t o g e t h e r a t t h e i r j u n c t i o n s . In e n g i n e e r i n g d e s i g n , s u c h compound s u r f a c e s a r e u s u a l l y c o m p r i s e d o f e l e m e n t s o f v a r i o u s common a n a l y t i c a l t y p e s i n t e r s e c t i n g one a n o t h e r a t b o u n d a r i e s of d i s c o n t i n u i t y where t h e e l e m e n t s i n t e r p e n e t r a t e . U s u a l l y t h e s e s u r f a c e t y p e s a r e s e c o n d d e g r e e q u a d r i c s u r f a c e s , t h e most common b e i n g s p h e r e s and c y l i n d e r s ( F i g u r e s 4.1 ). S i n c e t h e s e s u r f a c e s a r e r e p r e s e n t e d by we l l - k n o w n a n a l y t i c a l e q u a t i o n s , s u i t a b l e a l g o r i t h m s c a n be d e v e l o p e d t o model t h e r e q u i r e d compound s u r f a c e s f o r many e n g i n e e r i n g a p p l i c a t i o n s . 2. MODELLING OF COMPOUND SURFACES USING THE METHOD OF HIGHEST  POINT A compound a n a l y t i c a l s u r f a c e i s u s u a l l y g e n e r a t e d by s i m p l e s u r f a c e e l e m e n t s i n t e r p e n e t r a t i n g one a n o t h e r . E a c h i n d i v i d u a l e l e m e n t i s bounded by t w i s t e d s p a c e - c u r v e s of i n t e r s e c t i o n . A l t h o u g h e x p l i c i t s o l u t i o n s f o r t h e s e c u r v e s o f i n t e r p e n e t r a t i o n c a n be f o u n d by s o l v i n g t h e e q u a t i o n s o f t h e i n t e r s e c t i n g s u r f a c e - p i e c e s , t h e m a t h e m a t i c s i n v o l v e d a r e u s u a l l y t e d i o u s and c o m p l i c a t e d , and t h e s o l u t i o n s one can e x p e c t may n o t y i e l d any s i m p l e f o r m s . 28 3" | 1/4" JL_ . 1/4" 5/8' . . . . f . . . . 1/4 Rod. ; 1/4 Diam i Holes T 1188 ± 0 0 2 3/8 Diam. I 188 ± 0 0 2 cl 000 ±002 000 ±002 1/16 r r r IT 1/4" r r F i g u r e 4.1 T y p i c a l e n g i n e e r i n g component c o n t a i n i n g s i m p l e a n a l y t i c e l e m e n t s o f p r i s m s and c y l i n d e r s 29 A s i m p l e r a p p r o a c h , known as t h e Method o f H i g h e s t P o i n t [Duncan & M a i r , 1982], has been d e v e l o p e d t o d e f i n e t h e s e compound s u r f a c e s . The b a s i c a p p r o a c h i s t o t a k e t h e h i g h e s t p o i n t c a l c u l a t e d from any s e t of s u r f a c e - e l e m e n t e q u a t i o n s i n t h e domain o f i n t e r e s t . I f t h e s u r f a c e - p i e c e i s d e f i n e d by : z ± = F ± ( x,y ) i = 1 , 2 , 3 , . . . . A t e a c h l o c a t i o n ( x,y ) o v e r a f i n e r e c t a n g u l a r g r i d i n t h e p l a n v i e w , t h e h e i g h t z ( when d e f i n e d ) of e a c h p i e c e can t h e n be f o u n d . The h e i g h t o f t h e g l o b a l s u r f a c e a t ( x,y ) i s t a k e n t o be t h e maximum ( i e t h e h i g h e s t p o i n t ) among t h e z s. A two d i m e n s i o n a l a n a l o g y i s shown i n F i g u r e 4.2. Suppose 3 s u r f a c e p i e c e s f 1 f f 2 and f 3 a r e d e f i n e d w i t h i n t h e g l o b a l domain. By s c a n n i n g a l o n g d i r e c t i o n X w i t h an i n c r e m e n t A and c a l c u l a t i n g z,, z 2 and z 3 a t e a c h g r i d p o i n t , t h e h e i g h t of t h e g l o b a l s u r f a c e z a t e a c h p o i n t i s t a k e n t o be t h e maximum of z,, z 2 and z 3 . I t c a n be seen t h a t t h i s method d o e s n o t e x p l i c i t l y c a l c u l a t e t h e e x a c t l o c a t i o n o f i n t e r s e c t i o n between t h e s u r f a c e - p i e c e s , and t h e a c t u a l i n t e r s e c t i o n may l i e between n e i g h b o r i n g g r i d p o i n t s of d i f f e r e n t s u r f a c e - p i e c e s . However, i f t h e i n c r e m e n t A i s s m a l l enough ( i e . , t h e r e c t a n g u l a r g r i d i s v e r y d e n s e ) , t h e c u r v e s o f i n t e r s e c t i o n c a n be c l o s e l y a p p r o x i m a t e d . When ma c h i n e d by a s p h e r i c a l c u t t e r , as u s e d i n t h e POLYHEDRAL NC s y s t e m , t h e s h a r p d i s c o n t i n u i t i e s a r e a u t o m a t i c a l l y f i l l e t e d and smoothed. 31 By p e r f o r m i n g t h e above c a l c u l a t i o n s o v e r t h e e n t i r e g l o b a l f i e l d , t h e t a b u l a t e d p o i n t s f o r m t h e v e r t i c e s of a m u l t i f a c t e d p o l y h e d r o n s u b t e n d i n g t h e r e q u i r e d c o n t i n u o u s compound s u r f a c e . M a c h i n i n g c a n t h e n be a u t o m a t i c a l l y p e r f o r m e d by t h e POLYHEDRAL NC s y s t e m . 2.1 Subdomains W i t h i n The G l o b a l Domain O f t e n d e s i g n e r s may w i s h t o impose a 'window' on a s p e c i f i c s u r f a c e - p i e c e beyond w h i c h t h e p i e c e does not e x i s t . U s u a l l y s u c h a window i s a r e c t a n g u l a r sub-domain w i t h i n t h e r e c t a n g u l a r g l o b a l domain, w i t h s i d e s p a r a l l e l t o t h e g l o b a l f i e l d . ( F i g u r e 4.3 ) In o t h e r i n s t a n c e s , s u r f a c e - a d j u s t m e n t s may have t o be p e r f o r m e d a t c e r t a i n r e g i o n s w i t h i n t h e g l o b a l f i e l d . S u r f a c e - a d j u s t i n g f u n c t i o n s , s u c h a s bi-/3 f u n c t i o n s , c a n be a p p l i e d o v e r any sub-domains s p e i c i f i e d by t h e d e s i g n e r . [Duncan & V i c k e r s , 1980] C o n s e q u e n t l y , any g e n e r a l p u r p o s e s u r f a c e d e f i n i t i o n p r o g r a m s h o u l d a l l o w a u s e r t o d e f i n e sub-domains i f so r e q u i r e d . 2.2 M u l t i v a l u e d S u r f a c e s And N a t u r a l L i m i t s When d e f i n i n g a s u r f a c e i n t h e form z = F( x,y ), i t i s p o s s i b l e t h a t a t any p o i n t ( x,y ), t h e r e i s more t h a n one z. ( eg. s p h e r e s and e l l i p s o i d s ) When u s i n g a m i l l i n g machine f o r whi c h t u r n i n g i s not p o s s i b l e , a m u l t i v a l u e d s u r f a c e c an n o t be ma c h i n e d . When s u c h c a s e s o c c u r , e n g i n e e r i n g judgement i s r e q u i r e d t o c h o o s e one z v a l u e among t h e p o s s i b i l i t i e s . F i g u r e 32 4.4 shows some examples of t h e l i m i t s o f e x i s t a n c e o f s u r f a c e -p i e c e s . S u r f a c e s s u c h a s n o n - v e r t i c a l p l a n e s and p a r a b o l o i d s e x i s t o v e r t h e e n t i r e X, Y domain; o t h e r s , s u c h as s p h e r e s and c y l i n d e r s , e x i s t o n l y w i t h i n c e r t a i n s p e c i f i c n a t u r a l l i m i t s . ( E g . A s p h e r e d o e s not e x i s t beyond i t s e q u a t o r . ) T h e s e n a t u r a l l i m i t s must be t e s t e d t o a v o i d u n d e f i n e d r e s u l t s when c o m p u t i n g z. Figu're 4 . B Subdomain w i t h i n g l o b a l domain 33 F i g u r e 4.4 N a t u r a l l i m i t s f o r s u r f a c e - e l e m e n t s 34 2.3 G e n e r a l P r o c e d u r e F o r E x e c u t i n g The Method Of H i g h e s t P o i n t A macro a l g o r i t h m f o r t h e e x e c u t i o n of t h e Method of H i g h e s t P o i n t i s shown below : 1 D e f i n e t h e f o l l o w i n g p a r a m e t e r s : a ) G l o b a l f i e l d l i m i t s : ( XMIN,YMIN ) & { XMAX, YMAX ); b ) I n c r e m e n t f o r s c a n : A c ) Number of s u r f a c e - p i e c e s N and t h e i r t y p e s ; d ) L i m i t s ( 'window' ) f o r e a c h i n d i v i d u a l p i e c e . 2 S t a r t s c a n n i n g FOR X:= XMIN t o XMAX ; X:= XMIN + A ; FOR Y:= YMIN t o YMAX ; Y:= YMIN + A ; FOR e a c h s u r f a c e - p i e c e i := 1 t o N ; - c h e c k u s e r d e f i n e d window ; - c h e c k n a t u r a l l i m i t ; IF o u t - o f - l i m i t s k i p t o n e x t s u r f a c e - p i e c e ; ELSE :- c a l c u l a t e Z ( i ) := F ( X,Y ) ; - c h e c k h i g h e s t p o i n t ; IF Z ( i ) > Z ( i - 1 ) keep z ; REPEAT f o r n e x t s u r f a c e - p i e c e i+1 ; REPEAT f o r n e x t Y ; REPEAT f o r n e x t X ; S t o r e ( X,Y,Z ) f o r e a c h d a t a p o i n t i n d a t a f i l e ; S t o p . 35 3. GEN7 : A GENERAL PROGRAM FOR EXECUTING THE METHOD OF HIGHEST  POINT A g e n e r a l p r o g r a m , known as GEN7, has been d e v e l o p e d t o e x e c u t e t h e Method o f H i g h e s t P o i n t f o r p i e c e w i s e a n a l y t i c a l s u r f a c e s . In i t s p r e s e n t form, t h e p r o g r a m c a n h a n d l e up t o 3 e a c h of t h e f o l l o w i n g t y p e s of s u r f a c e - p i e c e s ( F i g u r e 4.5 ) : Q u a d r i c : e l l i p s o i d s ( w h i c h i n c l u d e s p h e r e s ) ; e l l i p t i c ( c i r c u l a r ) p a r a b o l o i d s ; h y p e r b o l i c p a r a b o l o i d s ; q u a d r a t i c c o n e s ; e l l i p t i c ( c i r c u l a r ) c y l i n d e r s ; N o n - q u a d r i c : p l a n e s ; t o r i ; t u b u l a r s u r f a c e s of v a r y i n g s e c t i o n s . The u s e r i s prompted i n t e r a c t i v e l y f o r i n p u t s i n t h e form of c o n v e n i e n t i d e n t i f i a b l e d a t a , s u c h a s v e r t i c e s , s e m i - a x e s , e t c . In a d d i t i o n , r o t a t i o n s a b o u t t h e X, Y, Z ax e s f o r each q u a d r i c p i e c e , t h e 'window' f o r e a c h s u r f a c e , and t h e t r u n c a t i o n h e i g h t can be s p e c i f i e d . Ellipsoid E l l i p t i c Paraboloid Cylinder Tubular surface with vary ing c ross - sec t ions Plane Torus F i g u r e 4 . 5 S u r f a c e , e l e m e n t s f o r p r o g r a m GEN7 37 3.1 G e n e r a l E q u a t i o n Of A Q u a d r i c S u r f a c e Any q u a d r i c s u r f a c e , i n any o r i e n t a t i o n , c a n be r e p r e s e n t e d by t h e f o l l o w i n g e q u a t i o n : A x 2 + B y 2 + C z 2 + Dxy + E y z + F x z + Gx + Hy + Kz + L = 0 ( 4.1 ) Hence, f o r e v e r y known x and y, e q u a t i o n 4.1 c a n be s i m p l i f i e d i n t o : A 1 z 2 + B , z + C 1 = 0 ( 4 . 2 ) where: A, = C B, = Ey + Fx and: C, = A x 2 + B y 2 + Dxy + Gx + Hy + L F o r A, * 0, e q u a t i o n 4.2 i s a q u a d r a t i c e q u a t i o n w i t h v a r i a b l e z. To s o l v e f o r z : z = (. -B, ± J B, 2 - 4A,C, ) / 2A, ( 4.3 ) Two t h i n g s c an be n o t e d from e q u a t i o n 4.3 : i ) F o r B, 2 - 4A,C! > 0, z has two v a l u e s f o r e v e r y ( x,y ). S i n c e POLYHEDRAL NC does n o t a l l o w m u l t i v a l u e d s u r f a c e s , GEN7 c h o o s e s t h e maximum ( h i g h e s t ) between t h e two s o l u t i o n s f o r z. Thus e q u a t i o n ( 4.3 ) becomes: z = ( -B, + J B, 2 - 4A ,«C, ) ) / 2A, ( 4.4 ) i i ) E q u a t i o n s 4.3 and 4.4 a r e u n d e f i n e d when: B, 2 - 4A,C, < 0 These c o r r e s p o n d s t o t h e r e g i o n b eyond t h e n a t u r a l b o u n d a r y of the s u r f a c e . F o r A, = 0, z = -C, / B, ( 4.5 ) 38 E q u a t i o n 4.5 g i v e s a s i n g l e v a l u e d s u r f a c e , n a t u r a l b oundary i s e x c e e d e d when B, = 0 3.2 G e n e r a l T r a n s f o r m a t i o n Of Axes L e t t h e c o - o r d i n a t e a x e s X, Y and Z i n t h e C a r t e s i a n s y s t e m be r o t a t e d by an a n g l e of 63 a b o u t t h e x - a x i s , f o l l o w e d by a r o t a t i o n o f 62 a b o u t t h e y - a x i s , and t h e n by 0, about t h e z-a x i s , a s shown i n F i g u r e 4.6; and l e t t h e r o t a t e d a x e s be X', Y' and Z' r e p e c t i v e l y . A p o i n t P w i t h c o o r d i n a t e s ( x , y , z ) would have c o o r d i n a t e s ( x ' , y ' , z ' ) i n t h e X'Y'Z' fra m e . They a r e r e l a t e d by e q u a t i o n 4.6 . F i g u r e 4.6 T r a n s f o r m a t i o n of axes f o r a q u a d r i c s u r f a c e p i e c e 39 X y z \ h h m. n l n 2 n 3 x ' y' z 1 ( 4.6 ) h ]2 ] 3 The m a t r i x m ^  m 3 i s the r o t a t i o n a l t r a n s f o r m a t i o n n i N 2 n 3 m a t r i x and i s d e r i v e d from e q u a t i o n 4.7 : ] 3 m 3 n l N 2 n 3 _ cosOj -s i n e s i n9^ C O S 0 0 0 0 0 1 cos6 2 s i n8 2 0 s i n 9 2 cos8 2 1 1 cos9 sin6 3 • S i n 9 3 0 0 3 C O s 6 3 1 ( 4.7 ) 40 o r : 1, = c o s 0 , c o s 0 2 m, = s i n e , c o s 0 2 n, = - s i n 0 2 1 2 = c o s f l , s i n 0 2 s i n 0 3 - s i n e , c o s 0 3 m 2 = s i n 0 , s i n 0 2 s i n 0 3 + cosf?, c o s 0 3 n 2 = c o s # 2 s i n 0 , ( 4.8 ) • 1 3 = c o s f i , s i n f ? 2 c o s 0 3 + s i n e , s i n f 5 3 m3 = sine?, s i n 0 2 c o s 0 3 - cos#, c o s 0 3 n 3 = c o s f ? 2 c o s 0 3 t can be shown t h a t x ' y' = z ' h m 3 n 3 4.6 can be r e w r i t t e n as JL X y z ( 4.9 ) I f the c o o r d i n a t e axes X, Y, Z have a l s o been t r a n s l a t e d to ( x n , y 0 , z Q ) i n a d d i t i o n to r o t a t i o n ( Figure 4.6 ), e q u a t i o n 4.9 becomes : x y z 12 ^2 ] 3 m 3 m. x - x Q y - y 0  z " z o ( 4.10 ) 41 An example i s g i v e n below to i l l u s t r a t e the a p p l i c a t i o n of ge n e r a l t r a n s f o r m a t i o n to a q u a d r i c s u r f a c e . The case of an e l l i p s o i d i s shown h e r e . The c h a r a c t e r i s t i c e q u a t i o n o f an e l l i p s o i d i s : 2 2 2 x y z - + - + - - 1 = 0 ( 4.11 ) 2 .2 2 a b c A p p l y i n g g e n e r a l t r a n s f o r m a t i o n to 4.11 : x ' 2 y ' 2 z ' 2 - + - + - - 1 = 0 ( 4 . 1 2 ) , 2 h 2 r 1 a b c With r e s p e c t e d to the o r i g i n a l axes X Y Z, 4.12 becomes : ( 1 l X l + m l V n l Z l ) 2 ( V l + n i 2 V n 2 z l ) 2 ' ( V i ^ W l ^ ^ = ^  2 .2 2 a b C ( 4.13 ) where : = x - XQ y l = y ~ y 0 Zl = 2 " Z 0 C o n v e r t i n g i n t o the form s i m i l a r to e q u a t i o n 4.2 : Al zl 2 + Bl zl + Cl = 0 ( 4*1 4 ) 42 A . ( ! i >* .• ( Ii ,2 + ( Ol ,2 a a b e 2 n l ^ l x l + m l y l ^ 2 n 2 ( 1 2 x l + m 2 y l ^ 2 n 3 ( 1 3 x 1 + m 3 y 1 ) B l = + -y + -1 ( 4 . 1 5 ) c = ( V l + V l } 2 ^ ( V l + V i }2 , ( h W l }2 _ ^  Put : D = B 2 2 - 4 A a C 2 For D ^ 0 and t a k i n g the p o s i t i v e s q u a r e r o o t o f D : z = —I ( 4.16 ) z = zx + z 0 ( 4.17 ) T r a n s f o r m a t i o n s to o t h e r q u a d r i c s u r f a c e s can be performed s i m i l a r l y . Appendix A i n c l u d e s the g e n e r a l t r a n s f o r m a t i o n s to the q u a d r i c s u r f a c e p i e c e s t h a t a r e ha n d l e d by the program GEN7. 43 3.3 S t r u c t u r e Of GEN7 GEN7 has been w r i t t e n i n s u c h a way t h a t a u s e r has t o s p e c i f y o n l y t h e b a s i c p a r a m e t e r s o f e a c h s u r f a c e - p i e c e ( e g . f o r an e l l i p s o i d , t h e s e m i - a x e s a, b and c ); t r a n s l a t i o n s and r o t a t i o n s ; as w e l l as u s e r - d e f i n e d sub-domain and t r u n c a t i o n h e i g h t . The i n p u t phase i s p e r f o r m e d i n t e r a c t i v e l y i n a s t e p -b y - s t e p manner g u i d e d by e a s y - t o - u n d e r s t a n d p r o m p t s . The p r o g r a m f i r s t a s k s f o r t h e g l o b a l f i e l d d i m e n s i o n s and t h e i n c r e m e n t between g r i d p o i n t s . N e x t , f o r e a c h s u r f a c e t y p e , t h e u s e r i s p r o mpted f o r t h e number o f p i e c e s ( maximum of 3 ). F o r e a c h q u a d r i c p i e c e , i n p u t i n c l u d e s t h e 3 b a s i c p a r a m e t e r s a, b and c d e f i n e d by t h e c h a r a c t e r i s t i c e q u a t i o n o f t h e s u r f a c e -t y p e ; t h e n t r a n s l a t i o n s ( x 0 , y 0 ; z 0 ) and r o t a t i o n s ( 6yiB2,03 ); window f o r t h e p i e c e (Xmin,Xmax), (Ymin,Ymax); the u s e r -d e f i n e d h e i g h t of t h e p i e c e beyond i t s n a t u r a l l i m i t ( o f f - l i m i t h e i g h t ) ; and t h e t r u n c a t i o n h e i g h t . I n p u t s f o r t h e n o n - q u a d r i c t y p e s o f p l a n e and t o r u s a r e s i m i l a r , e x c e p t t h a t no r o t a t i o n s a r e a l l o w e d . A summary of t h e i n p u t p a r a m e t e r s f o r e a c h s u r f a c e - p i e c e i s g i v e n i n T a b l e I ; and F i g u r e 4.7 shows a t y p i c a l p r o m p t i n g s e q u e n c e when r u n n i n g t h e p r o g r a m . Once t h e i n p u t p h a s e i s c o m p l e t e d , th e p r o g r a m s c a n s a l o n g e a c h g r i d p o i n t . A t e a c h p o i n t , t h e x and y c o o r d i n a t e s a r e f i r s t t e s t e d t o c h e c k i f t h e sub-domain i s e x c e e d e d ; i f n o t , g e n e r a l t r a n s f o r m a t i o n t o t h e s u r f a c e - p i e c e i s a p p l i e d and t h e n a t u r a l l i m i t i s c h e c k e d . I f t h e s u r f a c e i s w i t h i n t h i s l i m i t , t h e v a l u e z i s c a l c u l a t e d u s i n g e q u a t i o n 4.4; whereas i f i t i s beyond t h e n a t u r a l b o u n d a r y , z i s s e t t o t h e u s e r - d e f i n e d ' o f f -44 l i m i t h e i g h t ' . The p r o c e s s i s r e p e a t e d f o r e v e r y s u r f a c e - p i e c e a t e v e r y g r i d p o i n t , and t h e maximum z c a l c u l a t e d i s r e t a i n e d and w r i t t e n o n t o a d a t a f i l e . The o u t p u t f i l e c a n t h e n be p r o c e s s e d f o r g r a p h i c s o r m a c h i n i n g p u r p o s e s . TABLE I INPUT PARAMETERS FOR GEN7 Surface Type Characteristic Equation ' User Inputs E l l i p s o i d x 2 + y 2 + z 2 _ l Centroid x Q, y Q , z Q a2 b 2 c 2 Semi-axes a, b, c For spheres : Rotations 9^. 9 2 ' 9 3 a = b = c = r ( radius ) Subdomain Limits .... x m i n> Y m i n xmax' ymax O f f l i m i t Height z Q f f Truncation Height ... z^ r E l l i p t i c x 2 + y 2 = c z Vertex x Q, y Q , z Q Paraboloid a b Semi-axes a, b, c Rotations 8 j , e 2 > e 3 Subdomain Limits xmin' ymin xmax' ymax O f f l i m i t Height ..... z Q f f Truncation Height ... z t Hyperbolic x 2 _ y_2 = c z V e r t e x V V Z0 Paraboloid a 2 " b 2 Major & Minor Axes .. a, b, c Rotations 8 j , 9 2, e 3 Subdomain Limits x m i n ' ymin xmax' ymax O f f l i m i t Height z Q f f Truncation Height ... z TABLE I ( cont'd ) Surface Type Characteristic Equation Quadratic Cone 2 2 2 * + y. + * 2 K 2 . 2 a b c 0 For c i r c u l a r Cones : a = b = tan <j> (\> = semi-angle c = 1 2 2 Quadratic x , y _ •, ~2 2 Cylinder a b Length = 2r 0 Plane a b c User Inputs Centroid x^, y^, Zg Semi-axes a, b, c Rotations 8 ^ , 8 3 Subdomain Limits xmin' ymin x v max' •'max O f f l i m i t Height z Q f f Truncation Height ... z^ r Centroid x Q, y Q , z Q Semi -axes a, b Half Length r Q Rotations 9^ > 9 2 ' e 3 Subdomain Limits .... xm-jn» y m i n xmax' ymax Of f l i m i t Height z Q f f Truncation Height ... z t r Intercepts a, b, c Subdomain Limits ... *m-jn» y m i- n x , y max •'max Truncation Height .. z ^ TABLE I ( cont'd ) Surface Type Characteristic Equation User Inputs Torus (yy 2 + / - a ) 2 + z 2 = b 2 Centroid x Q, y Q , z Q Ring Radius a Tube Radius b Subdomain Limits x m i n > y m i n xmax' ymax Tubular surface j J Centroid x Q, y Q , z Q with parabolic z = ( cx 2 + b ) i l - \ Parameters a, b, c pr o f i l e a Rotation ( z-axis ) . 6^  Subdomain Limits x m i n ' ^min xmax' ^ max O f f l i m i t Height ..... z Q f f Truncation Height ... z. 48 1 BAS 2 3 IAS/RSX BASIC V02-01 4 5 READY 6 RUN GEN7 7 8 ENTER FIELD DIMENSION X AND Y ? 6 0 0 . . 2 3 0 . 9 ENTER INCREMENT D 7 15. 10 11 NUMBER OF ELLIPSOIDS ( max 3 ) 7 0 12 NUMBER Or ELL IPT IC PARABOLOIDS ( max 3 ) 7 2 13 14 ENTER ( X O . Y O . r O ) FOR ELLIP-PARA(1) 7 14 1 . 7 , 1 2 9 . 4 , 9 8 . 15 ENTER A, B. C FOR ELLIP-PARA( 1) 7 1 . . 1 . . - 1 9 2 . 16 FNTER ROTATIONS 1. 2 AND 3 FOR ELLIP-PARA(1 ) . . . 7 0 . . - 7 . 3 . 0 . 17 ENTER L0WLIMX, UPLIMX FOR ELLIP-PARA(1 ) 7 0 . , 6 0 0 . 18 ENTER L0WLIMY, UPLIMY FOR ELLIP-PARA( 1) 7 2 0 . . 2 3 0 . 19 ENTER OFFLIMIT HEIGHT FOR ELLIP-PARA( 1) 7 0 . 20 ENTER TRUNCATION HEIGHT FOR ELLIP-PARA( 1 ) 7 9 9 9 . 21 22 ENTER (XO.YO.ZO) FOR ELLIP-PARA(2) 7 4 3 0 . 4 . 1 2 9 . 4 . 1 8 3 . 23 ENTER A, B. C FOR ELLIP-PARA( 2) 7 1 . . 1 . . - 1 9 2 . 24 ENTER ROTATIONS 1, 2 AND 3 FOR ELLIP-PARA(2) . . . 7 0 . . - 7 . 3 . 0 . 25 ENTER LOWLIMX, UPLIMX FOR ELLIP-PARA(2) 7 0 . 6 0 0 . 26 ENTER LOWLIMY, UPLIMY FOR ELLIP-PARA(2) 7 2 0 . , 2 3 0 . 27 ENTER OFFLIMIT HEIGHT FOR ELLIP-PARA(2) 7 0 . 28 ENTER TRUNCATION HEIGHT FOR ELLIP-PARA(2 ) 7 999 . 29 30 P a u s i n g , t ype 1 t o a l t e r i n p u t , any n o . t o c o n t i n u e 7 999 31 32 33 NUMBER OF HYPERBOLIC PARABOLOIDS ( max 3 ) 7 O 34 NUMBER OF QUADRATIC CONES ( max 3 ) 7 0 35 36 ENTER (XO.YO.ZO) FOR C0NE(1) 7 1 4 8 . 1 . 1 2 9 . 4 , 5 9 1 . 6 37 ENTER A. B. AND C FOR C0NE( 1) 7 0 . 0 8 7 5 , 0 . 0 8 7 5 . 0 . 38 ENTER ROTATIONS 1. 2 AND 3 FOR C0NE(1) . . . . . . . . . 7 0 . . 0 . . 0 . 39 ENTER LOWLIMX. UPLIMX FOR CONE(I ) 7 0 . . 6 0 0 . 40 ENTER LOWLIMY. UPLIMY FOR C0NE(1) ? 2 0 . . 2 3 0 . 41 ENTER OFFLIMIT HEIGHT FOR C O N E O ) 7 0. 42 ENTER TRUNCATION HEIGHT FOR CONE(1) 7 110. 4 3 44 ENTER (XO.YO.ZO) FOR C0NE(2) 7 4 4 7 . 5 . 1 2 9 . 4 . 8 4 7 . 45 ENTER A, B. AND C FOR C0NE(2) 7 0 . 0 8 7 5 , 0 . 0 8 7 5 . 0 . 46 ENTER ROTATIONS 1. 2 AND 3 FOR C0NE(2) 7 0 . . 0 . . 0 . 47 ENTER LOWLIMX, UPLIMX FOR C0NE(2) 7 0 . . 6 0 0 . 48 ENTER LOWLIMY, UPLIMY FOR C0NE(2) 7 2 0 . . 2 3 0 . 49 ENTER OFFLIMIT HEIGHT FOR C0NE(2) 7 0 . 50 ENTER TRUNCATION HEIGHT FOR C0NE(2) 7 230. 51 52 P a u s i n g , t y p e 1 t o a l t e r i n p u t , any no . to c o n t i n u e 7 999 53 54 NUMBER 0* ELL IPT IC ( CIRCULAR ) CYLINDER ( max 3 ) 7 O 55 NUMBER OF PLANES 7 1 56 57 ENTER INTERCEPTS X, Y AND Z FOR PLANE(1) 7 1 . E 9 9 , 2 0 . , - 6 5 . 7 6 58 ENTER LOWLIMX, UPLIMX FOR PLANE(1 ) 7 0 . . 6 0 0 . 59 ENTER LOWLIMY. UPLIMY FOR PLANE(1) ? 0 . . 2 3 0 . 60 ENTER TRUNCATION HEIGHT FOR PLANE( 1 ) 7 999 . 61 62 P a u s i n g , t y p e 1 t o a l t e r Input , any no . to c o n t i n u e 7 999 63 64 NUMBER OF TORUS ( max 3 ) 7 0 65 NUMBER OF PARABOLIC ELL IPT ICAL CYLINDER 7 0 6C 67 End of f i l e Figure 4.7 T y p i c a l Prompting Sequence of GEN7 49 3.4 Sample Runs Of GEN7 3.4.1 P i p e - T e e P a t t e r n F i g u r e 4.8 shows a t y p i c a l p i p e - t e e p a t t e r n . A t t h e j u n c t i o n , two c i r c u l a r c y l i n d e r s i n t e r p e n e t r a t e a t r i g h t a n g l e t o e a c h o t h e r . The j u n c t i o n c a n be m o d e l l e d by GEN7 from i n p u t s s p e c i f y i n g t h e p a r a m e t e r s o f t h e two c y l i n d e r s and t h e i r r o t a t i o n s . The g e n e r a t e d s u r f a c e i s shown i n F i g u r e 4.9. F i g u r e 4.8 S k e t c h of a p i p e - t e e p a t t e r n 50 F i g u r e 4.9 P i p e j u n t i o n t e e m o d e l l e d by G E N 7 51 3.4.2 A u t o m o b i l e Rear Lamp Punch Model F i g u r e 4.10 shows a c o m m e r c i a l d r a w i n g o f an a u t o m o b i l e t a i l lamp p u n c h . The s u r f a c e i s an i n t e r p e n e t r a t i o n o f 5 r e g u l a r a n a l y t i c a l p i e c e s : 2 skewed p a r a b o l o i d s ( r e f l e c t o r s ), 2 t r u n c a t e d c o n e s ( lamp s o c k e t s ), and one i n c l i n e d p l a n e ( t o s u i t t h e a u t o m o b i l e body d e s i g n ). A l l n e c e s s a r y d i m e n s i o n s a r e p r o v i d e d f r o m t h e d r a w i n g and c o n v e r t e d i n t o i n p u t s f o r GEN7. F i g u r e 4.11 shows t h e o u t p u t g e n e r a t e d from GEN7 F i g u r e 4.10 P r i n c i p a l s e c t i o n s of an a u t o m o b i l e r e a r lamp punch 52 F i g u r e 4 . 1 1 T a i l lamp p u n c h m o d e l l e d by GEN7 53 3.4.3 Vacuum C l e a n e r H o u s i n g Punch Model The i n i t i a l d e s i g n o f a vacuum c l e a n e r h o u s i n g was done i n t h e form of f r e e - h a n d s k e t c h e s , as shown i n F i g u r e 4.12. In t h e n e x t s t a g e , t h e p r i n c i p a l d i m e n s i o n s were c h o s e n and o r t h o g o n a l p r o j e c t i o n s s k e t c h e d ( F i g u r e 4.13 ). S u i t a b l e c o n i c s e c t i o n s were t h e n a d o p t e d as e l e m e n t a r y s u r f a c e - p i e c e s t o be b l e n d e d . F o r t h e h a l f - s e c t i o n , t h e s e p i e c e s i n c l u d e : 2 e l l i p s o i d s , 2 e l l i p t i c a l c y l i n d e r s , 3 p l a n e s , 1 c o n e , and 1 t u b u l a r s u r f a c e w i t h v a r i a b l e c r o s s - s e c t i o n . The t u b u l a r s u r f a c e was c o n s i d e r e d a s a c o m b i n a t i o n of 2 d u c t - t y p e s u r f a c e s w i t h v e r t i c a l s e c t i o n s v a r y i n g p a r a b o l i c a l l y . A s p e c i a l f u n c t i o n was d e v e l o p e d t o h a n d l e t h i s s u r f a c e t y p e and i t s c h a r a c t e r s t i c e q u a t i o n i s shown i n F i g u r e 4.14. A l l l i n e s of i n t e r p e n e t r a t i o n s were g e n e r a t e d a u t o m a t i c a l l y . The r e s u l t i s shown i n F i g u r e 4.15 54 F i g u r e 4.13 P r i n c i p a l s e c t i o n s a d o p t e d f o r vacuum c l e a n e r h o u s i n g mould 55 F i g u re 4.14 Tubular s u r f a c e with p a r a b o l i c p r o f i l e 56 F i g u r e 4.15 Vacuum c l e a n e r h o u s i n g mould m o d e l l e d by GEN7 57 3.4.4 O t h e r Examples O t h e r sample o u t p u t s from GEN7 a r e shown i n t h e f o l l o w i n g f i g u r e s : F i g u r e 4.16 : i n c l i n e d c y l i n d e r w i t h e l l i p s o i d F i g u r e 4.17 : s p h e r e , i n c l i n e d c y l i n d e r , e l l i p t i c p a r a b o l o i d and c o n e . F i g u r e 4.16 I n c l i n e d c y l i n d e r w i t h e l l i p s o i d 59 4. COMPOUND SURFACES WITH NON-ANALYTICAL SURFACE-PIECES The d i s c u s s i o n s so f a r have been d e a l i n g w i t h compound a n a l y t i c a l s u r f a c e s . I t does n o t , however, mean t h a t t h e Method of H i g h e s t P o i n t i s l i m i t e d t o s u c h c a s e s . N o n - a n a l y t i c a l s u r f a c e s c a n a l s o be a s s o c i a t e d w i t h a n a l y t i c a l a s l o n g as t h e y a r e d e f i n e d by t a b u l a t e d p o i n t s a r r a n g e d i n r e c t a n g u l a r a r r a y s . F i g u r e 4.18 shows an example of ' m a r r y i n g 1 a n o n - a n a l y t i c a l s u r f a c e w i t h an e l l i p s o i d u s i n g t h e Method of H i g h e s t P o i n t . F i g u r e 4.18 ' M a r r i a g e ' of a n a l y t i c a l and n o n - a n a l y t i c a l p i e c e s by Method of H i g h e s t P o i n t s u r f a c e -60 5. MACHINING OF A DIE To make t h e c o r r e s p o n d i n g c a v i t y d i e t o t h e g e n e r a t e d s u r f a c e , one c an machine t h e model and t h e n make t h e d i e u s i n g r e v e r s a l t e c h n i q u e s . As d i s c u s s e d e a r l i e r , i t i s more s a t i s f a c t o r y t o machine t h e c a v i t y d i e d i r e c t l y i n s t e a d . The s u r f a c e s of t h e f e m a l e mould c a n be g e n e r a t e d by o b t a i n i n g t h e m i r r o r image of t h e male model. T h i s c a n be done by s i m p l y r o t a t i n g t h e t a b u l a t e d p o i n t s c a l c u l a t e d by program GEN7 by 180 d e g r e e s . In c a s e s where t h e c h a r a c t e r i s t i c s u r f a c e of a d i e i s d i f f e r e n t f r o m t h e f i n a l p r o d u c t due t o c o n s t r a i n t s imposed by d i f f e r e n t m a n u f a c t u r i n g p r o c e s s e s , a d d i t i o n a l m a n i p u l a t i o n s of d a t a a r e n e c e s s a r y . T h e s e may i n c l u d e d i l a t i o n of volume o r s u r f a c e a r e a , s u r f a c e - a d j u s t m e n t o v e r a sub-domain, or d r a f t i n g of c a v i t y w a l l s . The p o l y h e d r a l c o n c e p t p r o v i d e s e a s y means of c a l c u l a t i n g t h e g e o m e t r i c a l p r o p e r t i e s o f p h y s i c a l s u r f a c e s ( see C h a p t e r i l l ), and t h e s e m a n i p u l a t i o n s may be p e r f o r m e d u s i n g s i m p l e a l g o r i t h m s f o l l o w i n g p a t t e r n - m a k e r ' s r u l e s The g e n e r a t e d d i e s u r f a c e c an be v i e w e d o v e r a g r a p h i c s t e r m i n a l o r computer g e n e r a t e d p l o t s , and p r o p e r t i e s can be c a l c u l a t e d and a n a l y s e d . Once a s a t i s f a c t o r y s u r f a c e has been o b t a i n e d , t h e t o o l p a t h c a n be g e n e r a t e d by program SUMAIR o r NEWSU of t h e POLYHEDRAL NC s y s t e m . 61 V. THE NON-ANALYTICAL DIE 1. ARBITRARY ( FREE FORM ) SURFACES Most n a t u r a l l y o c c u r i n g s u r f a c e s , s u c h as human anatonmy and g e o g r a p h i c a l l a n d s c a p e s , c a n n o t be r e p r e s e n t e d by s i m p l e a n a l y t i c a l f u n c t i o n s . O t h e r s , s u c h as a r t i s t s ' s c u l p t u r e s , a r e o f t e n r e p l i c a t i o n s o f n a t u r a l o b j e c t s t h a t a r e a r b i t r a r y i n f o r m . T h e s e s u r f a c e s must be d e f i n e d by m easured d a t a and s u b s e q u e n t l y f u n c t i o n a l i z e d i n o r d e r t o g e n e r a t e c u t t e r l o c a t i o n d a t a f o r m a c h i n i n g p u r p o s e s . 1.1 Measurement Of A r b i t r a r y S u r f a c e s A r b i t r a r y s u r f a c e s a r e o f t e n d e f i n e d by measured d a t a o b t a i n e d from v a r i o u s m e c h a n i c a l , o p t i c a l , a c o u s t i c a l o r e l e c t r o m a g n e t i c t e c h n i q u e s . These. i n c l u d e m e c h a n i c a l measurements of p h y s i c a l o b j e c t s or m a r i n e s o u n d i n g s of s e a -b eds, y i e l d i n g r andomly measured d a t a p o i n t s . O p t i c a l measurements, s u c h as shadow m o i r e T e c h n i q u e o r p h otogrammetry, g i v e p a r t i a l l y o r g a n i z e d d a t a i n t h e form o f c o n t o u r maps. One of t h e modern t e c h n i q u e s o f v i e w i n g and m e a s u r i n g c o n c e a l e d s u r f a c e s i s t h e method of computed a x i a l tomography (CAT s c a n n i n g o r CT s c a n ) . I t s c h i e f a p p l i c a t i o n i s t o p e r f o r m d i a g n o s i s of i n t e r n a l o r g a n s o f human b o d i e s . F i g u r e 5.1 shows a s c h e m a t i c c o n f i g u r a t i o n o f a t y p i c a l CAT s c a n n e r . The p a t i e n t i s p l a c e d on a t a b l e w h i c h moves t h r o u g h an X-Ray s c a n n i n g d e v i c e . An X-Ray s o u r c e r o t a t e s r a p i d l y a r o u n d t h e p a t i e n t , making i n d i v i d u a l measurements of t h e d e n s i t i e s o f t h i n s l i c e s o f c r o s s - s e c t i o n s as t h e t a b l e moves. The d a t a a r e s t o r e d i n a computer and r e a s s e m b l e d t o form t h e image o f t h e p a t i e n t ' s 62 i n t e r i o r . F u r t h e r p r o c e s s i n g s i s o l a t e and d i s p l a y a d e s i r e d i n t e r n a l s t r u c t u r e o r o r g a n , p r o v i d i n g a d a t a base f o r a n a l y s i s and s u r f a c e r e p l i c a t i o n s when n e c e s s a r y . CRT Terminal Memory CAT scanner 0 COMPUTER 7S F i g u r e 5.1 S c h e m a t i c C o n f i g u r a t i o n o f a C A T s c a n n e r 6 3 A s u r f a c e c a n a l s o be i n i t i a l l y d e f i n e d by two d i m e n s i o n a l p r o f i l e p r o j e c t i o n s of i t s s p a t i a l b o u n d a r i e s . In t h i s c a s e , a l g o r i t h m s must be d e v e l o p e d t o 'span' a t h r e e - d i m e n s i o n a l s u r f a c e from t h e s e b o u n d a r i e s . F i g u r e 5.2 shows t h e measured b o u n d a r y - c u r v e s o f a v i o l i n t o p p l a t e f r o m w h i c h a t h r e e -d i m e n s i o n a l s u r f a c e i s spanned u s i n g b i - b e t a f u n c t i o n s . When t h e measured d a t a i s i n a n a l o g form, s u c h as c o n t o u r maps o r o u t l i n e s of c r o s s - s e c t i o n s , d i g i t i z a t i o n i s n e c e s s a r y . In o r d e r t o s t o r e d a t a i n a d i g i t a l c omputer, d i s c r e t e p o i n t s must be m easured u s i n g a . d i g i t i z e r pad o r o t h e r a n a l o g - t o -d i g i t a l c o n v e r t e r s . F i g u r e 5.2 G o v e r n i n g b o u n d a r y c u r v e o f a v i o l i n t o p - p l a t e t h e spanned s u r f a c e u s i n g b i - b e t a f u n c t i o n and ' 64 -< 2. MACHINING OF CAVITY MOULDS FOR MEASURED SURFACES To r e p r o d u c e a measured s u r f a c e u s i n g NC m a c h i n i n g , d a t a must f i r s t be s o r t e d and o r g a n i z e d b e f o r e t h e t o o l - p a t h c a n be c a l c u l a t e d . S i n c e a l l of t h e s e s u r f a c e s a p pear t o be smooth and s l o p e - c o n t i n u o u s , i t c a n be assumed t h a t t h e y can be r e p r e s e n t e d , a t l e a s t l o c a l l y , by m a t h e m a t i c a l e x p r e s s i o n s . By f i t t i n g a n a l y t i c a l s u r f a c e - p i e c e s t o t h e measured d a t a , random p o i n t s c an be t r a n s f o r m e d and t a b u l a t e d i n t o an o r t h o g o n a l g r i d and s u b s e q u e n t l y a p p r o x i m a t e d as a m u l t i f a c e t p o l y h e d r o n . A l t h o u g h t h e measured d a t a c an e i t h e r be t o t a l l y random o r p a r t i a l l y o r g a n i z e d , i t i s a d v a n t a g e o u s t o t r e a t them a l l as random so t h a t one g e n e r a l p u r p o s e s u r f a c e - f i t t i n g r o u t i n e c a n be u s e d t o h a n d l e a l l c a s e s . A pr o g r a m , known as TRUEPERS ( p r o p r i e t a r y , ' by T a y l o r , R i c h a r d s and H a l s t e a d ; E n e r g y , M i n e s and R e s o u r c e s , Canada, 1971), has been u s e d t o t r a n s f o r m t h e d a t a p o i n t s i n t o an o r t h o g o n a l g r i d . I t i n c o r p o r a t e s f e a t u r e s t h a t e n a b l e a u s e r t o s p e c i f y t h e d e g r e e of smoothness of t h e f i t t e d s u r f a c e , and t o view i t i n t h e f o r m of p e r s p e c t i v e p l o t s . Once t h e d a t a i s o r g a n i z e d i n t o an o r t h o g o n a l g r i d , t h e t o o l - p a t h c a n be g e n e r a t e d u s i n g p r o g r a m SUMAIR o r NEWSU o f t h e POLYHEDRAL NC s y s t e m . B e f o r e m a c h i n i n g , a d d i t i o n a l s t e p s must be t a k e n t o c h e c k whether t h e o r i e n t a t i o n of t h e s u r f a c e i s s u i t a b l e f o r e n d - m i l l i n g , whether t h e p a r t i n g p l a n e i s p r o p e r l y d e f i n e d , and whether s u r f a c e - a d j u s t m e n t s a r e n e c e s s a r y . 65 3. EXAMPLES ON REPLICATING MEASURED SURFACES 3.1 R a d i u s Bone P a r t o f a human r a d i u s bone was t o be r e p l i c a t e d , f o r r e s e a r c h p u r p o s e s . The s u r f a c e was measured u s i n g Shadow M o i r e T e c h n i q u e g i v i n g a c o n t o u r map as shown i n F i g u r e 5.@. The c o n t o u r l i n e s w i t h i n t h e r e g i o n o f i n t e r e s t were d i g i t i z e d i n t o d i s c r e t e p o i n t s u s i n g a d i g i t i z e r p a d . The d i s c r e t e p o i n t s were r e p l o t t e d and compared w i t h t h e o r i g i n a l c o n t o u r s t o ch e c k f o r d i s c r e p e n c i e s . ( F i g u r e 5.4) F i g u r e 5.3 Shadow M o i r e f r i n g e s of human r a d i u s bones ( f r o m T e r a d a , The S k e l e t a l A t l a s ) [ R e f 23] 66 10. 00 F i g u r e 5 .4 Computer r e p l o t of the r e g i o n of i n t e r e s t of the contour-map 67 The d i g i t i z e d p o i n t s were the n t r e a t e d as random d a t a f o r i n p u t t o t h e s u r f a c e - f i t t i n g r o u t i n e TRUEPERS. A p e r s p e c t i v e p l o t o f t h e f i t t e d s u r f a c e i s shown i n F i g u r e 5.5. S i n c e t h e o r i e n t a t i o n o f t h e d a t a p r e s e n t s no d i f f i c u l t y f o r e n d - m i l l i n g o p e r a t i o n s , no t r a n s f o r m a t i o n o f d a t a was n e c e s s a r y . To o b t a i n t h e f e m a l e mould, t h e f i t t e d s u r f a c e was r o t a t e d by 180 d e g r e e s . ( F i g u r e 5.6) T h i s was i n p u t t o t h e m a c h i n i n g p r o g r a m SUMAIR t o g e n e r a t e t h e c u t t e r l o c a t i o n d a t a . F o r c o m p a r i s i o n p u r p o s e s , t h e male s u r f a c e was m a c h i n e d u s i n g t h e same p r o c e d u r e . F i g u r e 5 . 5 " P e r s p e c t i v e p l o t of bone s u r f a c e f i t t e d by TRUEPERS 68 69 M a c h i n i n g was p e r f o r m e d u s i n g a h a l f - i n c h d i a m e t e r s p h e r i c a l l y - e n d e d m i l l i n g c u t t e r on p o l y u r e t h a n e foam and a l s o on a r e s i n - b a s e d s y n t a c t i c p l a s t i c c a l l e d SYNCAST'. The m o u l d i n g m a t e r i a l s u s e d were s i l i c o n e r u b b e r (on t h e foam mould) and d e n t a l p l a s t e r (on t h e p l a s t i c m o u l d ) . The s u r f a c e s and moulds a r e shown i n F i g u r e 5.7. F i g u r e 5.7 M a c h i n e d m o d e l s f o r male and f e m a l e s u r f a c e s 70 As e x p e c t e d , t h e moulded s u r f a c e s g i v e a b e t t e r s u r f a c e -f i n i s h t h a n t h e m a c h i n e d ones when no h a n d - f i n i s h i n g was done. To a c h i e v e a v e r y smooth s u r f a c e , " h a n d - f i n i s h i n g t o w i t n e s s " was r e q u i r e d . T h i s h a n d - f i n i s h i n g c a n be m i n i m i z e d by r e d u c i n g t h e s t e p - s i z e and by u s i n g a l a r g e t o o l . E x p e r i e n c e shows t h a t a s t e p - s i z e c o r r e s p o n d i n g t o o n e - t e n t h of t h e t o o l - d i a m e t e r g i v e s good r e s u l t s f o r most a p p l i c a t i o n s . F i g u r e 5.5 i n d i c a t e s t h a t t h e f i t t e d s u r f a c e from TRUEPERS does n o t y i e l d a p e r f e c t l y f l a t b a s e - p l a n e a t r e g i o n s j u s t beyond t h e b o u n d a r y of t h e c a v i t y , w h i c h i m p l i e s t h a t t h e c o r r e s p o n d i n g p a r t i n g s u r f a c e of t h e f e m a l e mould i s not a p l a n e . T h i s i s due t o t h e f a c t t h a t no d a t a was p r o v i d e d t o d e f i n e t h e b a s e - p l a n e as i n p u t t o TRUEPERS. Thus an u n d e f i n e d r e g i o n was c r e a t e d beyond t h e b o u n d a r y , t h e r e s u l t i s t h a t o v e r -and u n d e r - s h o o t i n g s , p l u s o s c i l l a t i o n s , a p p e a r i n i n t e r p o l a t i o n . To remedy t h i s , d a t a f o r t h e b a s e - p l a n e must a l s o be i n c l u d e d , as w i l l now be e x p l a i n e d . 71 3.2 F a c i a l M o u l d A model o f a human f a c e was r e q u i r e d f o r s u r g i c a l a p p l i c a t i o n s . D a t a was o b t a i n e d f r o m s t e r e o p h o t o g r a p h y and a c o n t o u r map was g e n e r a t e d . T h i s was d i g i t i z e d and r e p l o t t e d as shown i n F i g u r e 5.8. To p r e v e n t o s c i l l a t i o n s a t r e g i o n s b e y o n d t h e b o u n d a r y , a d d i t i o n a l d a t a d e f i n i n g t h e b a s e - p l a n e was r e q u i r e d . ^ T h i s was o b t a i n e d by g e n e r a t i n g a r t i f i c i a l ' c o n t o u r ' l i n e s w h i c h were a c t u a l l y o f f s e t c u r v e s a t v a r i o u s d i s t a n c e s from t h e b o u n d a r y - c u r v e . A r o u t i n e c a l l e d ATKIN [Law, 1984] was u s e d t o a u t o m a t i c a l l y g e n e r a t e t h e d a t a . Thus i n p u t d a t a f o r TRUEPERS a p p e a r s as shown i n F i g u r e 5.9. 72 5.00, 0. flfll ! i i i i i i i i i i 1 i i i i i i i I i i i ! i i i ! i i i i i i i I i i i E. 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4 4.50 5.00 F i g u r e 5.8 Computer r e p l o t of c o n t o u r map d e f i n i n g human f a c e 73 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 F i g u r e 5.9 Contour-map w i t h added d a t a f o r base p l a n e 74 The f i t t e d s u r f a c e by TRUEPERS i s shown i n F i g u r e 5.10. S i n c e TRUEPERS assumes p o s i t i o n - and s l o p e - c o n t i n u i t y o v e r t h e e n t i r e g l o b a l f i e l d , t h e j u n c t i o n between t h e b a s e - p l a n e and t h e model i s f i l l e t e d . When t r a n s f o r m e d i n t o t h e f e m a l e mould, t h e p a r t i n g c u r v e , w h i c h i s t h e i n t e r s e c t i o n between t h e c a v i t y -s u r f a c e and t h e p a r t i n g p l a n e and t h u s r e p r e s e n t s a j u n c t i o n of d i s c o n t i n u i t y , i s n o t d i s t i n c t . T h i s may o r may not be d e s i r a b l e , a c c o r d i n g t o d i f f e r e n t m a n u f a c t u r i n g p r o c e s s e s . One method o f g e n e r a t i n g t h e d i s c o n t i n u i t y i s t o s e t t h e d a t a of t h e b a s e - p l a n e a t a l e v e l l o w e r t h a n t h e a c t u a l b a s e - p l a n e . A f t e r s u r f a c e - f i t t i n g by TRUEPERS, t h i s p l a n e c a n be r a i s e d back t o i t s o r i g i n a l l e v e l , i n e f f e c t a r t i f i c i a l l y c r e a t i n g t h e p a r t i n g c u r v e . F i g u r e 5.11 shows t h e f e m a l e mould s u r f a c e . I t was m a c h i n e d u s i n g CLD g e n e r a t e d from SUMAIR on b o t h p o l y u r e t h a n e foam and d e n t a l p l a s t e r . F i g u r e 5.12 shows t h e c a v i t y - m o u l d and a p l a s t e r mould o f t h e model. 75 F i g u r e 5.10 F i t t e d s u r f a c e o f human f a c e by TRUEPERS F i g u r e 5.11 P e r s p e c t i v e p l o t of f e m a l e mould 77 F i g u r e 5.12 Machined face cavity-mould and p l a s t e r - m o d e l of a human 78 3.3 Ox T i b i a Bone W i t h t h e advance o f CAT s c a n n i n g and o t h e r s i m i l a r modern i m a g i n g t e c h n i q u e s , i t i s now p o s s i b l e t o r e p l i c a t e i n t e r n a l o r g a n s o r b o n e - s t r u c t u r e s t h a t u n t i l q u i t e r e c e n t l y have been u n a b l e t o be measured a c c u r a t e l y . S u c c e s s f u l e f f o r t s were made t o machine a human s k u l l from CAT s c a n n i n g d a t a [ P a r v i t i & c o l l e a g u e s , 1983], and s i m i l a r work c a n be done on o t h e r bone-s t r u c t u r e s . One major o b s t a c l e i s t h a t t h e o r i e n t a t i o n s of a p a t i e n t ' s i n t e r n a l s t r u c t u r e s a r e c o n s t r a i n e d by t h e i r p o s i t i o n and h i s p o s t u r e d u r i n g s c a n n i n g . I t may be i n c o n v e n i e n t o r even i m p o s s i b l e t o o r i e n t a s p e c i f i c s t r u c t u r e o f i n t e r e s t i n o r d e r t o make t h e measured d a t a c o r r e s p o n d t o t h e o r i e n t a t i o n d e s i r a b l e f o r m a c h i n i n g . T h i s p r o b l e m d i d not a r i s e when a s k u l l was machined, f a r - t h e o b v i o u s r e a s o n t h a t a human s k u l l model c a n be r o t a t e d q u i t e f r e e l y . However, t h i s s i t u a t i o n i s an e x c e p t i o n t o t h e g e n e r a l r u l e . S i n c e e x p e n s i v e CAT s c a n n i n g e q uipment was not a v a i l a b l e t o t h e a u t h o r f o r t h e p u r p o s e o f t h i s r e s e a r c h , an a l t e r n a t e mean of o b t a i n i n g d a t a was a d o p t e d . To s i m u l a t e t h e s l i c i n g o f c r o s s - s e c t i o n s , an ox t i b i a bone was p l a c e i n a box and r i g i d l y embedded w i t h p o l y u r e t h a n e foam. The whole a s s e m b l y ; box, foam and bone; was t h e n c u t a l o n g p a r a l l e l p l a n e s w i t h r e g u l a r i n t e r v a l s t o r e v e a l 35 p a r a l l e l c u t s t h r o u g h t h e i n c l i n e d bone. The c u t s were t h e n d i g i t i z e d t o p r o d u c e s l i c e s of c r o s s -s e c t i o n s . A few examples o f r e p l o t s a r e shown i n F i g u r e 5.13. The r e p l o t s were t h e n s u p e r i m p o s e d t o r e v e a l t h e p r o j e c t e d shape of t h e i n c l i n e d bone. ( F i g u r e 5.14) The o r i e n t a t i o n was 79 a r r a n g e d i n s u c h a way t h a t t r a n s f o r m a t i o n was n e c e s s a r y t o r o t a t e t h e bone t o a p o s i t i o n s u i t a b l e f o r m a c h i n i n g ( i e . , w i t h no n e g a t i v e d r a f t ) . G e n e r a l t r a n s f o r m a t i o n ( e q u a t i o n s 4.6 4.10 ) was a p p l i e d t o a c h i e v e t h e d e s i r e d o r i e n t a t i o n and a p a r t i n g p l a n e was s e l e c t e d . ( F i g u r e 5.15) The p o i n t s above and below t h e p a r t i n g p l a n e were s e p a r a t e l y s t o r e d i n t o two d i f f e r e n t d a t a f i l e s f o r i n p u t t o TRUEPERS t o g e n e r a t e t h e two s e p a r a t e h a l f m o u l d s . A s p e r i t i e s and h o l e s w h i c h c a n n o t be m a c h i n e d by e n d - m i l l i n g were b l a n k e d o u t , and d a t a f o r t h e b a s e -p l a n e was a d d e d . ( F i g u r e s 5.16 and 5.17) 1 0 . 0 0 _ 10.00, 9.00 8 . 0 0 L 7 . 0 0 L 5.00 4.00 3.00 i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I 0. flfll i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i oo O 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10. Figure 5.13 Replots of s l i c e s of c r o s s - s e c t i o n s of a t i b i a bone 81 -H- +-H—h-H- ++ •H--4.50 -3.00 -1.50 0.00 1.50 3. 4.50 6.00 7.50 9.00 (a) F i g u r e 5.14 (a) - (c) T h r e e v i e w s of t h e p r o j e c t e d shape of t i b i a bone by s u p e r i m p o s i t i o n of s l i c e s 82 111 111—I-H—HH—r-H—H-f -4.50 -3.00 -1.50 0.00 1.50 4.50 6.00 7.50 9.00 ( b ) 8 3 - 1 . 5 0 - 3 . 00 - 4 . 50 - 4 . 5 0 - 3 . 0 0 - 1 . 5 0 0 . 0 0 1 . 5 0 3 . 0 0 4 . 5 0 6 . 0 0 7 . 5 0 9 . 0 0 ( c ) 14. 00 12. 00 10. 8. 00 6. 00 4. 00 2.00 0.00 1 I | | | -2.00 -4. 00 -4.00 -2.00 0 > 0 0 2.00 4.00 6.00 8.00 10.00 12.00 14.-00 (a) F i g u r e 5.15 (a) - ( c ) T h r e e v i e w s of s u p e r i m p o s e d s l i c e s a f t e r t r a n s f o r m a t i o n f o r m a c h i n i n g w i t h no n e g a t i v e d r a f t ( b ) ( c ) 87 -H-B l a n k e d Out -4.00 -2.00 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 F i g u r e 5.16 D a t a f o r ' h o l e s ' be m a c h i n e d a r e b l a n k e d out s i n c e they c a n n o t 88 14. 00 -12^0~T0.¥0 -8.00 -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00 F i g u r e 5.17 D a t a f o r base p l a n e 89 F i g u r e 5.18 shows t h e f i t t e d s u r f a c e o f a h a l f mould. The mould was m a c h i n e d by t h e POLYHEDRAL NC System s i m i l a r t o t h e p r e v i o u s e x a m p l e s . F i g u r e 5.18 F i t t e d s u r f a c e f o r bone and c o r r e s p o n d i n g f e m a l e mould 90 -•<« 4. GENERAL PROCEDURE FOR REPLICATING A MEASURED SURFACE A g e n e r a l p r o c e d u r e f o r making a mould of an a r b i t r a r y s u r f a c e i s d e s c r i b e d below: 1) Measure s u r f a c e g e o m e t r y by one o f many m e a s u r i n g t e c h n i q u e s ; 2) D i g i t i z e m easured d a t a ; 3) Check i f s u r f a c e o r i e n t a t i o n i s s u i t a b l e f o r m a c h i n i n g , a p p l y g e n e r a l t r a n s f o r m a t i o n ( r o t a t i o n s and t r a n s l a t i o n s ) i f n e c e s s a r y ; 4) Add d a t a f o r t h e p a r t i n g p l a n e ; 5) O r g a n i z e d a t a i n t o a r e c t a n g u l a r a r r a y by s u r f a c e - f i t t i n g ; 6) A p p l y s u r f a c e - a d j u s t m e n t s i f n e c e s s a r y ; 7) M a c h i n e mould u s i n g t h e POLYHEDRAL NC s y s t e m . 5. OTHER CONSIDERATIONS In t h e above examples, t h e a s s u m p t i o n was made t h a t t h e c a v i t y -s u r f a c e s were i d e n t i c a l t o t h e measured o n e s , i e . , no a l l o w a n c e s were made f o r s h r i n k a g e s . T h i s a s s u m p t i o n h o l d s t r u e when su c h m a t e r i a l s a s s i l i c o n e r u b b e r , d e n t a l p l a s t e r or SYNCAST a r e used as m o u l d i n g m a t e r i a l s . The d e g r e e of s h r i n k a g e i s l a r g e l y d e p e n d a n t on t h e n a t u r e o f t h e m o u l d - m a t e r i a l a s w e l l a s t h e s u r f a c e - a r e a o f d i e s and m o u l d s . E s t i m a t e has t o be b a s e d on d e t a i l e d a n a l y s i s a s w e l l a s e n g i n e e r i n g judgement and e x p e r i e n c e , and i s beyond t h e s c o p e o f t h i s work. 91 V I . SPECIAL DIE-CAVITY SURFACES 1. SPECIALIZED DIES Many d i e - c a v i t i e s have c h a r a c t e r i s t i c s u r f a c e s w h i c h l e n d t h e m s e l v e s t o s p e c i a l t r e a t m e n t s and r e q u i r e n o • t r a n s f o r m a t i o n o f s u r f a c e - p o i n t s i n t o an o r t h o g o n a l g r i d . T h e s e s u r f a c e s a r e u s u a l l y a n a l y t i c a l i n n a t u r e and m a c h i n i n g t o o l - p a t h s c an be a n a l y t i c a l l y d e t e r m i n e d . F o r example, t h e t o o l - p a t h f o r m a c h i n i n g a c i r c u l a r c y l i n d e r f o l l o w s a c i r c u l a r p a t h t h a t i s c o n c e n t r i c w i t h t h e c y l i n d e r i t s e l f . ( F i g u r e 6.1) A c a v i t y -s u r f a c e may have t h e shape of a d u c t w h i c h f o l l o w s a g u i d i n g c u r v e c a l l e d a s p i n e , n o r m a l t o w h i c h i s a c r o s s - s e c t i o n h a v i n g a shape t h a t i s a f u n c t i o n of a r c - l e n g t h a l o n g t h e s p i n e . In t h i s c a s e , s u r f a c e - n o r m a l s c an be c a l c u l a t e d t o d e t e r m i n e t h e t o o l - p o s i t i o n s t o g u i d e a c u t t e r a l o n g t h e s p i n e . ( F i g u r e 6.2) I f an i t e m w i t h t h e s e t y p e s o f s u r f a c e s a r e t o be made i n l a r g e numbers o v e r a l o n g p r o d u c t i o n p e r i o d , i t p a y s t o d e v e l o p s p e c i a l i z e d t r e a t m e n t s t o make d i e s f o r a p a r t i c u l a r m a n u f a c t u r i n g r u n . S o f t w a r e may have t o be w r i t t e n f o r m o d e l l i n g o f c a v i t y - s u r f a c e s and o r g a n i z a t i o n o f m a c h i n i n g o p e r a t i o n s . F o r i n s t a n c e , b o t t l e s o f d i f f e r e n t s i z e s and volumes f o l l o w i n g one s t a n d a r d shape may be r e q u i r e d f o r mass m a n u f a c t u r i n g . An a n a l y t i c a l model o f t h i s s t a n d a r d shape may be c r e a t e d f o r a u t o m a t i c m a c h i n i n g o f d i e s . By s p e c i f y i n g g e n e r a l p a r a m e t e r s of t h e model, c h a r a c t e r i s t i c s u r f a c e s o f d i f f e r e n t s i z e s c a n be m o d e l l e d and s u b s e q u e n t l y m a c h i n e d . A n o t h e r example i s i n t h e making of s h o e - m o u l d s . F i g u r e 6.3 shows t h e g o v e r n i n g b o u n d a r y - c u r v e s of a s h o e - m o u l d i n g 92 c a v i t y - d i e . From t h e s e c u r v e s t h e d i e s u r f a c e s c a n b e . d e v e l o p e d and t o o l p a t h g e n e r a t e d . ( F i g u r e 6 . 3 ) [Duncan & F o r s y t h , 1977] D u r i n g t h e c o u r s e of t h i s r e s e a r c h , a s p e c i a l i z e d t r e a t m e n t was d e v e l o p e d f o r m o d e l l i n g and m a c h i n i n g o f a s p e c i a l d i e f o r m o u l d i n g o f s h e l l - t y p e components; t h e a p p r o a c h i s d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n s . 1 C y l i n d r i c a l M i l l i n g Cutter C y l i n d r i c a l Workpiece Path of c u t t e r f o l l o w s a c i r c l e c o n c e n t r i c to the workpiece F i g u r e 6.1 T o o l p a t h f o r m a c h i n i n g a c i r c u l a r c y l i n d e r 94 2. SPECIALIZED MOULD FOR A SHELL 2.1 I n t r o d u c t i o n F i g u r e 6.4 shows two h a l f - d i e s o f a c a v i t y - m o u l d f o r a s h e l l - l i k e component r e q u i r e d f o r a s p e c i a l a p p l i c a t i o n i n r e c o n s t r u c t i v e s u r g e r y . The l o w e r c a v i t y ( f l o o r ) has a shape o f an e l l i p t i c , p a r a b o l o i d and t h e u p p e r s u r f a c e ( c e i l i n g ) i s an o f f s e t s u r f a c e a t a s p e c i f i e d d i s t a n c e away f r o m t h e f l o o r . The b o u n d a r y w a l l s a r e n o r m a l t o t h e p a r a b o l o i d e v e r y w h e r e , and t h e p a r t i n g p l a n e i s i n c l i n e d t o f a c i l i t a t e t h e r e m o v a l of t h e moulded s h e l l . I t was d e c i d e d t h a t t h e l o w e r d i e - b l o c k c o u l d be m a c h i n e d d i r e c t l y . To o b t a i n good e d g e - d e f i n i t i o n , t h e t o p s u r f a c e can be m a c h i n e d as a c o n c a v e - u p w a r d c a v i t y and t h e u p p e r d i e made by c a s t i n g i n t o t h e c a v i t y t o form t h e c e i l i n g s u r f a c e . ( F i g u r e 6.5) 2.2 O r g a n i z a t i o n Of M a c h i n i n g P r o c e s s The r e q u i r e m e n t s o f t h e m a c h i n i n g p r o c e s s c a n be s t a t e d a s f o l l o w s : -a) upper and l o w e r s u r f a c e s s h o u l d be m a c h i n e d as c o n c a v e -upwards c a v i t i e s ; b) b o u n d a r y - w a l l s n o r m a l t o t h e l o w e r s u r f a c e s h o u l d be m a c h i n e d t o g i v e good edge d e f i n i t i o n ( t h e r i m , see F i g u r e 6.4) and f l a s h - l i n e ; c ) t h e i n c l i n e d p a r t i n g p l a n e s and t h e upper and l o w e r b a s e -p l a n e s r e q u i r e m a c h i n i n g . upper base-plane incl i n e d parting plane lower base-plane as a cavity F i g u r e 6.4 C a v i t y - d i e of a s h e l l - m o u l d 96 Good e d g e - d e f i n i t i o n can be o b t a i n e d by m a c h i n i n g the m i r r o r s u r f a c e of the upper d i e , from w h i c h the d i e b l o c k can be o b t a i n e d by r e v e r s a l p r o c e s s F i g u r e 6.5 Upper d i e - b l o c k i s made by c a s t i n g i n t o m achined c a v i t y 97 C u t t e r l o c a t i o n d a t a c an be computed f o r b o t h t h e upper and l o w e r d i e - s u r f a c e s u s i n g a s i n g l e p r o g r a m . The upper d i e -s u r f a c e i s an o f f s e t s u r f a c e t o an e l l i p t i c p a r a b o l o i d . T h i s i s an a n a l y t i c s u r f a c e and t o o l - c e n t e r p o s i t i o n s c an be c a l c u l a t e d by e m p l o y i n g g e n e r a l o f f s e t t h e o r i e s . The lo w e r c a v i t y can be c o n s i d e r e d a s t h e same t y p e of s u r f a c e b u t w i t h z e r o o f f s e t . P r o gram CAVITY6 ( A p p e n d i x C) was d e v e l o p e d t o g e n e r a t e t h e CLD f o r b o t h d i e - b l o c k s . U s e r i n p u t s i n c l u d e t h e g e n e r a l p a r a m e t e r s of t h e c h a r a c t e r i s t i c e q u a t i o n of an e l l i p t i c p a r a b o l o i d , t o o l - r a d i u s , s h e l l - t h i c k n e s s , and s t e p - s i z e f o r m a c h i n i n g . In t h i s way, d i f f e r e n t s i z e s and t h i c k n e s s e s o f t h e p a r t i c u l a r shape can be h a n d l e d , p r o v i d i n g f l e x i b i l i t y when d i f f e r e n t s i z e s a r e r e q u i r e d f o r d i f f e r e n t p r o d u c t i o n r u n s . A s p h e r i c a l l y ended m i l l i n g c u t t e r was u s e d , t h e s i z e of w h i c h was d e t e r m i n e d by t h e minimum r a d i u s of c u r v a t u r e of t h e d i e - s u r f a c e , as t h i s w ould e n s u r e t h e b e s t s u r f a c e - f i n i s h by u s i n g t h e b i g g e s t t o o l p o s s i b l e w i t h o u t t h e r i s k of u n d e r c u t t i n g . The s t a r t i n g p o s i t i o n f o r m a c h i n i n g was t h e v e r t e x of t h e p a r a b o l o i d , f r o m w h i c h t h e t o o l moved a l o n g a t a c o n s t a n t i n c r e m e n t o v e r t h e c a v i t y - s u r f a c e . Once t h e l i m i t o f t h e c a v i t y was r e a c h e d , t h e t o o l moved f i r s t o u t w a r d s ( t o a v o i d c u t t i n g t h e c a v i t y - s u r f a c e ) , and t h e n downwards t o c u t t h e edge s u r f a c e ( s i d e - w a l l s ) , t h e d i r e c t i o n s o f t h e c u t t e r b e i n g d e t e r m i n e d by th e v e c t o r p r o d u c t s o f t h e s u r f a c e - n o r m a l and t h e b o u n d a r y - c u r v e ( f l a s h - l i n e ) t a n g e n t v e c t o r . A f t e r t h e s i d e - w a l l was c u t , t h e t o o l was g u i d e d a c r o s s t h e p a r t i n g p l a n e t o g e n e r a t e t h e 98 i n c l i n e d p a r t i n g s u r f a c e . T h i s s c a n r e p e a t e d a t i n c r e m e n t s o f y, u n t i l t h e whole d i e was m a c h i n e d . F i g u r e 6.6 shows t h e c u t t e r p a t h f o r a s i n g l e s c a n . Tool Path Sequence 1) Tool O f f s e t Path f o r Upper Cavity-Surface 2) Tool Moves Outwards to Avoid Undercut of Edge 3) Tool Moves Downwards to Cut the Edge Wall 4) Tool Moves Outwards to Generate P a r t i n g Plane F i g u r e 6.6 C u t t i n g p a t h f o r upper c a v i t y 99 •2.2.1 Calculation of Cutter Location Data The c h a r a c t e r i s t i c equation of an e l l i p t i c paraboloid is : c'z = 0 2 2 x + y 2 . 2 a b or 2 • 2 + b c ±- = 0 F(x,y,z) The o f f s e t tool-centre position (x t»y t z^ .) i s found by where and x t : = X + ajR h : ; y + 6jR z t = = z + Y 1R = I 3F_ 3_F -2x a i : "7 S 3X °x a 6 i : 1_ 3F_ 3_F -2y s sy sy b 2 1 i f . 3F_ 1_ Y i = s 3Z 3Z c s = •• ( <5l>2 + 3x ay 3Z ( 6.1 ) ( 6.2 ) ( 6.3 ) ( 6.4 ) When the tool reaches the edge ( i e . when : x > a / ( 1 ,2 "2 y-)) ; i t moves outwards along the binomial vector to avoid cutting the cavity-surface. The binormal i s the cross product of the boundary tangent and the surface-normal. 100 The b o u n d a r y - c u r v e ( f l a s h - l i n e ) i s an e l l i p s e ( F i g u r e 6.7 ) w i t h t h e f o r m u l a : 2 2 2 . 2 1 a b T a k i n g p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o x and y , t h e d i r e c t i o n c o s i n e s o f t h e t a n g e n t v e c t o r t^  i s : 1 c H " H l Bo = - ( 6.5 ) H \ ' o 2 where : H = 1 + H . .2 1 and : = - 0 * -a y The b i n o r m a l v e c t o r b_ i s t h e c r o s s p r o d u c t o f t h e s u r f a c e normal n_ and t h e edge t a n g e n t t^  where : i = a i i + &ii+ Y j j i t = a 2 j_ + 6 2 j_ + Y2J< Thus : b^  = n X t ( 6.6 ) = « 3 l + B3J + Y3J1 Y 1 B 2 '3 B 3 = Ii!2 ( 6.7 ) Y _ a l B 2 - g l a 2 3 H 2 H2 = v A ^ l B2^ 2 + ^ l a 2 ^ + ^ a l 6 2 " e l a 2 ) ' 101 F i g u r e 6.7 F l a s h i s an L i n e o f E l l i p s e D i e ( B o u n d a r y o f Lower C a v i t y ) 102 F i g u r e 6.8 T o o l - m o t i o n f o r c u t t i n g t h e e d g e - w a l l 103 2.2.2 M a c h i n i n g Of D i e s F i g u r e s 6.9 and 6.10 show t h e upper and l o w e r d i e - s u r f a c e s w i t h t h e i r c o r r e s p o n d i n g CLD p a t h computed by program CAVITY6, and F i g u r e s 6.11 and 6.12 show t h e f i n i s h e d d i e s . M a c h i n i n g was p e r f o r m e d u s i n g d i f f e r e n t m a t e r i a l s r a n g i n g from p o l y u r e t h a n e foam, SYNCAST, d e n t a l p l a s t e r , t o p l e x i g l a s s and z i n c - a l u m i n i u m a l l o y . S u r f a c e a s p e r i t i e s were i m p e r c e p t i b l e when m a t e r i a l s w i t h c o a r s e s u r f a c e s , s u c h as foam, were u s e d . Cusps were o b s e r v a b l e on m e t a l and p l e x i g l a s s , but no h a n d - f i n i s h i n g was n e c e s s a r y f o r t h e p a r t i c u l a r a p p l i c a t i o n i n wh i c h t h e d i e s a r e t o be u s e d . The s t e p - s i z e u s e d was o n e - t e n t h t h e d i a m e t e r of t h e m i l l i n g c u t t e r , and t h e t o t a l m a c h i n i n g t i m e f o r one d i e was o f t h e o r d e r o f t h r e e h o u r s . S u r f a c e - f i n i s h would be f u r t h e r i m p r o v e d when m a c h i n i n g i s done i n ' v e c t o r ' mode as o p p o s e d t o 2 - 1/2 D mode u s e d f o r t h i s r e s e a r c h , a l i m i t a t i o n imposed by t h e machine w i t h w h i c h t h e a u t h o r p e r f o r m e d a l l h i s work. T h i s was s u b s e q u e n t l y p r o v e n when t h e same m a c h i n i n g p r o c e d u r e was p e r f o r m e d i n a n o t h e r i n s t a l l a t i o n . 1 The a d v a n t a g e o f t h e good s u r f a c e - f i n i s h p r o v i d e d by a l a r g e c u t t i n g t o o l was p a r t i a l l y o f f s e t by t h e l a r g e f i l l e t i t c r e a t e d a t t h e f l a s h - l i n e ( F i g u r e 6.13). In p r a c t i c e , a ' r e t o u c h i n g ' o p e r a t i o n m i g h t be n e c e s s a r y by g u i d i n g a s m a l l e r c u t t e r a r o u n d t h e edge t o m i n i m i z e any e x c e s s i v e ' f l a s h ' t h a t V e c t o r Mode i m p l i e s s i m u l t a n e o u s m o t i o n on 3 a x e s ; whereas an 2 1/2 D machine o n l y a l l o w s s i m u l t a n e o u s m o t i o n o f 2 a x e s a t one t i m e . 1 04 may o c c u r d u r i n g t h e m o u l d i n g p r o c e s s . F i g u r e 6 . 9 Lower c a v i t y - s u r f a c e and c o r r e s p o n d i n g t o o l - p a t h 105 F i g u r e 6.10 Upper c a v i t y - s u r f a c e and c o r r e s p o n d i n g t o o l - p a t h 106 F i g u r e 6.12 M a c h i n e d s u r f a c e f o r upper c a v i t y 107 Large f i l l e t , or ' f l a s h 1 , appears when a large tool i s used; this can be cleaned off by retouching with a small t o o l . F i g u r e 6.13 M a c h i n i n g upper l a r g e amount o f c a v i t y w i t h a b i g t o o l f l a s h a t f l a s h - l i n e r e s u l t s i n 108 2 . 3 E x t e n s i o n Of Method The p r o c e d u r e d e s c r i b e d above d e a l s w i t h c a v i t y - s u r f a c e s t h a t a r e a n a l y t i c a l , and w i t h p a r t i n g s u r f a c e s a s p l a n e s . The same p r o c e d u r e c an be a p p l i e d t o n o n - a n a l y t i c a l c a v i t y - s u r f a c e s and p a r t i n g l i n e s . F i g u r e 6.14 shows a c a v i t y - d i e f o r a s h e l l - t y p e component s u c h as a p i e c e of human s k u l l . B o t h t h e upper and l o w e r s u r f a c e s a r e n o n - a n a l y t i c a l , and t h e p a r t i n g l i n e i s a t h r e e -d i m e n s i o n a l s p a c e - c u r v e o f an a r b i t r a r y shape ( F i g u r e 6.15). To g e n e r a t e t h e c u t t e r l o c a t i o n d a t a , i t i s n e c e s s a r y t o c a l c u l a t e t h e s u r f a c e - n o r m a l and t h e t a n g e n t of t h e b o u n d a r y -c u r v e ( i e . , f l a s h l i n e ) . The s u r f a c e - n o r m a l c an be computed u s i n g t h e p o l y h e d r a l c o n c e p t (see C h a p t e r I I , S e c t i o n 4 ) ; whereas t h e a r b i t r a r y s p a c e - c u r v e r e p r e s e n t i n g t h e f l a s h - l i n e c an be f u n c t i o n a l i z e d u s i n g one of t h e many a v a i l a b l e c u r v e -f i t t i n g r o u t i n e s and i t s t a n g e n t c a l c u l a t e d by f i n d i n g t h e p a r t i a l d e r i v a t i v e s of t h e f i t t e d c u r v e . 109 Both f l o o r and c e i l i n g are non-analytical surfaces that can be machined using the polyhedral concept Bounding curve for edge i s a three-dimensional space-curve which r e s u l t s i n a two piece c y l i n d r i c a l l y curved parting surface F i g u r e 6.14 S h e l l - m o u l d w i t h a r b i t r a r y and c u r v e d p a r t i n g s u r f a c e b o u n d i n g s u r f a c e s 110 -Flash-line defined by a r b i t r a r y 3-D space-curve Non-symmetric, non-planar parting surfaces F i g u r e 6.15 A r b i t r a r y f l a s h - l i n e g i v e s n o n - s y m m e t r i c , c y l i n d r i c a l l y c u r v e d p a r t i n g s u r f a c e s 111 V I I . DISCUSSIONS AND CONCLUSIONS 1. CONSIDERATIONS IN DIE AND MOULD MAKING When d e s i g n i n g d i e s and mould s , many f a c t o r s must be c o n s i d e r e d . In a d d i t i o n t o t h e p r o b l e m s o f s h r i n k a g e , s p r i n g - b a c k , e t c . , t h e methods of d i e and mould making a r e a l s o d e p e n d a n t on t h e m a n u f a c t u r i n g p r o c e s s e s . T r a d i t i o n a l l y , a p a t t e r n o f d e s i r e d shape i s made f i r s t and u s e d t o mould a c a v i t y o f m a t c h i n g s h a p e . In s a n d - c a s t i n g , t h i s p a t t e r n i s u s u a l l y made i n wood, or more r e c e n t l y , m a c h i n e d i n p o l y s t y r e n e foam. I t i s t h e n b u r i e d i n f o u n d r y sand t o g i v e t h e r e q u i r e d c a v i t y - m o u l d . In i n v e s t m e n t - c a s t i n g ( a l s o known as l o s t - w a x p r o c e s s ), t h e p a t t e r n i s made o f wax. T h i s i s d i p p e d i n t o s l u r r i e s t o form a s h e l l a r o u n d i t . The wax i s removed by b a k i n g t h e s h e l l - m o u l d and b u r n i n g o u t t h e p a t t e r n . • D i e s made o f h a r d and t o u g h m a t e r i a l s , s u c h as t h o s e f o r f o r g i n g and i n j e c t i o n - m o u l d i n g c a n be made from p r e s s i n g a h a r d male m a s t e r model i n t o a t e m p o r a r y s o f t e n e d ( h e a t e d ) b l o c k o f m e t a l . T h i s i s known as h o b b i n g , and i t has t h e a d v a n t a g e of b e i n g a b l e t o make m u l t i p l e d i e - c a v i t i e s from one s i n g l e m a s t e r model; b ut w i t h t h e a d v e n t o f modern a u t o m a t i c machine t o o l s , t h e making o f s u c h d i e s may be more e f f i c i e n t l y done by d i r e c t m a c h i n i n g o f t h e c a v i t i e s , a s d i s c u s s e d i n C h a p t e r s I I I , V and VI . The a d v a n t a g e s of d i r e c t m a c h i n i n g of moulds, s u c h as t h o s e d i s c u s s e d i n C h a p t e r I I I , may be o f f s e t by t h e r e q u i r e m e n t t h a t a l a r g e number of moulds have t o be made. In t h i s c a s e , i t i s more e f f i c i e n t t o make a male m a s t e r p a t t e r n f r o m w h i c h any 1 12 number o f moulds can be d e r i v e d . T h i s i s e s p e c i a l l y a p p a r e n t i n p r o c e s s e s l i k e s a n d - c a s t i n g and i n v e s t m e n t - c a s t i n g , i n w h i c h moulds a r e d e s t r o y e d d u r i n g t h e r e m o v a l o f c a s t p r o d u c t s , and t h u s c a n be u s e d o n l y o n c e . M o r e o v e r , m a t e r i a l s commonly u s e d i n c a s t i n g p r o c e s s e s , s u c h a s f o u n d r y sand, a r e i m p o s s i b l e t o m a c h i n e . D i r e c t m a c h i n i n g of d i e s i s a d v a n t a g e o u s when many r e p l i c a t i o n s a r e t o be c a s t f r o m one s i n g l e mould i n p r o c e s s e s s u c h as i n j e c t i o n - m o u l d i n g , f o r g i n g o r t h e l a y i n g - u p of f i b r e -g l a s s m a t e r i a l s . 2. CASTING AND MOULDING OF MODELS The s i m p l e s t f o r m of m o u l d i n g p r o c e s s i s what i s known as open m o u l d i n g — i n w h i c h l i q u i d m a t e r i a l i s p o u r e d i n t o a s i n g l e mould - c a v i t y . T h i s method was u s e d f o r making most of t h e m o d e l s f o r t h i s r e s e a r c h . More e l a b o r a t e m o d e ls r e q u i r e two o r even more s e p e r a t e d i e - b l o c k s , s u c h as t h e s h e l l - m o u l d and t h e s hoe-mould d i s c u s s e d i n p r e v i o u s c h a p t e r s . To f a c i l i t a t e t h e r e m o v a l o f m a t e r i a l a f t e r t h e m o u l d i n g p r o c e s s , s i d e - w a l l s of a d i e a r e u s u a l l y ' d r a f t e d ' , i e . , t h e y have a s l o p e t o t h e v e r t i c a l a t a s m a l l a n g l e . (See F i g u r e 3.1) N e g a t i v e d r a f t i s u s u a l l y n o t a l l o w e d , s i n c e t h i s c a n n o t be m a c h i n e d u s i n g s i m p l e m i l l i n g o p e r a t i o n s . I n a d d i t i o n , i t l o c k s t h e component i n , a l t h o u g h t h i s may n o t p r e s e n t any p r o b l e m f o r f l e x i b l e and p l i a b l e m a t e r i a l s . To p r e v e n t t h e moulded p i e c e f r o m s t i c k i n g t o t h e c a v i t y s u r f a c e s , s u i t a b l e r e l e a s e a g e n t may have t o be a p p l i e d . 1 13 The open m o u l d i n g a p p r o a c h i s most s u i t a b l e i n ' s u r f a c e -r e a c t i v e ' c a s t i n g . F l u i d o r f i b r o u s m a t e r i a l i s a p p l i e d t o t h e m o u l d by p o u r i n g o r s p r a y i n g where i t s o l i d i f i e s by some s u r f a c e - r e l a t e d m e c h a n i s m . An e x a m p l e o f t h i s i s t h e method o f ' s l i p - c a s t i n g ' . A f e m a l e c a v i t y made o f p o r o u s m a t e r i a l ( u s u a l l y some f o r m o f p l a s t e r ) i s f i l l e d w i t h a c e r a m i c s l u r r y ( u s u a l l y c l a y ) . C a p i l l a r y a c t i o n removes w a t e r f r o m t h e s l u r r y , l e a v i n g a u n i f o r m s e m i - r i g i d s h e l l o f d e w a t e r e d s l u r r y ( a c a k e ) on t h e c a v i t y - w a l l , i t s t h i c k n e s s d e p e n d s on t i m e a l l o w e d . S u r p l u s s l u r r y , w h i c h i s n o t y e t d e w a t e r e d , i s t h e n p o u r e d o u t , p r o d u c i n g a s h e l l - m o u l d w i t h o u t t h e use o f a m a l e f o r m e r m o del ( c o r e ) w i t h i n t h e c a v i t y . T h i s method was e x a m i n e d u s i n g t h e p l a s t e r m o u l d o f t h e f a c i a l m o d e l ( s e e C h a p t e r V ) , a n d t h e r e s u l t i n g s h e l l - m o u l d i s shown i n F i g u r e 7.1 F i g u r e 7.1 S h e l l - m o u l d o f f a c i a l mask f r o m s l i p - c a s t i n g 1 14 3. DIE DESIGN AND MACHINING SYSTEM BASED ON POLYHEDRAL NC  SYSTEM The u l t i m a t e o b j e c t i v e o f t h i s r e s e a r c h i s t o d e v e l o p a g e n e r a l d i e d e s i g n and m a n u f a c t u r i n g s y s t e m u s i n g modern h i g h s p e e d c o m p u t e r s and a u t o m a t i c machine t o o l s . Such a s y s t e m s h o u l d have t h e f o l l o w i n g c a p a b i l i t i e s . a) I t s h o u l d a l l o w d e s i g n e r s t o d e s i g n and model s u r f a c e s f r o m a n a l y t i c a l e q u a t i o n s and measured p h y s i c a l m o d e l s , a s w e l l as f r o m t w o - d i m e n s i o n a l s k e t c h e s o f c h a r a c t e r i s t i c b o u n d a r y - c u r v e s . b) I t s h o u l d i n c o r p o r a t e f e a t u r e s f o r v i s u a l i z a t i o n and m a n i p u l a t i o n of s u r f a c e s so t h a t d e s i g n e r s can i n t e r a c t i v e l y a d j u s t and m o d i f y d e s i g n e d s u r f a c e - s h a p e s and p r o p e r t i e s . c ) I t s h o u l d be a b l e t o s u p p o r t d i f f e r e n t t y p e s of machine t o o l s , f r o m th e s i m p l e s t t o t h e most s o p h i s t i c a t e d . M o r e o v e r , i t s h o u l d i n c o r p o r a t e r e a l - t i m e c o n t r o l o f m a c h i n e s t o p e r m i t a f a s t t u r n - a r o u n d t i m e . d) I t s h o u l d be ' u s e r - f r i e n d l y ' , by n o t r e q u i r i n g e x p e r t s i n computer programming t o o p e r a t e t h e s y s t e m . On t h e o t h e r hand, i t s h o u l d a l l o w s p e c i a l p r o g r a m s t o be d e v e l o p e d f o r s p e c i f i c t y p e s o f d i e s s i m i l a r t o t h o s e d i s c u s s e d i n C h a p t e r V I . 115 3.1 Work A c h i e v e d In T h i s R e s e a r c h E l e m e n t s o f a p r o p o s e d a u t o m a t i c d i e d e s i g n and m a c h i n i n g s y s t e m , b a s e d on t h e p o l y h e d r a l c o n c e p t , have been d e v e l o p e d . They i n c o r p o r a t e programs w r i t t e n o v e r t h e p a s t few y e a r s a s w e l l a s new r o u t i n e s t h a t were d e v e l o p e d f o r t h e p u r p o s e of t h i s r e s e a r c h . To summarize, t h r e e m a j o r g o a l s were a c h i e v e d : a) The Method o f H i g h e s t P o i n t was e x t e n d e d i n t o a g e n e r a l g e o m e t r i c m o d e l l i n g r o u t i n e f o r p i e c e w i s e compound a n a l y t i c s u r f a c e s w i t h t h e d e v e l o p m e n t o f program GEN7. b) A. g e n e r a l a p p r o a c h i n r e p l i c a t i n g a r b i t r a r y s u r f a c e s by c a s t i n g i n t o m a c h i n e d c a v i t y - m o u l d s were d e v e l o p e d by u t i l i z i n g s u r f a c e - f i t t i n g .program TRUEPERS and m a c h i n i n g p r o g r am SUMAIR. M o r e o v e r , t e c h i q u e s were d e v e l o p e d t o h a n d l e c o m p l i c a t e d s u r f a c e s ( s u c h as t h o s e measured from CAT s c a n n i n g ) and t o t r a n s f o r m them i n t o an o r i e n t a t i o n most s u i t a b l e f o r m a c h i n i n g . A new a p p r o a c h t o f o r m a t i o n o f a ' s h a r p ' p a r t i n g p l a n e and f l a s h l i n e was d e v e l o p e d . c) A s p e c i a l i z e d a p p r o a c h i n t h e making of d i e s f o r s h e l l - t y p e components was p r o p o s e d and t e s t e d . In p a r t i c u l a r , good e d g e - d e f i n i t i o n ( i e . , s h a r p edge) was o b t a i n e d from r e v e r s a l t e c h n i q u e s . Good s u r f a c e - f i n i s h and minimum f l a s h were a c h i e v e d by c a s c a d i n g l a r g e and s m a l l t o o l s d u r i n g t h e m a c h i n i n g p r o c e s s . T h i s a p p r o a c h a l s o p e r m i t s a r b i t r a r y b o u n d i n g s u r f a c e s w i t h n o n - s y m m e t r i c , s p a t i a l l y c u r v e d f l a s h - l i n e s . 1 16 3.2 Scheme F o r P r o p o s e d D i e D e s i g n And M a c h i n i n g System F i g u r e 7.2 shows scheme f o r an i n t e g r a t e d g e n e r a l a p p r o a c h f o r d i e and mould making. D i f f e r e n t c l a s s e s of s u r f a c e s , w h ether s p e c i f i e d by e q u a t i o n s , measured d a t a , or p r o j e c t e d b o u n d a r i e s , c a n be m o d e l l e d by r o u t i n e s GEN7, TRUEPERS and PROPDEV r e s p e c t i v e l y . The p o i n t - d e f i n e d s u r f a c e s t h u s g e n e r a t e d c a n t h e n be p r o c e s s e d by program SUMAIR or NEWSU t o g i v e t h e m a c h i n i n g p a t h o f a s p h e r i c a l l y ended m i l l i n g c u t t e r . P rograms s u c h as TRUEPERS and GEN7 a l s o i n c o r p o r a t e g r a p h i c s s u b r o u t i n e s so t h a t t h e m o d e l l e d s u r f a c e s can be v i e w e d on a CRT s c r e e n o r from h a r d c o p y p l o t s . T h i s f a c i l i t a t e s e r r o r c h e c k i n g and s u r f a c e - a d j u s t m e n t s (when n e c e s s a r y ) by p r o v i d i n g e a s y means f o r u s e r s t o v i s u a l i z e any i n t e r m e d i a t e or f i n a l r e s u l t s . 117 Piecewise A n a l y t i c a l SURFACE SPECIFICATION Define General Parameters for Surface -Pieces POINT-DEFINED SURFACE Modelled by GEN7 Measured Data Transformation of Data ( i f necessary) Surface-Fitting by TRUEPERS Surface Defined by Boundaries Functionalized Boundary Curves by CRVFIT Span 3-D Surface by PROPDEV Surface Adjustment ( i f necessary) CALCULATE TOOL PATH BY POLYHEDRAL NC FINAL PRODUCT Rotation by 180 for female cavity ( i f necessary) Comput by S e CLD UMAIR Machine Die F i g u r e S c h e m a t i c a p p r o a c h of a g e n e r a l d i e d e s i g n and m a c h i n i n g s y s t e m b a s e d on POLYHEDRAL NC 118 M a c h i n i n g p r o g r a m s SUMAIR and NEWSU have been m o d i f i e d so t h a t t h e y a r e now ' m a c h i n e - i n d e p e n d e n t ' , i e . , t h e fo r m a t of t h e i r o u t p u t s does n o t l i m i t them t o be use d by o n l y c e r t a i n p a r t i c u l a r CNC m a c h i n e s . P r e v i o u s l y SUMAIR and NEWSU wr o t e t h e m a c h i n i n g commands o n t o E I A - f o r m a t t e d p a p e r t a p e s . The t a p e s had t o be p h y s i c a l l y t r a n s f e r r e d and mounted o n t o a SLO-SYN NC machine b e f o r e m a c h i n i n g . T h i s was b o t h u n r e l i a b l e ( p a p e r t a p e s t e n d t o b r e a k o r ja m ) , i n e f f i c i e n t and r e s u l t e d i n l o n g t u r n -a r o u n d t i m e . The programs have been m o d i f i e d so t h a t CLDs a r e now w r i t t e n o n t o d a t a f i l e s as x , y , z c o o r d i n a t e s of t o o l c e n t r e p o s i t i o n s . T h i s d a t a can t h e n be e l e c t r o n i c a l l y t r a n s m i t t e d v i a d a t a - l i n k from an Amdahl 470/V8 mainframe computer (where t h e POLYHEDRAL NC s y s t e m r e s i d e s ) t o a PDP 11/34 m i n i c o m p u t e r w h i c h c o n t r o l s t h e m i l l i n g m a c h i n e . A u t o m a t i c r o u t i n e s i n t h e PDP c o n v e r t t h e c u t t e r c o - o r d i n a t e s i n t o c o d e d machine commands. The main a d v a n t a g e of s p e c i f y i n g t h e c u t t e r l o c a t i o n d a t a as C a r t e s i a n p o i n t s i s t h a t t h e y can be p r o c e s s e d f o r use on d i f f e r e n t m a c h i n e s u s i n g d i f f e r e n t command c o d e s . Such c o d e s can t h e n be l o a d e d o n t o a NC machine v i a e l e c t r o n i c l i n k s (as th e a u t h o r u s e d ) , m a g n e t i c o r p a p e r t a p e s , o r f l o p p y d i s k s e t c . , d e p e n d i n g upon t h e f a c i l i t i e s o f a p a r t i c u l a r i n s t a l l a t i o n . E l e c t r o n i c t r a n s m i s s i o n of d a t a d r a m a t i c a l l y s h o r t e n s t h e ti m e l a g between c o m p u t a t i o n and a c t u a l m a c h i n i n g . T h i s i s e s p e c i a l l y i m p o r t a n t when t h e p h y s i c a l d i s t a n c e between t h e c o m p u t i n g and m a c h i n i n g s i t e i s v e r y l o n g . T h i s would be done i n a few h o u r s i n an e s t a b l i s h e d s e t - u p . T a b l e II shows a summary of v a r i o u s computer r o u t i n e s t h a t 119 c a n be i n c o r p o r a t e d i n t o t h e p r o p o s e d s y s t e m . As m e n t i o n e d p r e v i o u s l y , t h i s s y s t e m s h o u l d a l l o w r o u t i n e s t o be d e v e l o p e d f o r s p e c i a l i z e d d i e - s u r f a c e s , s u c h as program CAVITY6 f o r t h e s h e l l - m o u l d ( s e e C h a p t e r V I ) . To s i m p l i f y t h e t a s k of p r o g r a m d e v e l o p m e n t , s t a n d a r d m o dules s u c h a s PLTXYZ ( f o r p l o t t i n g ) and CNCPKG ( f o r g e n e r a t i n g machine commands, see T a b l e I I ) have been w r i t t e n so t h a t t h e y can be l i n k e d t o a n a l y s i s r o u t i n e s t h a t a r e r e q u i r e d f o r s p e c i a l i z e d s u r f a c e s . < 120 T a b l e II - Summary of R o u t i n e s t o be u s e d f o r P r o p o s e d D i e D e s i g n and M a c h i n i n g System SURFACE MODELLING GEN7 m o d e l l i n g p i e c e w i s e compound a n a l y t i c a l s u r f a c e s TRUEPERS .. g e n e r a l s u r f a c e - f i t t i n g p r o g r a m ( w i t h g r a p h i c s ) CRVFIT .... g e n e r a l c u r v e - f i t t i n g p r o g r a m e m p l o y i n g c o n i c - f i t PROPDEV ... s p a n n i n g 3-D s u r f a c e f r o m p r o j e c t i o n of s u r f a c e b o u n d a r i e s by p r o p o r t i o n a l d e v e l o p m e n t SURFACE VISUALIZATION AND MANIPULATION TRUEPERS .. g e n e r a l s u r f a c e - f i t t i n g p r o g r a m ( w i t h g r a p h i c s ) PLTXYZ .... t r i m e t r i c p l o t t i n g of p o i n t - d e f i n e d s u r f a c e s from o u t p u t s o f GEN7, CAVITY6, TRANSFORM e t c . (no h i d -den l i n e r e m o v e l ) PBONE3D ... t r i m e t r i c p l o t t i n g of d i g i t i z e d c o n t o u r l i n e s TRANSFORM . g e n e r a l t r a n s f o r m a t i o n o f d a t a MACHINING SUMAIR, NEWSU . CNCNEWSU CNCPKG compute CLD p a t h by p o l y h e d r a l c o n c e p t , i n -c o r p o r a t i n g a n t i - i n t e r f e r e n c e f e a t u r e g e n e r a t e command code f o r SLO-SYN machine from o u t p u t o f SUMAIR or NEWSU package o f FORTRAN c a l l a b l e s u b r o u t i n e s f o r g e n e r a t i n g command c o d e s f o r m a c h i n i n g by SLO-SYN NC m i l l i n g m achine 121 4. PROPOSED FURTHER WORK To i n c o r p o r a t e t h e e l e m e n t s d e v e l o p e d f o r t h e p r o p o s e d d i e d e s i g n and m a c h i n i n g s y s t e m , f u r t h e r work i s n e c e s s a r y t o merge them i n t o one s i n g l e u n i t so t h a t a ' t u r n - k e y ' s y s t e m , w h i c h i n c l u d e s b o t h hardware and s o f t w a r e , c an be made. Recommended i t e m s o f work a r e as f o l l o w s : a) d e v e l o p m e n t o f 'master c o n t r o l program' t o d i r e c t and a l l o c a t e t a s k s among v a r i o u s e l e m e n t s of t h e s y s t e m ; b) d e v e l o p m e n t o f i n t e r a c t i v e ' f r o n t - e n d s ' t o f a c i l i t a t e c o m m u n i c a t i o n between u s e r and computer, p e r h a p s i n t h e form of s c r e e n menus; c) d e v e l o p m e n t of d a t a a c c q u i s i t i o n a p p a r a t u s d i r e c t l y c o m p a t i b l e w i t h t h e s y s t e m t o e l i m i n a t e t h e need f o r d i g i t i z a t i o n ; d) d e v e l o p m e n t of a n a l y s i s r o u t i n e s f o r e v a l u a t i n g s u r f a c e p r o p e r t i e s as w e l l as s u r f a c e - a d j u s t m e n t s . 5. CONCLUSIONS C a v i t y - d i e s c o n s i s t of b o u n d i n g s u r f a c e s t h a t a r e e i t h e r a n a l y t i c a l o r n o n - a n a l y t i c a l . A n a l y t i c a l s u r f a c e s a r e u s u a l l y c o m b i n a t i o n s of v a r i o u s i n d i v i d u a l s u r f a c e - e l e m e n t s t h a t c a n be r e p r e s e n t e d by m a t h e m a t i c a l e q u a t i o n s . F o r s i m p l e , d e v e l o p a b l e s u r f a c e s , t o o l - p a t h s can be computed and s p e c i a l a l g o r i t h m s w r i t t e n t o o r g a n i z e t h e m a c h i n i n g p r o c e s s e s . O t h e r s may c o n t a i n s t a n d a r d a n a l y t i c a l s u r f a c e - p i e c e s i n t e r s e c t i n g and i n t e r p e n e t r a t i n g one a n o t h e r a t c u r v e s of d i s c o n t i n u i t y . T hese have t o be s c u l p t u r e d . Most of t h e s e t y p e s o f s u r f a c e s can be 122 •< m o d e l l e d by r o u t i n e GEN7 f o r s u b s e q u e n t m a c h i n i n g by t h e POLYHEDRAL NC s y s t e m . A r b i t r a r y s h apes d e f i n e d by measured d a t a c a n be t r a n s f o r m e d i n t o a p o i n t - d e f i n e d s u r f a c e o v e r a r e g u l a r o r t h o g o n a l g r i d u s i n g p r o g r a m TRUEPERS. V a r i o u s s h a p e s measured by d i f f e r e n t t e c h n i q u e s have been so t r e a t e d and r e p r o d u c e d s u c c e s s f u l l y by m o u l d i n g i n t o m a c h i n e d c a v i t y - m o u l d s . E l e m e n t s o f an a u t o m a t i c d i e d e s i g n and m a c h i n i n g s y s t e m have been d e v e l o p e d and t e s t e d . The r e s u l t s , as shown i n p r e v i o u s c h a p t e r s , p r o v e t h a t t h e p o l y h e d r a l a p p r o a c h p r o v i d e s a f e a s i b l e mean f o r a u t o m a t i c m o d e l l i n g and m a c h i n i n g of d i e s and moul d s . T h i s i n t u r n c an be d e v e l o p e d i n t o an i n t e g r a t e d and e f f i c i e n t m a n u f a c t u r i n g s y s t e m . 123 APPENDIX A G e n e r a l T r a n s f o r m a t i o n o f Q u a d r i c S u r f a c e s E l l i p s o i d s C h a r a c t e r i s t i c E q u a t i o n : 2 2 2 x y z - + - + - - 1 = 0 2 ,2 2 a b e A f t e r t r a n s f o r m a t i o n o n t o t h e X'Y'Z' c o o r d i n a t e s y s t e m : ,2 ,2 ,2 x y z - + - + - - 1 = 0 2 , 2 2 a b c In t erms o f the o r i g i n a l XYZ c o o r d i n a t e s y s t e m : 2 2 ( 1 j x j+m^ y j+nj z j ) ( 1 ^ + m 2 y ^ + n 2 z ^ ) (1 3 x ^ + m 3 y ^ + n 3 z ^ ) 2 2 2 a b c C o n v e r t i n g i n t o t h e f o r m : 2 A i z r + B I Z L + c i = ° n n 2 n A n = ( - 1 ) Z + ( — ) + ( — ) a b c "1 2 n ^ ( l ^ x ^ + m ^ y j ) 2 n 2 ( l 2 x j + m 2 y ^ ) 2 n 3 ( 1 3 x ^ + m 3 y j ) B i a 2 + b 2 + ' r 2 _ C = ( W Y l ) 2 + ( V l ^ 2 l l ) 2 + ( 1 3 X ] + m 3 y ] ) 2 b where : x l = x " x 0 y i • y ~ ^o 124 E l l i p t i c P a r a b o l o i d s C h a r a c t e r i s t i c Equation ( i n X'Y'Z' coordinate system ) : ,2 ,2 x y - 2 + — 2 - cz' = 0 a b In terms of the XYZ coor d i n a t e system : 2 2 (1.x..+m y-.+n-z. ) (1 x +m y,+n z.) L J J_J i_ J_ + £_J f_J ^_J_ _ c ( l 0 x 1 + m o y .-n z . ) = 0 3 1 3 ^ 1 3 1 ' 2 v 2 a b Converting i n t o the form : V i 2 + V i +c i n l 2 n 2 2 J a b 2 " 1 ( l ] ^ 1 ^ 1 y 1 ) 2 n 2 d 2 x ] - r m 2 y ] ) _ ] 2 L 2 c n 3 a b 1 x +m y 1 x +m y C ] = ( - J - J LJ- ) Z + ( 2 ' 2 J ) / - c ( l 3 x 1 + m 3 y i ) 125 E l l i p t i c a l ( C i r c u l a r ) C y l i n d e r s C h a r a c t e r i s t i c E q u a t i o n ( i n X ' Y ' Z ' c o o r d i n a t e s y s t e m ) .2 ,2 x y F o r : - 2 + - 2 ^ 1 z ' = c a b In t e r m s o f t h e XYZ c o o r d i n a t e s y s t e m : ( 1 ] V m l y 1 + V l ) 2 + ( 1 2 X l + m 2 y 1 + n 2 Z 1 ) 2 _ 1 m Q 2 i 2 a b C o n v e r t i n g i n t o t h e f o r m : 2 A 1 Z 1 + B 1 Z 1 + C l = ° n. A . - < -J- > 2 + (--* > 2 1 a b 2 n 1 ( 1 1 X 1 + m l y 1 ) 2 + 2 n 2 ( 1 2 X ] + m 2 y l ) 2 1 2 , 2 a b l x + m . y . 9 1 x +m.y. c - ( X T l i l } 2 + ( - 2 - 1 - 2 ^ ] ) 2  1 a b The l i m i t o f t h e c y l i n d e r i s d e f i n e d by c . T h i s c a n be t e s t e d by s u b s t i t u t i n g ( X j , y ^ , z ^ ) i n t o : V l + m 3 y l + n 3 Z l " C L i m i t of t h e c y l i n d e r i s e x c e e d e d i f the a b o v e c o n d i t i o n i s n o t s a t i s f i e d . 126 C h a r a c t e r i s t i c E q u a t i o n ( i n X'Y'Z' c o o r d i n a t e s y s t e m ) : |2 ,2 x' y - 2 ~ —2 ~ c z ' = 0 a b In terms of the XYZ c o o r d i n a t e s y s t e m : 2 2 (1 .x.+m. y ,+n. z . ) ( l x . . + m y , + n z , ) 1 3 1 I  ] ] 1 I \ ~ ^ W V f V l * = ° a b C o n v e r t i n g i n t o the form : 2 A 1 Z 1 + B 1 Z 1 + c i = 0 n ] 2 n 2 2 1 a b B = z y ^ w i ) 2 " 2 ( 1 2 x ] ^ 2 y i ) c u 1 2 L 2 c n 3 a b 1 x +m y 1 x +m y 2 C 1 = ( - J - J 1-1 ( 2 ] 2 J ^ _ c d ^ + ^ y ^ 127 Q u a d r a t i c Cones C h a r a c t e r i s t i c E q u a t i o n ( i n X'Y'Z' s y s t e m ) : ,2 ,2 ,2 x' y z' _ + - + - = o 2 , 2 2 a b e In terms of the XYZ c o o r d i n a t e s y s t e m : ( 1 l X l + m i y i + n i 2 l ) 2 A ( 1 2 X l + m 2 y i - f n 2 Z 1 ) 2 A ( 1 3 X l + m 3 y i + n 3 Z l ) 2 + : * — + : 2 2 2 a b c C o n v e r t i n g i n t o the form : V i 2 + V ] + c i = 0 n n 2 n 2 a b c 2 n 1 ( l 1 x 1 + m 1 y 1 ) 2 n 2 ( 1 2 x 1 + m 2 y 1 ) 2 n 3 ( 1 3 x ^ + m 3 y 1 ) B i . , __ + + _ a b c c = ( 1 l X l + m l y l } 2 , ( W V l } 2 + ( 1 3 x l + m 3 y i ,2 = 0 128 APPENDIX B - USER MANUAL FOR PROGRAM GEN7 Program Name P u r p o s e .... GEN7 A v a i l a b i l i t y S o f t w a r e R e q u i r e d G e o m e t r i c a l m o d e l l i n g p r o g r a m f o r p i e c e w i s e compound a n a l y t i c a l s u r f a c e s u s i n g t h e Method of H i g h e s t P o i n t A v a i l a b l e f r o m t h e D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h C o l u m b i a . IAS/RSX BASIC V02-01, r u n n i n g under D i g i t a l E q u i p m e n t C o r p o r a t i o n RSX-11M o p e r a t i n g s y s t e m . 1. HOW TO RUN Program GEN7 has been r u n n i n g on a PDP11/34 computer a t t h e Department o f M e c h a n i c a l E n g i n e e r i n g , UBC. I t i s s t o r e d i n a m a g n e t i c t a p e named LAU ( ANSI f o r m a t t e d , 1600 b p i ). To r e t r i e v e t h e program from t h e t a p e , t h e f o l l o w i n g p r o c e d u r e must be f o l l o w e d : 1) Mount m a g n e t i c t a p e o n t o t a p e - d r i v e a c c o r d i n g t o i n s t r u c t i o n s p r o v i d e d w i t h t h e t a p e - d r i v e ; 2) Log on t o PDP u s i n g t h e HELLO command, t h e n t y p e i n a c c o u n t name and p a s s w o r d ; 3) Mount t a p e u s i n g t h e mount command : MOUNT MTO:LAU 4) Copy t a p e o n t o s y s t e m by t y p i n g : PIP =MT0:GEN7.BAS 5) Invoke BASIC i n t e r p r e t e r by t y p i n g BAS 6) To r u n t h e program, t y p e t h e command : RUN GEN7 2. USER-INPUT Program GEN7 a c c e p t s u s e r - i n p u t s i n an i n t e r a c t i v e manner. I t f i r s t prompts f o r t h e g l o b a l f i e l d d i m e n s i o n X and Y, f o l l o w e d by t h e i n c r e m e n t o f s c a n . Then, f o r e a c h of t h e s u r f a c e - t y p e , i t a s k s f o r t h e number o f p i e c e s , and f o r e a c h o f t h e p i e c e s , u s e r - i n p u t s a r e t r a n s l a t i o n s , c h a r a c t e r i s t i c p a r a m e t e r s , r o t a t i o n s , subdomain l i m i t s , o f f l i m i t h e i g h t as w e l l as t r u n c a t i o n h e i g h t . A t y p i c a l p r o m p t i n g s e q u e n c e i s shown i n F i g u r e 4.7, and t h e r e q u i r e d i n p u t f o r e a c h s u r f a c e p i e c e i s shown i n T a b l e I . 3. PROGRAM OUTPUT O u t p u t from GEN7 i s c o n t a i n e d i n t h e f i l e DATA.DAT. O u t p u t f o r m a t s a r e a s f o l l o w s : 1st l i n e : NPNTS ( f o r m a t ### ) - number of p o i n t s p e r X - s c a n Next (NPNTS+1) l i n e s : x, y , z ( f o r m a t ###.###,###.###,###.### ) - c a r t e s i a n c o o r d i n a t e s o f n o d a l p o i n t 129 Repeat 1 st t o (NP N T S + l ) t h l i n e f o r e a c h X - s c a n 4. SAMPLE INPUTS Sample i n p u t s f o r t h e v a c um c l e a n e r h o u s i n g mould d i s c u s s e d C h a p t e r IV a r e a s f o l l o w s : F i e l d D i m e n s i o n s : I n c r e m e n t f o r Scan : Number of E l l i p s o i d s 3.5 # 7.0 0.1 E l l i p s o i d ( 1 ) E l l i p s o i d ( 2 ) : ( xo , y 0 , z 0 ) ( a, b, c ) ( 6l , 62, 6 3 ) ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t : ( xo , yo, z 0 ) ( a, b, c ) ( 6i , 62, 63 ) ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t Number o f C y l i n d e r s : C y l i n d e r ( 1 ) C y l i n d e r ( 2 ) ( Xo » Y o i z 0 ( a, b, c ) ( 6i , 62, 63 ) ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t ( Xo , y D , z 0 ) ( a, b, c ) ( e l f e 2 , e 3 ) ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t ) = Number o f c o n e s : 1 C o n e d ) ( xo , y 0 , z 0 ) ( a, b, c ) ( 6i, 8 2 , 63 ) ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t ( 1 .50, 1 .25, 0. ) ( 1.25, 1.60, 1. ) ( 0 ° , 0 ° , 0 ° ) ( 1.5, 2.75 ) ( 0.65, 2.25 ) 0 99 ( 1.50, 4.95, 0. ) ( 1 .25, 1 .35, 1 .75 ) ( 0 ° , 0 ° , 0 ° ) ( 1.5, 2.75 ) ( 4.95, 6.30 ) 0 99 = ( ( 0., 2.25, 0. ) ( 1.0, 1.6, 1.5 ) 0 ° , - 9 0 o , 0 ° ) ( 0., 1.50 ) ( 0.65, 2.25 ) 0 99 ( 0., 4.95, 0. ) ( 1.75, 1.35, 1.50 ) ( - 9 0 ° , 0 ° , 0 ° ) ( 1.5, 2.0 ) ( 4.95, 6.30 ) 0 99 ( 1.5, 2.25, 1.75 ) ( 0.46631, 0.46631, 1 ) ( 0 ° , 0 ° , 0 ° ) ( 1.5, 2.0 ) ( 1.8, 2.25 ) 0 130 T r u n c a t i o n H e i g h t = 9 9 Number o f V a r i a b l e C y l i n d e r s : 2 V a r i - C y l O ) V a r i - C y l ( 2 ) : ( xo , yo , zo ) ( a, b, c ) 6 ( Xmin, Xmax ) ( Ymin, Yamx ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t : ( xo , yo , zo ) ( a, b, c ) 6 ( Xmin, Xmax ) ( Ymin, Ymax ) O f f - l i m i t H e i g h t T r u n c a t i o n H e i g h t Number o f P l a n e s P l a n e d ) : P l a n e ( 2 ) : ( 1.5, 2.25, 0. ) ( 1.25, 1.0, 0.127463 ) 90° ( 1.5, 2.75 ) ( 2.25, 4.429 ) 0 1.6054 ( 1.5, 4.95, 0. ) ( 1.25, 1.75, -0.5335 ) 90° ( 1.5, 2.75 ) ( 4.429, 4.90 ) 0 1 .75 ( a, b, c ) = = ( 1 0 s , 1.4339, -3.0751 ) ( Xm i n, Xmax ) = = ( 0. , 1.5 ) ( Ymin, Ymax ) = = ( 0. , 4.95 ) T r u n c a t i o n H e i g h t = = 1 .75 ( a, b, c ) = = ( 2. 1361, 10 9 , 4.9668 ) ( Xmin, Xmax ) = -- ( 0. , 2.75, ) ( Ymin, Ymax ) = = ( 2.25, 4.95 ) O f f - l i m i t H e i g h t = = 0 T r u n c a t i o n H e i g h t = = 1 .75 131 5. PROGRAM LISTING FOR GEN7 1 5 1 2 • 2 0 1 3 4 0 1 4 5 0 1 5 5 1 I G 5 2 I 7 6 1 I 8 6 2 9 6 5 I to G G 1 1 6 9 1 2 7 0 1 3 7 5 I 1 4 9 6 1 5 1 0 0 1 6 1 0 1 1 7 l O P 1 0 1 0 5 1 9 1 0 6 2 0 1 0 7 2 1 1 0 8 2 2 1 1 1 2 3 1 1 2 2 4 1 1 3 2 5 1 1 4 2 6 1 1 5 2 7 1 1 6 2 8 1 1 7 2 9 1 I H 3 0 1 1 9 3 1 1 2 0 3 2 1 2 1 3 3 1 2 2 3 4 1 2 3 3 5 1 2 4 3 G 1 2 5 3 7 1 2 6 3 8 1 2 7 3 9 1 2 « 4 0 1 2 9 4 1 1 3 0 4 2 1 3 1 4 3 1 3 8 4 4 1 3 9 4 5 1 4 0 4 6 1 4 2 4 7 1 4 5 4.", 1 5 0 4 0 1 5 1 5 0 • 15:? 5 1 1 8 9 5 2 1 9 0 5 3 1 9 1 5 4 2 G O 5 5 2 0 1 5 0 2 0 5 5 7 2 1 0 5 R 2 1 5 5 9 2 2 ( ) 6 0 R F M G E N E R A L P R O G R A M F O R E X E C U T I N G T H E M E T H O D O F H I G H E S T P O I N T O P E N " D A T A " F O R O U T P U T A S F I L E # 1 0 I M C ( 4 4 ) . E ( 4 4 ) , G ( 4 4 ) . P ( 4 4 ) , R ( 4 4 ) , T ( 4 4 ) . V ( 4 4 ) , F ( 2 6 ) R E M R F M I N I T I A L I Z A T I O N R E M " E N T E R F I E L D D I M E N S I O N X A N D Y " ; A 1 , A 2 P R I M r I N P U I P R I N I I N P U T L E T M-INCREMENT D " E N T E R A 3 ] N T ( A 1 / A 3 ) I. E 1 N " I N T ( A 2 / A 3 ) P R I N T 1 . U S I N G • liiiti. ' ; N L E T D 1 - . 0 1 7 4 5 3 2 9 2 5 2 P R I N T " N U M B E R O F E L L I P S O I D S ( max 3 ) " ; I N P U T C I F O O G O T O 1 0 6 G O S U P , 2 0 0 P R I N T " N U M B E R O F E L . P A R A B ( max 3 ) " ; I N P U T F I E F " O G O T O 1 1 2 G O S U B 4 0 0 P R I N T " N U M B E R O F H Y P . P A R A B ( max 3 ) " ; I N P U T G I F C.-O G O T O 1 1 6 G O S U B 6 0 0 P R I N T " N U M B E R O F Q U A D R A T I C C O N E ( max 3 ) " ; I N P U T P I F P = 0 G O r o 1 2 0 G O S U B 8 0 0 P R I N T " N U M B E R O F E L L I P T I C ( C I R C U L A R ) C Y L I N D E R ( max 3 ) I N P U T R I F R - 0 G O T O 1 2 4 G O S U P , l O O O P R I N T " N U M B E R O F P L A N E S " ; I N P U T T I F T = 0 G O T O 1 2 8 G O S U P , 12'. >0 P R I N T " N U M B E R O F T O R U S ( max 3 ) " ; I N P U T V I F V--0 G O TO 1 3 8 G O S U P . 1 4 H O P R I N T " N U M B E R U F P A R A B O L I C E L L I P T I C A L C Y L ( max 3 I"; I N P U T F I F F - 0 G O T O 1 4 5 G O S U B 14 5 0 G O T O 1 5 0 0 R F M S T A R I R E M R F M R F M R r M F O R PI O F D A T A E N T R Y "4JB 2 0 0 I S F O R A N E L L I P S O I D 1 - O TO ! I N T <;•!• - S T P ' ! ; ( I ' A T- " F N I I! R P R I N I I N P U T G< I ' 1 A $ - " E N 1 E R A C - 1 1 ) I X O , Y O . Z O I F O R El . L I P ( " * B ' | , <-" ) 5 + 3 B-,0(1'! A N D C . i + 4 I F O R C ( 1 ' 1 5 E L L I P ( •5 ) ' + P , V 1 32 61 230 PRINT A$; 62 235 INPUT C(I * 15),C(I + 15+1),C(I * 15 + 2 ) 63 240 AT. ~ " ENT E R ROTATIONS 1, 2 AND 3. FOR E L LIP ( " + B$ + " ) ... " 64 245 PRINT At; 65 246 INPUT C( I + 15+10).C( I * 15+1 1) .C ( I * 15+12) 66 300 A $ ™"E N T E R LOWLIMX, UPLIMX FOR ELLIP("+B$ +") ... " 67 305 PRINT Af,; 68 310 INPUT C I I M 5 + 6 ) . C U M 5 + 7) 69 315 A$ - "ENT ER LOWLIMY. UPLIMY FOR ELLIP("+Bt+") ... " 70 316 PRINT A$; 71 320 INPUT C1I+15+8), C U M 5 + 9) 72 322 A $ ="ENT E R OFFLIMIT HEIGHT FOR ELL IP( " +B$ + " ) ... " 73 324 PRINT A$; 74 326 INPUT C( I * 15+13) 75 328 A$ ="ENT ER TRUNCATION HEIGHT FOR E LLIP( "t B$ 4 " ) . " 76 330 PRINT A$; 77 332 INPUT C( I * 15+14) 78 345 NEXT I 79 346 A$="Pnusing .. Type 1 to a l t e r i n p u t , any no. to c o n t i n u e 80 347 PRINT A$ ; 8 1 348 INPUT 0 82 349 IF 0=1 GOTO 200 83 350 RETURN 84 389 REM 85 390 REM SUB 400 IS FOR EL. PARAB 86 391 REM 87 400 FOR 1=0 TO E-1 88 401 PRINT 89 405 B$ = STR$(1 + 1 ) 90 410 A$="ENTER VERTEX (XO.YO.ZO) FOR EL PARAB( "+ BT.+" ) .. " 91 415 PPINT AT.; 92 420 INPUT E(I*15).E(I'15+1),E(I*15+2) 93 455 A$ = " E NT ER A, E AND C. FOR EL PARAB( "+B$+" ) " 94 460 PRINT A$; 95 465 INPUT E( I ' 1 5 + 3 ) , E ( I ' 15 + 4 ).E( I * 15+5) 96 470 A'f. ="ENTER ROT 1, 2 AMD 3. FOR EL PARAB( "+B$+" ) .... " 97 475 PRINT AS; 93 480 INPUT E( I * 15+10),E(I * 15+1 1 ).E( I ' 15* 12) 99 500 A $ = "E NT E R LOWLIMX, UPLIMX FOR EL PARAB< "+B$+" ) .. " 100 505 PRINT A$; 101 510 INPUT E( I* 15 + 6) . E ( I • 15 + 7) 102 520 AJ. = " ENT E R LOWLIMY. UPLIMY FOR EL PARAC( " H3$+" ) 103 525 PRINT At; 104 530 INPUT E(I*7015+8).E(I'15+9) 105 53 1 A $ = " F. N T E R OFFLIMIT HT FOR EL PARAB ("+B$+" ) " 106 532 PRINT A$; 107 533 INPUT E(I•15+13) 108 54 1 AT,= " ENTER TRUNCATION HT FOR EL PARAB(" + B$+") .... " 109 542 PR INI A$; 1 10 543 INPUT El I + 15+14 ) 1 1 1 545 NE X T I 112 54G A$="Paus ing .. Type 1 to a l t e r input, any no. to c o n t i n u e 113 54 7 PR INT A$; 114 548 INPUT O 115 5 19 IF 0^1 GOTO 400 116 550 RE~1 URN 117 589 REM 118 590 REM SUB 600 IS FOR HYP. PARABOLOID 119 59 1 REM 120 600 FOR I=0 TO G-1 133 12 1 601 PRINT 122 605 BT=STRT<1+1) 123 610 AT = " ENT ER CENTER (XO.YO.ZO) FOR HYP PARAB("+B$+") . . " 124 615 PRINT AT; 125 620 INPUT G( I • 15) . G( I + 15+1) .G( I• 15 + 2 ) 126 655 AT="ENTER A. B AND C, FOR HYP PARAB( "+BT +" ) " 127 660 PRINT AT; 128 665 INPUT G(I•15 + 3 ) ,G ( I * 15 + 4 ) ,G ( I•15 + 5) 129 670 AT="ENTER ROT 1. 2 AND 3. FOR HYP PARAB("+B$+") . . . . " 130 675 PRINT AT; 131 680 INPUT G( I>15+10) .G( I•15+11) .G( I * 15+12) 132 700 AT ="ENT ER LOWLIMX AND UPLIMX. FOR HYP PARAB(" + BT+") . . . . 133 705 PRINT AT: 134 7 10 INPUT G( I * 15 + 6 ) ,G ( I•15 + 7) 135 720 AT="ENTER LOWLIMY AND UPLIMY. FOR HYP PARABf "+BT+" ) 136 725 PRINT AT; 137 730 INPUT G1 I * 15+8 ) .G(I * 15+9) 138 732 AT = "ENTER OFFLIM HT FOR HYP PARAB( " + BT +") 139 733 PRINT AT; 140 735 INPUT G ( l + 1 5 + 1 3 ) 141 737 AT="ENTER TRUNCATION HT FOR HYP PARAB("+BT+") 142 739 PRINT A$; 143 740 INPUT G( I*15+14) 144 745 NEXT I 145 746 A T - " P n u s l n g . . Type 1 to a l t e r Input , any no. to c o n t i n u e 146 747 PRINT AT; 147 748 INrUT 0 148 749 IF 0=1 GOTO 600 149 750 RETURN 150 789 REM 151 790 REM SUB 800 IS FOR QUADRATIC CONE 152 79 1 REM 153 800 TOR 1=0 TO P-1 154 801 PRINT 155 805 UT-STRTf1+1) 156 810 AT""ENTER VERTEX ( XO .YO .ZO ) . FOR CONE("+B$+") . . " 157 8 15 PRINT AT; 158 820 INPUT P( I •» 1 5 ) . P( I • 15+ 1 ) , P( I • 15+2 ) 159 825 AT = "ENTF.R A. B AND C. FOR CONE("+BT+") " 160 830 PRINT AT; 161 835 INPUT P( I•15 + 3 ) ,P ( I• 15 + 4 ) . P ( I * 15 + 5) 162 8 10 AT="ENTER ROT 1. 2 AND 3. FOR CONE("+BT+") " 163 842 PRINT AT ; 164 814 INPUT P( I + 15+10).P( I• 15+ 1 1 ) . P(I • 15+12) 165 860 A$-"ENTER TRUNCATION HT FOR CONE( "+BT+" ) " 166 865 PRINT AT: 167 866 INPU1 P( I * 15+ 14 ) 168 870 AT "• " ENTE R OFFLIMIT HT FOR CONE ( " + P.T + " ) " 169 875 PRINT AT: 170 88n INPUT P ( I ' 15+13 ) 171 9O0 AT="ENTER LOWLIMX AND UPLIMX FOR CONE("+BT+") . . " 17? 905 PRINT AT: 173 010 INPUT P( I * 15 + 6 ) .P! I * 15 + 7 ) 174 9T> AT-"ENTER LOWLIMY AND UPLIMY FOR CONE( "*BT +") 175 935 PPIN1 AT; 176 q . I N P U T PI I * 15 + 8 ) ,P( I • 15 + 9) 177 94 5 NF>T T 178 916 AT=" I 'nusing Type 1 to a l t e r i n p u t , any no. to c o n t i n u e 179 9 17 PR INI AT: 180 "'"I INPUT U 1 34 18 1 949 IF 0=1 GOTO 800 18? 950 RETURN 183 989 REM 184 990 REM SUB 1000 IS FOR ELL IPTICAL (CIRCULAR) CYLINDER 185 99 1 REM 186 1000 TOR 1=0 TO R~1 • 187 1001 PRINT 188 1O05 Bt=STRt( 1 + 1 ) 189 1010 AT- "ENTER CENTRE POINT (XO.YO.ZO) FOR CYL("+Bt+"> . " 190 1015 PRINT At.; 191 10?0 INPUT R(I * 15 ) ,R<I * 15+1 ) ,R ( I *15 + 2) 192 1025 AT-"ENTER A. B AND RO, FOR CYL( "+B$+") " 193 1027 PRINT A$; 194 1030 INPUT R( I* 15 + 3) ,R( I* 15+4),R(I * 15 + 5) 195 1035 At ~"ENTER ROT 1, 2 AND 3 FOR CYL ( "+Bt+" ) " 196 1040 PRINT A t ; 197 1045 INPUT R(I * 15+10 ) .R(I * 15+11).R( I * 15+12) 198 1100 At="ENTER LOWLIMX AND UPLIMX FOR CYL("+B$+") . . . . " 199 1105 PRINT AH: 200 1110 INPUT R (1*15+6) .R (1*15+7) 201 1115 At = "ENTER LOWLIMY AND UPLIMY FOR CYL("+B$+") . . . . " 202 1120 PRINT A t ; 203 1125 INPUT R( I • 15 + 8 ) ,R(I * 15+9) 204 1130 At ="ENTER OFFLIM HT FOR CYL( "+B$+") " 205 1135 PRINT A t ; 206 1140 INPUT R( I*15+13). 207 1141 At ="ENTER TRUNCATION HT FOR CYL ( "+Bt+" ) " 208 1142 PRINT A t ; 209 1143 INPUT R( I*15+14) 210 1145 NEXT' I 211 1146 A t = " P a u s i n g Type 1 to a l t e r i n p u t , any no. to c o n t i n u e " 212 1147 PRINT A t : 2 13 1148 INPUT O 214 1149 IF 0=1 GOTO 1000 2 15 1 150 RETURN 216 1189 REM 2 17 1190 REM SUB 1200 IS FOR A PLANE 218 1191 REM 2 19 1200 FOR 1=0 TO T-1 220 1201 PRINT 221 1205 BT = STRt< 1+1 ) 222 1210 At="ENTER INTERCEPTS X.Y AND Z FOR PLANE( " + B t+" ) . " 223 12 15 PRINT A t : 224 1220 INPUT T( I • 8 ) .T( I *8+ 1 ) ,T( I *8 + 2 ) 225 1255 A t " " ENTER LOWLIMX AND UPLIMX FOR PLANE( "+Bt+" ) . . . . " 226 1260 PRINT A t : 227 12S5 INPUT T( I*8 + 3 ) . T ( I * 8 + 4) 228 1285 AI-"ENTER LOWLIMY AND UPLIMY FOR PLANE( "+BT+ " ' . . . " 229 1290 PRINT A t ; 230 1295 INPUT T (1 *8+5 ) .T (1 *8+6 ) 231 1297 At-"ENTER TRUNCATION HT FOR PLANE("+BT+") " 232 1298 PR I NT A t : 233 1299 INPUT T(1*8+7) 234 1350 NEXT I 235 1360 A t - " P a u s i n g . . Type 1 to a l t e r i n p u t , any no. to c o n t i n u e " 2 36 136 1 PRINT A t ; 237 1362 INPUT 0 238 1363 IF 0-1 GOTO 1200 239 1365 RETURN 240 1389 RFM 1 35 24 1 131Q REM SUB 1400 TS FOR A TORUS 242 1391 RFM 243 MOO FOR 1 0 TO V- 1 244 1401 r,i. = snn< 1 + 1) 245 14-12 A'l- = " F NT F R CENTER (XO.YO.ZO) FOR TORUS I " +BT + " ) .." 246 1 103 PR INI PRINT AT: 247 I40'l INPUT V I [ * 9 ) , V ( I ' 9 4 1 ) , V ( I 4 9 + 2 ) 248 1 111 A'l. = "CLRADI UBE <"+13T + ") = " 249 14 12 PPINI AT; 250 14 13 INPUT V(I*9 + 3) 251 1414 A'l.-"SECRADTUBE <"+BT+")= " 252 14 15 PRINT AS; 253 14 1(5 INPUT V( 1*9 + 4 ) 254 14 17 AT-"ENTER LOWLIMX AND UPLIMX FOR TORUS( " + BT +" ) ... " 255 14 18 PRINT AT; 256 14 19 INPUT V ( 1*9 + 5),V( I *9 +6) 257 1423 AT ="ENTER LOWLIMY AND UPLIMY FOR TORUS("+BT + ") ... " 258 1424 PRINT AT: 259 1425 INPUT VI I*9+7 ) ,V( I*9+8) 260 1430 NEXT I 261 1436 AT="Pnus1ng .. Type 1 to a l t e r Input, any no. to c o n t i n u e " 262 1437 PRINT AT: 263 1438 INPUT 0 264 14 39 IF 0=1 GOTO 1400 265 1440 RETURN 266 144 1 RFM 267 1442 REM SUBROUTINE 1450 IS FOR PARABOLIC ELLIPTICAL CYLINDERS 268 1445 REM 269 1450 FOR 1=0 TO F-1 270 145 1 B T = S T R T( I + 1 ) 271 1453 AT ="ENTER (XO.YO.ZO) FOR PARA-EL-CYLf"+BT+") ... " 272 1455 PRINT " PRINT AT: 273 1457 INPUT F(I * 13 ) .F( I * 13+1 ) ,F(I * 1 3+2 ) 274 1459 AT" "ENT I'R a, b. c FOR PARA-EL-CYL ( "+ET + " ) " 275 14GO PRINT AT; 276 1462 INPUT F( I * 1 3 + 3 ) .F( I• 13 + 4 ) . F ( I • 1 3 « 5 ) 277 1465 AT="ENTER ROTATION FOR PARA-EL-CYL( " +BT+ " ) ..... " 278 1466 PRINT AT: 279 1468 INPUT F(1*13*6) ?KO M7U AT="ENTER LOWLIMX AND UPLIMX FDR CYL("+BT+") .. " 281 147 1 PRINT AT: 282 1173 INCUT f (1*13^7),F(I'13+8) 2H3 . 1475 A|,-"FNTFR LOWLIMY AND UPLIMY FOR CYL("+BT+") . . " 284 14 76 PR IN I AT; 285 1177 INPUT F ( I 1 1 3 + 9 ) . F (I*13*10) 286 MHO AT""CNIER OEFLIM HT FOR CYL(" + RT+") " 287 MP. 1 PRINT AT ; 288 1482 INPUT rI 1*13+11) 289 1483 AT""ENIER TRUNCATION HT FOR CYL("+RT*"> .." 290 1184 PRINT AT; 2'.") 1 1 185 INPUT ("(1*13+12) 292 1187 NEXT I 29 3 1 • If'n AT = " r \ i " 3 1 nci Type 1 to a l t e r input, any no. to c o n t i n u e " 294 Mflfl PPINI AT: 295 149Q INP1II 0 296 MM 1 IF O • ( GOTO 1450 297 1 l:i2 Rf.'T URN 2H8 1493 prr.i 299 14'.!4 Pi.M EMU OF INPUT SUBROUTINES .TOO 14 OH REM 1 36 301 149 7 REM 302 1493 REM START LOOKING FOR THE HIGHEST POINTS 303 1 4 99 REM 304 1500 PRINT " PRINT " Program r u n n i n g ... " PRINT 305 1502 FOR K-0 TO M 306 1505 LET X=K*A3 307 1507 PRINT "'Loop ":K 308 15 10 FOR u=0 TO N 309 15 15 LET Y=0'A3 3 10 1520 LET Z = 0 3 1 1 152 2 REM 3 12 1523 REM FIRST CHECK ELLIPSOIDS 3 13 1524 REM 314 1525 IF C=0 GO TO 1600 3 15- 1530 TOR 1=0 TO C-1 316 1535 IF X<C(I * 15+6) GO TO 1585 3 17 1540 IF X~-C< 1*15 + 7) GO TO 1585 3 18 154 5 IF Y<C(I*15+8) GO TO 1585 3 19 1550 IF Y>C(I*15*9) GO TO 1585 320 1555 LET R1=C(I * 15+10)*D1 " LET R2 = C(I * 15+1 1 ) *D 1 LET P3=C(1*15+12) *D 1 32 1 1556 LET A = C(I*15) " LET B = C(l*15+1) LET C2 = C( I * 15 + 2) 322 1557 LET XO = C(I*15+3) LET YO = C(I*15+4) " LET ZO = C( I * 15+5) 323 15G0 GOSUB 3000 324 1562 LET A1=(N1/A)a2 + (N2/B)a2 + (N3/C2)a2 325 1563 LET B1 = (2*N1*(L1*X1+M1*Y1) )/(Aa2) +(2*N2*(L2* X1+M2*Y1))/(Ba2) 326 1564 LET B1=B1 + (2*N3*(L3*X1+M3*Y1))/(C2a2) 327 1565 LET C1=((L1*X1+M1*Y1)/A)a2+((L2*X1+M2*Y1)/B)(i2+((L3'X1+M3*Y1)/C2)a2-1 328 1566 LET D = B1a2 - 4*A1*C1 329 1567 IF 0<0 THEN LET, Z2 = C ( I * 15+13) " GOTO 1580 330 1568 LET Z2 = ((S0R(D)-B1 )/(2*A1 )) + ZO 331 1570 IF Z2>C(I * 15+14) THEN LET Z2 = C(I * 15+14) 332 1580 IF Z2>Z THEN LET Z=Z2 333 1585 NEXT I 334 1589 REM 335 1590 REM CHECK EL. PARAB 336 1591 REM 337 1600 IF E=0 GO TO 1700 338 1605 FOR 1=0 TO E-1 339 16 10 IF X<E(I*15'6) GO TO 1686 340 1G 15 IF X>E(1*15+7) GO TO 1686 34 1 1620 IF Y<E(1*15+8) GO TO 1686 342 1625 IF Y.-EIIM5 + 9) GO TO 1686 343 1630 LET R 1=E( I * 15+10) *D1 " LET R2 = E( I* 15+1 1 ) * D 1 LET R3=E(1*15+12) *D 1 34 4 1632 LET A=E(I'15+3) LET R=E(I*15+4) LET C2=E(I * 15 + 5 1 345 163 1 LET X0=E(I'15) " LET Y0=E(I*15+1) " LET ZO=E(I * 15+2 ) 346 1638 IF (R1+R2 + R3)=0 THEN LET Z3=(((X-XO)/A)a2M (Y-YO)/B )<i2 1/C2 + Z0 GOTO 1675 34 7 1640 GOSUB 3 O O O 348 16 15 LET A1=(N1/A)n2 + (N2/B)a2 34 9 1650 LET E1 = (2 ,N1ML1*X1+Mt*Y1))/(A«2)+(2*N2*(L2'*X1*M2*Y1) )/(8a2 )-C2 •N3 350 1655 LET C1=((L1*X1+M1*Y1)/A)«2+((L2*X 1+M2*Y 1 )/B )<»2 -(L3'X 1 + M3*Y1)*C2 35 1 16C0 LET 0 =B1n2 - 4*A1'C1 352 1665 IF P<0 THEN LET Z3 = F( I* 15+13) GOTO 1685 353 1667 IF Al=0 THEN LET Z3=-C1/B1+Z0 ' GOTO 1675 354 16 70 LET Z3=M SOR(D)-B1)/(2*A1 ) + ZO 355 1675 IF Z3 -E( 1 ' 15+14 ) THEN LET Z3=•E( I * 1 5+ 1 4 ) 356 1 685 IF 73>7. THEN LET Z = Z3 357 1686 NEXT I 1 358 1689 REM 359 1690 REM CHECK HYP. PARAB 360 - 169 1 REM 1 37 36 1 1700 If G-0 0.0 TO 1800 362 i7or> F O R r =o T O G - 1 363 1710 IT X ' G I1*15+6) GO TO 1785 364 17 15 II" X -G i l ' 15+7 ) GO TO 1785 365 1720 IF Y-G(1*15+8) GO TO 1785 366 1725 IF Y G(T*15+9) GO TO 1785 3G7 1730 LIT R1=G< I * 15+10)*D1 LET R2 = G(I•15+1 1 ) *D 1 LET R3 = G ( I *15+ 1 2 ) *D 1 368 1732 LFT A=G(I*15+3) " LET B=G(I*15+4) LET C2 = G ( I ' 1 5 + 5) 369 1735 LET X0=G( I '15 ) " LET YO = G(I•15+1 ) LET ZO = G(I * 15 + 2) 370 1738 IF ( R H R 2 + R3)=0 THEN LET Z4 = ( ( ( X-XO) / A ) a?_ - ( ( Y - YO)/B ) a2 )/C2 + Z0 " GOTO 1775 37 1 1740 GOSUB 3000 372 1742 LET A 1 = ( N 1 / A )tt2 - ( N2/B ) <l2 373 1745 LET B1 = (2*N1 *(L1 *X 1 +M1 *Y 1 ) )/(Aa2 )-(2*N2*(L2*X1+M2*Y1) )/ (Bn2)-C2*N3 374 1747 LET C 1 = ( (L I 'X1+ M1 * Y1 ) /A )a2-((L2*X1+M2*Y1)/B)fJ2-C2*(L3*X1+M3*Y1) 375 1750 LET D = B1a2 - 4+A1+C1 376 1752 I F D<0 THEN LET Z4 =G( 1*15+13) " GOTO 1780 377 1753 IF A 1 = 0 THEN LET Z4 = -C1/B1 + Z0 " GOTO 1775 378 1755 LET Z4 = ( SQR ( D )-B 1 )/ ( 2 • A 1 ) + ZO 379 1775 IF Z4>G( I * 15+ 14) THEN LET Z4 =G(I * 15+14) 380 1780 IF Z4.-Z THEN LET Z = Z4 38 1 1785 NEXT I 382 1789 REM 383 1790 REM CHECK QUADRATIC CONE 384 1791 REM 385 1800 IF P = 0 GO TO 1900 386 1805 FOR 1=0 TO P-1 387 1810 I F X<P(1*15+6) GO TO 1885 388 1815 IF X>P<1*15+7) GO TO 1885 389 1820 IF Y ' . P ( I '15 + 8) GO TO 1885 390 1825 IF Y^P(1*15+9) GO TO 1885 391 1830 LET R1=P(1*15+101*01 LET R2 = P(I * 15+ 1 1 ) *D1 LET R3-P( I * 15+ 12 ) *D1 392 1832 LET A = P < I * 1 5 + 3.) LET B=P(I*15+4) LET C2 = P ( I ' 1 5 + 5) 393 1835 LET XO=P(I*15) " LET YO = P(1*15+1) LET ZO=P(1*15+2) 394 18 10 GOSUB 3000 395 1845 LET A1=(N1/A)a2 + (N2/B)<l2 - (N3/C2)n2 396 1847 LET E1 = <2*N1 *(L1 * X1+M1•Y1 ) )/(Art2 ) + 12 *N2*(L2* X1+M2*Y1 ) )/(Ba2) 397 1848 LET R 1 --B 1 - ( 2 *N3 * ( L3 *X 1 +M3* Y 1 ) )/ ( C2a2 ) 398 1850 LET C1 = ( (L 1 * X1+M 1 *Y1 )/A ) a2 + ( ( L 2 ' X 1+M2*Y1 )/B ) a2 - ( (L 3 * X 1+M3 *Y1)/C2)a2 399 1852 LEI D - B1a2 - 4*A1*C1 400 1855 IF 0 0 THEN LET Z5 = P( 1*15+13) GOTO 18RO 401 1860 LET Z5=(S0R(D)-B1 )/(2*A1 ) + ZO 102 1875 IT Z5 P ( I *15+14) THEN LET Z5 = P ( I * 1 5 H 4 ) 403 1880 IF Z5>Z THEN LET Z=Z5 404 1885 NEXT I 405 1889 REM 40G 18')0 REM CHECK ELL IPT ICAL (CIRCULAR) CYLINDERS 407 189 1 REM 4()8 19'JO IF R-O GO TO 2000 109 1905 FUR 1-0 TO R-1 110 19 10 IF X-.R1 I * 15 + G) GO 10 1985 411 19 15 IT X -PI I '15 + 7) GO TO 1985 4 12 1 9 " 0 IF Y<RII*15+8) GO TO 1985 413 1925 IF Y-R(I*15+9) GO TO 1985 414 1930 Ll ' l R 1 =R( I ' 15 * 10 ) *D 1 ' LET R2 = R( I * 1 5+1 1 ) * D 1 LEI R3 = R ( I ' 1 5+1 2 ) *D 1 415 1932 LFT A-R( I '15+3) LET B-R(I*15+4) LET R0'R( I*15+5> 416 1935 LET X 0 R ( I * 1 5 ) ' LET YO=R(I*15+1) LET 70=R(I*15*2) 417 1940 GOSUB 3O00 418 1945 LEI A1 = (Nf/A)rr2 + (N2/B)a2 419 1950 LFT Rl - ( 2 * N 1 ' ( I. 1 ' X 1 +M 1 * Y 1 ) ) / ( A i2 ) * ( 2 ' N2 * ( L 2 * X 1 •M2'Y 1 ) ) / ( B(l2 ) 420 1957 LET CI = I ( L 1 * X 1 +M 1 * Y 1 ) / A ) a2 + ( (L2*X1'M2*Y1 ) / R ) ir 2 - 1 1 38 421 1953 LET D - B1rr2 - 4*A1*C1 422 1960 I F D<0 THEN LET Z6 = R( I'15+13 ) " GOTO 1980 423 1962 IE A 1 '0 THEN LET ZG-RO+ZO GOTO 1972 424 1965 LET Z6 = (50R(0)-B 1)/(A1* 2) + Z0 4 25 1966 IF ACSIN3X1E-05 GOTO 1970 426 1968 IT ( 1.3'X1+M3 * Y1+N3 * Z6)>R0 THEN LET Z6 = ((RO-L3'X1-M3*Y1)/N3) GOTO 1980 427 1970 IF ABS( L3*X 1+M3*Y 1 )>R0 THEN LET Z6 = R< I ' 15+1 3 ) GOTO 1980 428 197 1 GOTO 1975 429 1972 IF ((X1/A)«2+(Y1/B)al)>1 THEN LET Z6 = R( I * 15+13) GOTO 1980 430 1975 IF Z6>R(IM5+14) THEN LET ZG = R(I * 15+14 ) 431 1980 IF Z6>Z THEN LET Z = 76 432 1985 NEXT I 433 1989 REM 434 1990 REM CHECK PLANES 435 1991 REM 4 36 2O00 IF T=0 GO 10 2 100 437 2005 FOR 1=0 TO T-1 438 2010 IF X<T(1*8+3) GO TO 2045 439 2015 IF X>T(1*8+4) GO TO 2045 440 2020 IF Y<T( 1*8 + 5) GO TO 2045 441 2025 IF Y>T(I*8+6) GO TO 2045 442 2030 LET Z7 = ( 1-X/T(I*8)-Y/T(1*8+1 ) )*T(1*8 + 2) 443 2035 IF Z7<0 THEN LET Z7=0 444 2038 IF Z7>T(I*8+7) THEN LET Z7=T(I*8+7) 445 2040 IF Z7>Z THEN LET Z = Z7 446 2045 NEXT I 447 2089 REM 448 2090 REM CHECK TORUS 449 209 1 REM 450 2100 IF V=0 GO TO 2200 45 1 2 105 FOR 1=0 TO V-1 452 2110 IF X<V(I'9+5) GO TO 2180 453 2 115 IF X>V(1*9+6) GO TO 2 180 454 2120 IT Y<V(I*9+7) GO TO 2180 455 2125 IF Y>V( I*9 + 8 ) GO TO 2180 456 2130 IF y,<-V( I *9)-V( I *9+3)-V( 1*9 + 4) GO TO 2 180 457 ' 2135 IF X = V ( I ' 9 ) + V ( I ' 9 + 3 ) + V ( I * 9 + 4 ) GO TO 2180 458 2140 IT Y > = V (I'9+1)+V(I*9+3)+V(1+9+4) GO TO 2 180 459 2145 IF Y'. = V ( I * 9+1 ) - V ( I • 9 +3 ) - V ( I * 9 + 4 ) GO TO 2180 4GO 2150 LET R1=5QR( (X-V( 1*9) )n2 + (Y-V( 1*9+1 ) )a2) 461 • 2155 IF R K = V( I '9 + 3)-V( 1*9 + 4) GO TO 2180 462. 2160 IF R 1 •» = V ( I*9 + 3) + V( 1*9 + 4) GO TO 2180 463 2165 LET R2=R1 -V(I'9 +3 ) 464 2166 LET Z8 = 5QR(V(I •9+4 )a2-R2a2)+V( I•9 + 2 I 465 2170 IF ZR-'O THEN LET Z8~0 466 2175 IF Z8-Z THEN LET Z=Z8 467 . 21RO NEXT I 468 2190 REM 469 2 19 1 REM CHECK PARABOLIC ELLIPTICAL CYLINDERS 470 2 192 REM 47 1 • 22GO IF F=0 GOTO 2600 472 2205 FOR 1=0 TO F-1 473 2210 IF X'F(I•13<7) GOTO 2500 474 2212 IF X>F(1*13+8) GOTO 2500 475 2214 IF Y<F(I*13+9) GOTO 2500 476 2216 IF Y>F(I*13+10) GOTO 2500 477 2220 LET R 1 =F( I * 13 + 6)*D1 478 2222 LET X0"=F(I'13) " LET Y0 = F(I*13+1) LET Z0 = F(I*13 + 2) 479 2224 LET A 1 =F( I * 13 + 3 ) B1 = F(I*13 + 4) C 1 -F( I ' 13+5 I 480 2226 LET XI = (X-XO)* COS(R1 ) + (Y-YO)'SIN(R1 ) 1 39 40 1 2 2 28 L [-" I Y 1 = -(X-XO)'SINIR1) + 1Y-YO)'COS(R1) 482 2230 IF A B S ( Y 1 )>A1 THEN LET Z9 = F( I• 1 3+1 1 ) GOTO 2250 483 2232 LFT 7 I - ( X 1<i2 *C 1+B 1 ) * SORI 1 - ( ( Y 1 )/A 1 )c2 I 484 22.35 LF.T 79 = Z1 + 70 485 2240 IF Z9> F ( I •13+12) THEN LET Z9 = F(I * 1 3+ 1 2 ) 486 2250 IF 79 > 7 THEN LET Z = Z9 487 2500 NFXT I 488 2502 RIM 489 2503 RFM HIGHEST POINT FOUND 490 2504 REM 491 2600 PRINT USING ' Hill . Hilll . HHH . HHH , HHH . HUH . ' ; X ; Y ; 7 492 2630 NEXT J 493 264 0 NEXT K 494 2650 STOP 495 2997 REM 496 2998 REM SUBROUTINE 3000 FINDS DIRECTION COSINES OT ROTATED 497 2999 REM 490 3000 LET L 1 = C0S(R1)*C0S(R2) 499 3040 LET L2 = C0SIR1 ) tSIN(R2)*SIN(R3) - SIN ( R1 )+COS(R3) 500 3050 LET L3 = C0SIR1)*SIN(R2) +C0S(R3) - COS(R1)•SIN(R3) 501 3060 REM 502. 3070 LL I M1 = SINIR1) • C0S(R2) 503 3080 LET M2 = S1NIR1)*SIN(R2)'SIN(R3) + COS(R1)'COS(R3) 504 3090 LET M3 = SIN(R1 )•SIMR2 ) *SIN(R3) - COS(R1 ) *SIN(R3) 505 3 100 REM 506 3 1 10 LET N 1 = -SINIR2) .507 3 1 20 LE T N2 = COS!R2 ) *SIN<R3 ) 508 3 130 LEI N3 = COS!R2 ) <COS(R3 ) 509 3140 REM 510 3150 LET X 1 = X - X O 51 1 3 160 LET Y 1 Y '- YO 5 12 3 170 RETURN 513 5000 END End of f i l e 5. PROGRAM LISTING FOR CAVITY6 1 42 1 10 REM 2 20 REM Program to c a l c u l a t e the CLD pa th of a s p h e r i c a l end m i l l 3 30 REM to machine a d i e c a v i t y 1n the shape of an e l l i p t i c a l 4 40 REM p a r a b o l o i d 5 50 REM 6 GO OPEN "CAV ITY1 .DAT" FOR OUTPUT AS FILE01 7 65 . OPEN "CAVITY2.DAT" FOR OUTPUT AS FILE/C2 8 70 REM 9 80 PRINT " En t e r a , b and c f o r p a r a b o l o i d . . " ; 10 90 INPUT A, B, C 11 _ 100 PRINT " En t e r t i l t i n g a n g l e o f c a v i t y . . . . 12 ' 110 INPUT F1 13 120 PRINT " E n t e r t h i c k n e s s o f d i e c a v i t y . . . . " : 14 130 INPUT T 15 140 PRINT " E n t e r t o o l r a d i u s " ; 16 .150 INPUT RO 17 160 PRINT " En t e r inc rement f o r X-scan " ; 18 165 INPUT D1 19 170 PRINT " En t e r inc rement f o r Y-scan " ; 20 175 INPUT 02 2 1 180 REM 22 190 REM Set t o o l c e n t r e to be (RO+T) away from s u r f a c e of 23 200 REM p a r a b o l o i d 24 210 REM 25 220 LET R = RO + T 26 230 REM 27 240 REM I n i t i a l i z e pa rame te r s f o r s can 28 250 REM 29 260 LET G = ABS(INT(B/D2 ) ) 30 270 LET F1 = F1 * 0 .01745329252 31 280 REM 32 290 REM S t a r t Y-scan 33 300 REM 34 305 PRINT " PRINT " Program r u n n i n g . . . . " " PRINT 35 310 FOR N = 0 TO G 36 LM2 LET Y = N * ABS(D2) 37 315 PRINT" Loop - " ; N 38 320 REM 39 330 REM F i n d b o u n d a r i e s f o r X-scan 40 340 REM 41 350 IF D K O THEN LET B 1 = - A * SOR ( 1 - ( Y / B ) « 2 ) 42 355 IF D1>0 THEN LET B1= A*SOR( 1 -(Y/B)a2) 43 360 LET B2 = Y 44 370 LET B3 = C 45 380 LET 0 = ABS( INT(B1/D1) ) 46 390 PRINT #1, USING "HMD";0+3 47 392 PRINT #2, USING "HHH";Q+3 48 400 REM 49 410 REM For each Y, s can a l o n g X 50 420 REM 51 430 FOR M = O TO 0 52 440 REM 53 450 REM U s i n g the e q u a t i o n of p a r a b o l o i d , c a l c u l a t e Z f o r 54 460 REM each X and Y 55 470 REM 56 480 LET X =• M * D1 57 490 LET Z = C * ( (X/A)a2 + (Y/B)a2 ) 58 500 REM 59 510 REM C a l c u l a t e d i r e c t i o n c o s i n e s and t oo l c e n t r e p o s i t i o n s 60 515 REM If t oo l on bounda ry , check f o r i n t e r f e r e n c e 143 e i 520 REM 62 530 GOSUB 1000 63 540 IF M=0 THEN GOSUB 1200 64 550 REM 65 560 REM C a l c u l a t e t o o l c e n t r e p o s i t i o n s w . r . t . t i l t e d base p l a n e 66 565 REM and w r i t e r e s u l t s on to d a t a f i l e 67 570 REM 68 580 LET U = Z1*S IN(F1) + X1*C0S(F1) 69 590 LET V = Y1 70 600 LET W = Z1*C0S (F1 ) - X1*SIN(F1) 7 1 605 IF M=Q THEN GOSUB 2000 72 610 PRINT/SM, USING " tttt. f/tttttttt, tttt. ttttffft/t ,ttft .HHHttH"; U, V, W 73 61 1 PRINT/V2. USING "tttt . tt It tt tt tt . tt tt . tt tt tt tt tt , tt It . It tt tt It It" ; U ,-V , W 74 612 REM 75 613 REM Move t o o l a l o n g i n c l i n e d p a r t i n g p l a n e i f t oo l i s beyond 76 614 REM boundary 77 615 REM 78 616 IF M=0 THEN GOSUB 2500 79 620 NEXT M 80 630 NEXT N 81 640 STOP 82 650 END 83 1000 REM 84 1010 REM S u b r o u t i n e 1000 f i n d s the d i r e c t i o n c o s i n e s of the t oo l o f f s e t 85 1020 REM p a t h at any p o i n t on the p a r a b o l o i d and computes the t o o l 86 1030 REM c e n t r e p o s i t i o n s 87 1040 REM 88 1050 LET S = SQR((2*X/Aa2)a2 + (2*Y/Ba2)a2 + 1/Ca2) 89 1060 LET L1 = -2 * X/ (A*A*S) 90 1070 LET M1 = -2 * Y/(B*B*S) 91 :080 LET N1 = 1/(C*S) 92 1090 REM 93 1 100 LET X1 = X + R*L1 94 1 1 10 LET Y1 = Y + R*M1 95 1 120 LET Z1 = Z + R*N1 96 1 130 RETURN 97 1200 REM 98 1210 REM S u b r o u t i n e 1200 moves t o o l a l o n g edge of boundary 99 1230 REM 100 1240 REM F i r s t f i n d d i r e c t i o n c o s i n e s of t o o l o f f s e t pa th 101 1250 REM 102 1260 LET X=B1 " LET Y=B2 " LET Z=B3 103 1270 GOSUB 1000 104 1280 REM 105 1290 REM Then f i n d d i r e c t i o n c o s i n e s of tangent 106 1300 REM 107 1305 IF Y=0 THEN LET L2=0 " GOTO 1345 108 1310 LET H1 = ( (B/A)a2 * (X/Y) ) 109 1320 LET H2 = 1 + (H1)a2 1 10 1330 LET L2 = 1 / S0R(H2) 1 1 1 1340 LET M2 = -H1/SQR(H2) " GOTO 1350 1 12 1345 IF X<0 THEN LET M2 = 1 1 13 1347 IF X>=0 THEN LET M2 = - 1 1 14 1350 LET N2 = 0 1 15 1360 REM 1 16 1370 REM The outward normal i s o b t a i n e d from the v e c t o r p r o d u c t 1 17 1380 REM of the t oo l o f f s e t pa th w i t h the tangent 1 18 1390 REM 1 19 1400 LET H5 = (N1*M2ta2 + (N1*L2)a2 + (L 1 *M2-M 1 • L 2 ) a l 120 14 10 LET H6 = 1/S0R(H5) 144 121 1420 LET L3 =-N1 * M2/H6 122 1422 LET M3 = N1 * L2/H6 123 1424 LET N3 = ( L1*M2 - M1*L2 ) / H6 124 1430 REM 125 1432 REM Move t o o l a l o n g outward normal to a v o i d the boundary 126 1436 REM 127 1438 LET X1 = X1 + R0*L3 128 1440 LET Y1 = Y1 + R0*M3 129 1442 LET Z1 = Z1 + R0*N3 130 1444 LET U1 = Z1*SIN(F1 ) + X1*C0S(F1 ) 131 1446 LET V1 = Y1 132 1448 LET W1 = Z1*C0S(F 1 ) - X1*SIN(F1) 133 1450 PRINT01, USING "HH.HHHHH,HH . HHHHH , HH .HHHHH" ; U 1 , V1 ,W1 134 1460 PR I HT Ml, USING "HH . HHH'HH , *'H . H M H H H , *M . H HHHH" ;U1 ,-V 1 ,W1 135 1480 REM 136 1486 REM Then I nove t o o l downward to c r e a t e the boundary 137 1487 REM 138 1488 REM F i r s t f i n d CLD f o r lower e l l i p s e 139 1489 REM 140 1500 LET X5 = B1 + R0*L3 14 1 1510 LET Y5 = B2 + R0*M3 142 1520 LET Z5 = B3 + R0*N3 143 1525 REM 144 1530 REM A v o i d I n t e r f e r e n c e w i t h i n c l i n e d p a r t i n g p l a n e 145 '535 REM 146 1540 LET Z7 = C + RO 147 1545 LET S7 = (Z7-Z5)/ (Z1-Z5) 148 1550 LET X7 = S7*(X1-X5) + X5 149 1555 LET Y7 = S7*(Y1-Y5) + Y5 150 1590 REM 151 1592 REM R e t u r n t o o l p o s i t i o n d a t a to main program 152 1595 REM 153 1600 LET X1 =X7 " LET Y1=Y7 " LET Z1=Z7 154 1620 RETURN 155 1630 END 156 2000 REM 157 2010 REM S u b r o u t i n e 2000 checks f o r I n t e r f e r e n c e w i t h h o r i z o n t a l 158 2020 REM p i ane 159. 2030 REM 160 2032 REM F 1 rs t save c o - o r d i n a t e s of t oo l 16 1 2033 REM 162 2034 LET UO =U " LET V0=V " LET W0=W 163 2035 164 2038 REM 165 2040 REM C a l c u l a t e Z f o r base p l a n e 166 2050 REM 167 2060 LET Z9 = C+C0S(F1) - A*S IN(F1) + RO 168 2070 REM 169 2080 REM Check f o r p o s s i b l e u n d e r c u t , i f n o t , r e t u r n 170 2090 REM 17 1 2100 IF WO => Z9 THEN GOTO 2200 172 2 1 10 REM 173 2 120 REM If t h e r e 1s u n d e r c u t , r a i s e too l to base p l a n e l e v e l 174 2 130 REM 175 2 140 LET S9 = (Z9-W0)/(W1-W0) 176 2 150 LET U = S9*(U1-U0) + UO 177 2 160 LET V = S9*(V1-V0) + VO 178 2 170 LET W = Z9 179 2200 RETURN 180 2500 REM 145 18 1 2510 REM 182 2520 REM S u b r o u t i n e 2500 moves t o o l a l o n g the I n c l i n e d pa r t 1ng 183 2530 REM 1f t o o l 1s beyond boundary of the p a r a b o l o i d 184 2550 REM 185 2570 LET V = B*1.25 186 2590 PRINT/C1, USING "HH.HHHHH,HH.HHHHH,HH.HHHHH" ;U , ,V,W 187 2592 PRINT#2, USING . "HH.HHHHH,HH.HHHHH , HH . HHHHH " ;U, -V,W 188 2600 RETURN End of f i l e 146 BIBLIOGRAPHY 1. B e z i e r , P., N u m e r i c a l C o n t r o l - M a t h e m a t i c s and  A p p l i c a t i o n s , John W i l e y and Sons, London, 1972. 2. B e e c r a f t , G., C a s t i n g T e c h n i q u e s f o r S c u l p t u r e , C h a r l e s S c r i b n e r ' s and Sons, New Y o r k , 1979. 3. Coons, S.A., " S u r f a c e s f o r Computer A i d e d D e s i g n o f Space Forms", P r o j e c t MAC, MAC-TR-41, M a s s a c h u s e t t s I n s i t u t e of T e c h n o l o g y , Cambridge, M a s s a c h u s e t t s , 1967. 4. D o y l e L . E . , K e y s e r , C.A., L e a c h J . L . , S c h r a d e r G.F., S i n g e r , M.B., M a n u f a c t u r i n g P r o c e s s e s and M a t e r i a l s f o r  E n g i n e e r s , P r e n t i c e H a l l I n c . , Englewood C l i f f s , New J e r s e y , 1961. 5. Duncan, J . P . and c o l l e a g u e s , "POLYHEDRAL NC : A Computer-A i d e d System f o r D i e D e s i g n and M a c h i n i n g " , P r o c e e d i n g s , 3 r d A n n u a l M e e t i n g of C o m p u t e r - A i d e d M a n u f a c t u r i n g I n t . I n c . , page 184-195, A r l i n g t o n , T e x a s , 1974. 6. Duncan, J . P . , F o r s y t h D.G., " M a c h i n i n g Shoe-Moulds by N u m e r i c a l C o n t r o l " , P r o c e e d i n g s , 14th A n n u a l M e e t i n g and T e c h n i c a l C o n f e r e n c e , N u m e r i c a l C o n t r o l S o c i e t y , P i t t s b u r g , 1977. 7. Duncan, J . P . , F o r s y t h D.G., "From A r t i s t ' s S k e t c h t o C a v i t y M o u l d " , P r o c e e d i n g s , CAM78, Glasgow, 1978. 8. Duncan, J . P . , Hanson, J . , B r i t i s h N u m e r i c a l C o n t r o l  S o c i e t y News , V o l . 3 , No.4, page 12-18, A u g u s t 1972. 9. Duncan, J . P . and Lau C.Y.K. " D e f i n i n g and M a c h i n i n g P i e c e w i s e D i e S u r f a c e s " , P r o c e e d i n g s , 2 1 s t A n n u a l M e e t i n g and T e c h n i c a l C o n f e r e n c e , N u m e r i c a l C o n t r o l S o c i e t y , Long Beach, C a l i f o r n i a , 1984. 10. Duncan, J . P . and Lau C.Y.K. " S c u l p t u r e d D i e C a v i t i e s " , P r o c e e d i n g s , 12th A n n u a l M e e t i n g and C o n f e r e n c e , N o r t h A m e r i c a n M a n u f a c t u r i n g R e s e a r c h I n s t i t u t i o n , M i c h i g a n T e c h n o l o g i c a l U n i v e r s i t y , Houghton, M i c h i g a n , 1984. 11. Duncan, J . P . , M a i r , S.G., "The A n t i - I n t e r f e r e n c e F e a t u r e s of P o l y h e d r a l M a c h i n i n g " , P r o c e e d i n g s , P r o l a m a t 1976, T h i r d I F I P / I F A C I n t e r n a t i o n a l C o n f e r e n c e on Programming L a n g u a g e s f o r N/C M a c h i n e T o o l s , S e s s . I I , 1-15, S t i r l i n g , S c o t l a n d . 12. Duncan, J . P . , M a i r , S.G., S c u l p t u r e d S u r f a c e s i n  E n g i n e e r i n g and M e d i c i n e , Cambridge U n i v e r s i t y P r e s s , 1 983. 147 13. Duncan, J . P . , V i c k e r s , G.W., "A S i m p l i f i e d Method f o r t h e A d j u s t m e n t of S u r f a c e s " , Computer A i d e d D e s i g n , V o l . 1 2 . , #6, 1980. 14. Duncan J . P . , Whybrew, K., S t e e v e s A.O., "The M o t i o n s of T o o l s i n M a c h i n i n g P i e c e w i s e A n a l y t i c a l S u r f a c e s " , P r o c e e d i n g s , 9 t h C a n a d i a n C o n g r e s s of A p p l i e d M e c h a n i c s , S a s k a t o o n , 1983. 15. F e r g u s o n J . C . , " M u l t i v a r i a b l e C u r v e I n t e r p o l a t i o n " , J o u r n a l o f ACM , A p r i l 1964. 16. Law, K.K.N., " A n t i - I n t e r f e r e n c e i n T e r r a c e M a c h i n i n g " , T y p e s c r i p t R e p o r t , Department of M e c h a n i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , 1984. 17. M a i r , S.G., Duncan, J.P., "POLYHEDRAL NC Program D o c u m e n t a t i o n " , T y p e s c r i p t R e p o r t , Department of M e c h a n i c a l E n g i n e e r i n g , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , 1978. 18. O a k l e y , C O . , A n a l y t i c a l Geometry , B a r n e s and N o b l e , New Yo r k , 1949. 19. P a r v i t i , D., Wood, S., Young, S.W., Duncan, J.P., "An I n t e r a c t i v e G r a p h i c s System f o r P l a n n i n g R e c o n s t r u c t i v e S u r g e r y " , P r o c e e d i n g s , N a t i o n a l Computer G r a p h i c s A s s o c i a t i o n , C h i c a g o , 1983. 20. P o r t u g a l , F.H., " T e c h n i c a l Imaging : The T e c h n o l o g y of Body A r t " , H i g h T e c h n o l o g y , V o l . 2 , #6, T e c h n o l o g y P u b l i s h i n g Co., B o s t o n , 1982. 21. Pressman, R.S., W i l l i a m s , J . E . , N u m e r i c a l C o n t r o l and  C o m p u t e r - A i d e d E n g i n e e r i n g , John W i l e y and Sons, New Yo r k , 1977. 22. T a y l o r , J . , R i c h a r d s , P., H a l s t e a d , R., "Computer R o u t i n e s f o r S u r f a c e G e n e r a t i o n and D i s p l a y " , T y p e s c r i p t R e p o r t , S e r i e s No. 16, M a r i n e S c i e n c e s B r a n c h , M i n i s t r y of E n e r g y , M i n e s and R e s o u r c e s , Ottawa, Canada, 1971. 23. T e r a d a , H., T o s h i k o , I . , A t l a s of Bones o f Human Body , Nanzando P u b l i s h i n g Company, Tokyo, 1980. ( T e x t i n ( J a p a n e s e ) 

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