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Geodesic focussing in parallel-plate systems Mosevich, Jack Walter 1972

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GEODESIC FOCUSSING IN PARALLEL-PLATE SYSTEMS by  JACK WALTER MOSEVICH B.S., University of I l l i n o i s , 1965 M.S., Northern I l l i n o i s University, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of Mathematics  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1972  In presenting 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 the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that copying or p u b l i c a t i o n  of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  SUPERVISOR: PROFESSOR Z. A. MELZAK ABSTRACT  i i .  This thesis i s concerned with the mathematics of the design of parallel-plate equivalents of optical systems, i n particular vith the parallel-plate equivalent of the parabolic mirror. A parallel-plate microwave system consists of a pair of metal plates, not necessarily plane, which are parallel in the sense that they share common normals at every point, and the normal separation i s constant throughout.  Consider the mean surface  M , which i s the locus of midpoints  of the double normals, and suppose that microwave radiation i s fed into the region between the plates.  If  M  i s not excessively curved the rays  travel along i t s geodesies and a natural problem arises of how to shape M  so that a l l rays from a point source between the plates emerge in a  parallel beam  Cthus  i s assumed that where  B  M  the  system is equivalent to a parabolic mirror).  consists of two flats connected by a focusing bend  i s part of a tubular surface with directrix  is to determine the curve by  It  A  A .  B ,  The problem  so that the geodesies on the tube generated  A. satisfy the focusing condition. The exact mathematical formulation of this problem yields  an  extremely involved functional differential equation in terms of the polar equation of  A, r = rC6) , which proves unsuitable for solving for  r(6) .  Methods are developed by which an approximate solution i s given in terms of an implicit non-linear integro-differential equation i n r(6) .  This  equation also proves too involved to solve exactly, but numerical approximations are calculated by two different schemes. One scheme i s an analog of Euler's method, and the other i s based on Galerkin's method of undetermined coefficients. The problem i s so involved that analytical error analysis appears too d i f f i c u l t to handle.  The best that can be achieved at this time i s to  iii . c a l c u l a t e numerically the deviation of a beam from true p a r a l l e l i s m .  The  r e s u l t s prove to be encouraging, the maximum deviation from true p a r a l l e l ! of a geodesic was 6/10 of one degree, at the periphery of the system. The necessary modifications of these methods f o r solving other o p t i c a l problems are also taken up.  iv. TABLE OF CONTENTS PAGE CHAPTER I  PARALLEL-PLATE OPTICS  1  CHAPTER I I  MATHEMATICAL  8  1. 2. 3. 4.  FORMULATION  The Mean S u r f a c e M G e o m e t r i c a l Framework Geodesies on B Exact Formulation  CHAPTER I I I  ANALYTICAL APPROXIMATIVE METHODS  1. I n t r o d u c t i o n : O s c u l a t i n g Tubes 2. Z e r o t h Order A p p r o x i m a t i o n : N e g l e c t i n g the Bend 3. F i r s t Order A p p r o x i m a t i o n : t h e O s c u l a t i n g Cylinder 4. Second Order A p p r o x i m a t i o n : the Osculating Torus 5. Higher Order A p p r o x i m a t i o n CHAPTER IV  NUMERICAL APPROXIMATIVE METHODS  1. I n t r o d u c t i o n 2. P i e c e w i s e O s c u l a t i n g P o l y n o m i a l s 3. G a l e r k i n ' s Method  8 10 14 18 20 20 22 23 27 34 36 .36 37 42  CHAPTER V  ERROR ANALYSIS  48  CHAPTER V I  GENERAL SYNTHESIS OF PARALLEL-PLATE SYSTEMS  51  1. Simple Systems 2. Complex Systems  51 54  3. T h r e e - D i m e n s i o n a l Systems  56  CHAPTER V I I  IMPLICIT FUNCTIONAL DIFFERENTIAL EQUATIONS  61  CONCLUSION  65  BIBLIOGRAPHY  66  APPENDIX  67  ILLUSTRATIONS  PAGE Figure 1  2  Figure 2  5  Figure 3  5  Figure 4  5  Figure 5  11  Figure 6  13  Figure 7  13  Figure 8  24  Figure 9  24  Figure 10  26  Figure 11  29  Figure 12  32  Figure 13  33  Figure 14  52  Figure 15  55  Figure 16  55  ACKNOWLEDGMENTS  The a u t h o r wishes to thank Dr. Z.A. Melzak f o r s u g g e s t i n g t h e topic of this  t h e s i s and f o r h i s e x c e l l e n t guidance d u r i n g  i t s preparation.  Dr. F.G.R. Warren o f t h e RCA r e s e a r c h group i s a l s o thanked f o r h i s thorough r e a d i n g and c o n s t r u c t i v e c r i t i c i s m o f t h e m a n u s c r i p t . Finally, is  appreciated.  the f i n a n c i a l  support  o f t h e Department  o f Mathematics  CHAPTER I  PARALLEL - PLATE OPTICS  This thesis i s concerned with the mathematics of the design of p a r a l l e l - p l a t e equivalents of o p t i c a l systems, i n p a r t i c u l a r with the p a r a l l e l - p l a t e equivalent of the parabolic mirror. A p a r a l l e l - p l a t e microwave system consists of a p a i r of metal plates  S^  and  S  2  , not n e c e s s a r i l y plane, which are p a r a l l e l i n the  sense that they share common normals at every point, and the normal separation  d  i s constant throughout (see F i g . l a ) .  To ensure a lower  bound on the p r i n c i p a l curvatures i t must be possible to move a sphere of radius  d/2  f r e e l y between the plates so that i t always touches both  p l a t e s , and every point of each plate can be touched. Consider the mean surface  M , which i s the locus of midpoints  of the double normals ( F i g . l b ) , and suppose that microwave r a d i a t i o n of wavelength that  X  d < A/2  i s fed into the region between the p l a t e s . so that only the TEM  mode can propagate, with the same  v e l o c i t y as i n free space and unit-index of r e f r a c t i o n . with  I t i s assumed  Note that systems  d > X/2 , where other modes may propagate with non-unit i n d i c e s ,  can a l s o be designed, 1.5 t o 2.0 times  I f the p r i n c i p a l r a d i i of curvature of  M  exceed  X , the r a d i a t i o n propagates along the geodesies of  M and we are i n the domain of geometrical o p t i c s , f o r the shape of M determines the focussing properties of the system. are a l s o known as geodesic lenses.  P a r a l l e l - p l a t e systems  FIGURE  1  3. A parallel-plate  system i s s a i d  t o be e q u i v a l e n t  to a given  o p t i c a l system i f i t has the same o p t i c a l c h a r a c t e r i z a t i o n as t h e g i v e n system.  F o r example, t h e p a r a l l e l - p l a t e  equivalent  of t h e parabolic  m i r r o r must s a t i s f y t h e c o n d i t i o n t h a t a l l r a y s emanating from a f i x e d point An  C t h e f o c u s ) w h i c h e n t e r t h e system, emerge i n a p a r a l l e l beam.  equivalent  focus  c o n d i t i o n i s t h a t t h e t o t a l o p t i c a l p a t h l e n g t h from t h e  to a fixed  line  Cor p l a n e ) be  A n a t u r a l problem a r i s e s system, i n p a r t i c u l a r system.  o f shaping  constant.  now o f d e s i g n i n g  the p a r a l l e l - p l a t e  M , so as t o a c h i e v e  a perfect  Such a system s a t i s f i e s two c o n d i t i o n s :  1) There i s a one - t o - one correspondence between o b j e c t and  points  image p o i n t s . 2) C o l l i n e a r  and  optical  object p o i n t s correspond to c o l l i n e a r  image p o i n t s ,  conversely. In p r a c t i c e i t i s i m p o s s i b l e  to achieve  a p e r f e c t system, f o r  t h e image o f a p o i n t i s a " p a t c h " o f f i n i t e s i z e which i s a c o m b i n a t i o n of c e r t a i n  aberrations.  F o r any o p t i c a l system an a b e r r a t i o n  i s d e f i n e d and e x p r e s s e d i n a power s e r i e s , the magnitude o f one o f these a b e r r a t i o n s In t h e p r a c t i c a l d e s i g n the d e s i r e t o eliminate c e r t a i n t h i s p r o b l e m has been a t t a c k e d and  how i t s s o l u t i o n  each term o f which  function represents  (see [ 1 ] ) .  o f an o p t i c a l system we a r e guided by  aberrations.  We s h a l l now examine how  i n t h e case o f r e f l e c t i n g o p t i c a l  can be a p p l i e d t o t h e d e s i g n  systems,  of p a r a l l e l - p l a t e  systems.  We c o n s i d e r r e f l e c t i n g o p t i c a l systems which a r e r o t a t i o n a l l y symmetric, n o t i n g t h a t t h e i r p a r a l l e l - p l a t e  e q u i v a l e n t s w i l l be a x i a l l y  symmetric but not r o t a t i o n a l l y  symmetric.  I t can be shown that f o r such  o p t i c a l systems, the even-order terms i n the aberration series must vanish. The f i r s t three odd-order terms, i n order of decreasing magnitude are known as primary s p h e r i c a l aberration (1st order), primary coma (3rd order), and astigmatism  (5th order).  A spheroidal mirror s u f f e r s from a l l of these, primary s p h e r i c a l aberration.being most serious, for no p a r a l l e l beam of rays focuses at a single point.  A one-mirror system free of primary s p h e r i c a l aberration  i s the paraboloid, which s t i l l suffers from primary coma and higher order aberrations; only one family of p a r a l l e l rays e x i s t s which focuses at a single point. The problem of the design of r e f l e c t i n g systems free of both primary spherical aberration and coma was f i r s t solved by K. Schwarzchild i n 1905 (reference [5]).  His s o l u t i o n consists of two r o t a t i o n a l l y  symmetric mirrors which are combined to give a system free of f i r s t and t h i r d order aberrations (see F i g . 2, where rays enter from the l e f t ) . Here there are i n f i n i t e l y many f a m i l i e s of p a r a l l e l beams, r e s t r i c t e d within a f i n i t e angular range of the axis of symmetry, each of which focuses at a point on the f o c a l sphere.  6. With t h i s s o l u t i o n i n mind we of order  k  as a  system of  k  f r e e o f a b e r r a t i o n s of o r d e r s  d e f i n e a Schwarzchild  m i r r o r s , the c o m b i n a t i o n of which i s 1  to  2k - 1 , i n c l u s i v e  A p a r a b o l o i d i s a system of o r d e r 1, and is of r  order  System  Schwarzchild's  (see F i g . 3 ) . original  2.  Note t h a t systems of o r d e r  3 or g r e a t e r , b e i n g  systems, a r e i m p r a c t i c a l as f a r as r e f l e c t i n g systems a r e  single-level concerned  because they a r e s e l f - o b s t r u c t i n g ( t h i s problem can be p a r t i a l l y for  k = 2  system).  system  by u s i n g a p o r t i o n o f the f r o n t m i r r o r . a s  eliminated  i n a Cassegnanian  T h i s i s not t r u e i n the case of the m u l t i - l e v e l p a r a l l e l - p l a t e  e q u i v a l e n t s o f these system o f o r d e r  k  systems, f o r the mean s u r f a c e o f the can be c o n s t r u c t e d  as f o l l o w s :  Each m i r r o r i s r e p l a c e d a t a d i f f e r e n t l e v e l by p l a t e e q u i v a l e n t , and (see F i g . 4 where  these a r e j o i n e d i n the proper  k = 2).  Schwarzchild  its parallel-  sequence by  planes  The major f e a t u r e here i s t h a t such systems,  b e i n g m u l t i - l e v e l , a r e not s e l f - o b s t r u c t i n g . The reasons.  Schwarzchild  First,  system of o r d e r  i t i s important  in itself,  k  i s proposed here f o r  f o r i t provides  the  two  kth  o r d e r a p p r o x i m a t i o n of a p e r f e c t o p t i c a l system, " t h e o r e t i c a l l y " f o r reflecting it  provides  systems but  " p r a c t i c a l l y " f o r p a r a l l e l - p l a t e systems.  an example o f how  t o a t t a c k the problem of d e s i g n i n g  Second, parallel-  p l a t e e q u i v a l e n t s of "complex" o p t i c a l systems: i f the g i v e n o p t i c a l  system  can be r e a l i z e d as a c o m b i n a t i o n of s i n g l e systems ( s i n g l e m i r r o r s  or  l e n s e s ) , t h e n the g e n e r a l problem i s reduced to the s i m p l e r one  designing  the p a r a l l e l - p l a t e e q u i v a l e n t s of simple : A n o t h e r important  of  systems.  f e a t u r e of p a r a l l e l - p l a t e systems i s t h a t  while  7.  t h e s e systems have a c t u a l i n d i c e s o f r e f r a c t i o n of u n i t y , by s h a p i n g they have e f f e c t i v e v a r i a b l e i n d i c e s o f r e f r a c t i o n .  Thus the problem  M of  t h e d e s i g n o f p a r a l l e l - p l a t e e q u i v a l e n t s o f v a r i a b l e i n d e x systems i s concerned  o n l y w i t h the problem  o f s h a p i n g the mean s u r f a c e so as to  achieve the e q u i v a l e n t e f f e c t i v e index of r e f r a c t i o n , while a v o i d i n g the problem  of d i e l e c t r i c m a t e r i a l s .  The  s t a r t i n g p l a c e , t h e n , f o r the d e s i g n of p a r a l l e l - p l a t e  systems i s w i t h the p a r a l l e l - p l a t e e q u i v a l e n t o f the p a r a b o l i c m i r r o r , t h e s u b j e c t o f t h i s work. but the m a t h e m a t i c a l  Not o n l y i s t h i s system  methods developed  employed i n the d e s i g n o f o t h e r I t may  seem odd  in itself,  f o r i t s d e s i g n can p o s s i b l y  be  systems.  t h a t a problem  as simple as the p a r a l l e l - p l a t e  e q u i v a l e n t o f the p a r a b o l i c m i r r o r appears problem  important  as y e t u n s o l v e d .  Although  i s s i m p l e , the s o l u t i o n i s i n v o l v e d , and even the e x a c t  the  mathematical  f o r m u l a t i o n i s g h a s t l y , because p a r a l l e l - p l a t e systems, w i t h the e x c e p t i o n of  systems w i t h r o t a t i o n a l symmetry  general d i f f i c u l t  Ce.g.  the T i n Hat  to h a n d l e m a t h e m a t i c a l l y .  [ 2 ] , [7 J ) a r e i n  The mathematical  fact  b e h i n d t h i s i s t h a t the o n l y g e n e r a l c l a s s o f s u r f a c e s whose g e o d e s i e s can be expressed by e x p l i c i t  formulae  i s the c l a s s o f L i o u v i l l e s u r f a c e s ,  w h i c h c o n t a i n s t h e c l a s s of s u r f a c e s o f r e v o l u t i o n .  CHAPTER I I  THE  1.  The  Mean S u r f a c e We  equivalent M  MATHEMATICAL FORMULATION  now  M  take up  the problem of the d e s i g n  of the p a r a b o l i c m i r r o r , where we  c o n s i s t s o f two  M^  and  parallel-plate  assume t h a t the mean M.^  surface  connected smoothly by  a  B.  f o c u s s i n g bend An  parallel flats  of the  important c o n s i d e r a t i o n B  f o r the f o c u s s i n g bend [3]).  , noting  that surfaces  of a f a m i l y of  surfaces  of r e v o l u t i o n cannot  (see  One,  the m e c h a n i c a l requirement of ease of c o n s t r u c t i o n of the p l a t e s S^ Two,  r a d i i of c u r v a t u r e and  standing  B  that  B  be  geodesies of  the e l e c t r o m a g n e t i c  Three, s i n c e we  as e x p l i c i t  severe  s u f f i c i e n t l y approximable by  choice: and  principal  reflections  e x p e c t to be unable to e x p r e s s  formulae, we  ( s o l v a b l e by e x p l i c i t  i n our  requirement t h a t the  be bounded below-'so as to a v o i d  waves.  g e o d e s i e s of  r e q u i r e m e n t s which g u i d e us  be  used  (Fig. l a ) .  There a r e t h r e e  i s the c h o i c e  the  have the m a t h e m a t i c a l r e q u i r e m e n t surfaces  of r e v o l u t i o n , whose  formulae) w i l l approximate the  geodesies  B. These r e q u i r e m e n t s a r e s a t i s f i e d e x a c t l y by  surfaces.  A tubular  the  f a m i l y of  s u r f a c e , a l s o known as a c a n a l s u r f a c e  of  constant  r a d i u s , i s the envelope of the f a m i l y of a l l spheres of c o n s t a n t whose c e n t r e s  l i e on a d i r e c t r i x c u r v e  requirement two exceeds  a  above, we  A  . For r e g u l a r i t y and  assume t h a t the r a d i u s of c u r v a t u r e  tubular  radius  to of  satisfy A  at every p o i n t . In our  tube o b t a i n e d  by  case  A  i s a p l a n e c u r v e and  t a k i n g spheres of r a d i u s  a  B  i s a p o r t i o n of  centred  on  A  a ,  , while  the the  9. bends o f a^  and  and  a.^ » r e s p e c t i v e l y . (a  and  a =  a r e s i m i l a r l y o b t a i n e d by t a k i n g spheres o f r a d i i separation i s  d = a^ - a^ ,  + a ) 1  2  .  Our problem  so t h a t a l l g e o d e s i e s on over  The normal  M  then, i s t o determine  which o r i g i n a t e a t  B , e x i t p a r a l l e l t o each o t h e r i n  .  F  in  the c u r v e and  pass  A  1 0 .  2.  G e o m e t r i c a l Framework L e t the mean s u r f a c e  M  l i e i n the r e c t a n g u l a r C a r t e s i a n  5a, With  c o o r d i n a t e system as shown i n F i g . the l e a d i n g edges o f symmetric,  lies  and  i n the  x - y  and as p o l a r a x i s , where 5b  contains C (0,  P A  determine The c u r v e s  P  £  $  on  of r a d i u s C  £  a  has p o l a r c o o r d i n a t e s (0,<j>)  so t h a t the p a i r  0 = c o n s t a n t a r e s e m i — c i r c l e s and  by  <j> = -ir/2 A  A  to  .  the c u r v e s  i s so t h a t the two  A^  One  B .  and  , the lower and upper rims of  which t e r m i n a t e the f l a t s  between t h e s e .  on  <j> = c o n s t a n t  In p a r t i c u l a r the c u r v e s  c>j = TT/2  and  which  whose c e n t r e  u n i q u e l y and w i l l be used as our c o o r d i n a t e system  l i e s half-way  system  (see F i g .  A  then the p l a n e normal to  The p o i n t  has l a t i t u d e  a r e the c o p i e s o f A  B  i n a semi-circle  are plane curves p a r a l l e l determined  .A  M).  (see F i g . 5c).  and P  B  A , which i s c l e a r l y  i s the p o l a r e q u a t i o n of  i s any p o i n t on  cuts  l i e s on r(0))  P  The d i r e c t r i x  p l a n e w i t h the y - a x i s as a x i s o f symmetry  r = r(6)  which i s a top view o f If  .  the o r i g i n midway between  and  so  B ,  that  reason f o r choosing t h i s coordinate  f a m i l i e s of c o o r d i n a t e c u r v e s a r e o r t h o g o n a l ,  which s i m p l i f i e s somewhat the a n a l y s i s to f o l l o w . A geodesic i n F i g . 6 , where  n^  a straight line i n e x i t s from  B  cutting  T  s t a r t i n g at  and  TI^  ,  cuts  F  with p o l a r angle  a r e normals to A^  at  the system v i a a s t r a i g h t l i n e i n M„  at  P^  A^  and  .  i s shown T  follows  with angle of i n c i d e n c e  with angle of e x i t .  0^  3 , and  leaves  a ,  FIGURE  5  12.  There a r e two e q u i v a l e n t beam, a n g l e  focussing conditions f o r a p a r a l l e l  f o c u s s i n g c o n d i t i o n denoted by AFC, and l e n g t h  focussing  c o n d i t i o n , denoted by LFC. AFC  + y = a + 3  Q  (1)  i s g i v e n by  1  > where  y  i s the a n g l e between LFC  and  n^ .  i s g i v e n by 1  where  r  = - ir/2 t o TT/2) , and LFC  'from  2  2  i s the p o l a r angle o f  B(from  2  n^  k = r ( 9 ) + rC6 ) cos(6 ) + s  (2)  9  >  F  = 0  expresses  2  , s  is a  i s the a r c length o f  M  2  i s constant.  For  0^ = 0  a more a c c u r a t e  this, refer  E  angle  length  we have  k =• 2r(P) + ira .'  than LFC , and thus p r o v i d e s to F i g . 7 where  l e n g t h from  the f o c a l p l a n e  on  constant.  We note t h a t i n g e n e r a l AFC i s more s e n s i t i v e t o d a t a  path  T  the c o n d i t i o n t h a t the t o t a l o p t i c a l path  t o t h e upper edge o f so t h a t  k  P  E  estimate  i s any o p t i c a l system,  to the f o c a l plane,  L^  AL  the l e n g t h i n c r i m e n t  2  cos Ai|> = 1 - — 2 " —  L  To see  i s the o p t i c a l  2  i s t h e l e n g t h from  f o r a perturbation i n direction,  increment and  of e r r o r .  perturbations  E  to  AI{J i s t h e c o r r e s p o n d i n g  (AL = | L ^ - L | ) . 2  Since  2  we have  AL = L^ —  2  —  .  Thus a s m a l l e r r o r i n a n g l e  c o r r e s p o n d s t o a much s m a l l e r e r r o r i n l e n g t h , v e r i f y i n g our a s s e r t i o n .  13.  14.  3. Geodesies  on  B  The r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e s of a p o i n t on  B  are:  x = r s i n 9 + a cos <j> sin(9-ot) y = r cos Q + a cos §  cos(0-a)  z = a sin$ . A g e o d e s i c on  B  i s a curve  T = (x,y,z)  where  a r e r e l a t e d by t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n  d^  22 V "  V  T /  12  0  and <j>  (see [ 6 ] ) :  22'  r  2  ii The..symmetric ..Chr.istof.fel .symbols  T  are functions of the  ij coefficients ds  2  E, F, G  of the f i r s t  = E d<{> + 2 F d<f> d0 + G d 0 2  fundamental form o f  where  2  E = x  2  + y  <p  and  F = 0  + z  2  <p  , G = x  2  E  and  G  x. = - a s i n <f> s i n ( 0 - a )  x  y, = - a s i n (fi cosC0-a) <p  y  z, =  z. = 0  a cos < j>  + z  2  O  Q  0 Q  we c a l c u l a t e the p a r t i a l  D  2  D  derivatives:  = r ' s i n 0 + r cos 0 + a cos <f> cos (0-a) (1-a') = r ' cos 9 - r s i n 0 - a cos  where prime (') denotes d i f f e r e n t i a t i o n w i t h r e s p e c t to  2  + y  <p  r  sin(G-a)  D  <J)  2  s i n c e the c o o r d i n a t e curves a r e o r t h o g o n a l . To e v a l u a t e  E = a  B ,  E i s e a s i l y evaluated: 2 2 2 2 2 s i n <f> s i n (0-a) + a s i n <j> cos (0-a) + a  2 cos  0 .  2 2 <j> = a  (1-a')  15.  To e v a l u a t e equation  Cr  =  (r We  R,  tan a = r ' / r of  and  the  polar  A, i s  ,2,3/2  + 2 r'  2  - rr")  2  that  2  - . • C  2  +  r  r  tt  We  note t h a t  + r' )  also find  1  we  of the r a d i u s of c u r v a t u r e  . 2 .  R  G  ' >  R  have  r G = r  -u  2  -  -•.  2  + r'  + a  (r + 2 a r  r' )  2  cos  +  2  A (r +  2  2  cj> —  2  -z  R  r' ) 2  1  7  2  [ s i n 0 cos  f  <J> cos(0-a) - cos 0 cos <f> s i n ( 0 - a ) ]  K  (r  +  2  + 2 a r  r' ) 2  1  /  2  [ cos 8 cos <f> cosC0-a) + s i n 0 cos <j> s i n C 9 - a ) J  K and  the b r a c k e t e d  expressions  [ J s i m p l i f y to  cos <|> s i n a  and  cos d) cos a , r e s p e c t i v e l y . F u r t h e r s i m p l i f i c a t i o n g i v e s „ , 2 , , 2 v ( R + a cos <f>),2 , „ , . , . , . , G = (r + r ' ) . Next, we compute the C h r i s t o f f e l :os ^<j>—; XT  2  symbols  1  1  (see  2  rn = ri2  = r n l  L  „1 T  2 2  [6])  •  x  and  =  u  o'  t 2 ,  = s i n <j) (r  get  + r  ,2.  1  (R + a cos <j>) ) —^~ a R  p2 _ - a^ s i n (j> 12 (R + a cos <{>)  r222  _  Crr  1  + r ' r")  , 2 ,  Cr  ,2.  + r' )  a cos <|> R' RCR + a cos <j>) Y  16.  r  Thus f o r the g e o d e s i c  C4)  d 9  _  2  s i n 9 Cr  a R  (  a cos 9 R'  RCR  |  The  is  2 a sin 9 (R + a cos  " ^2  IT/2  to  s u b j e c t a r e determined  2  +  r  (dj)  ,2  A^  as  (5)  =  0( The  (sin  0^  t  -rr I  =  (r*  r  1  &  1  II  |-rr|  4r  1 d9  n  (6)  t  T on  .  The  B  i s determined  initial  |1  .  at  $  c o n d i t i o n s to which  T  9  =  -  + r cos 0.. 4r 1 d9  0^  a t t h i s p o i n t has t o ( s i n 0^  TT/2  direction  , cos 0^  dr/dtf)  where  , 0)  = 1 .  i s calculated  Now  to be  + a sin(0., - a) ,  1  d0 -r— d9  , * . + a cos(0.. - a) , - a) .  1  find = TT/2  (4)  w i t h the p o l a r a x i s  the i n d i c a t e d dot p r o d u c t , e q u a t i n g to u n i t y and  9 = -  traverses  have  Hence  n  d0_ d<{>  when  follows:  to  d0 . . - r r - r sxn 0 d9 i  Upon p e r f o r m i n g we  1  , 0)  [d$]  .  Q  tangent  , cos 0^  sin 0  cos  TT/2)  -  we  /djV  J  }  S i n c e a r a y of r a d i a t i o n making a n g l e c u t s the lower r i m  3  9) (dj)  c o u r s e of  - TT/2  range  /dQ\  (rr' + r' r"j\  + a cos 9)  complete  becomes  2  2  +  C3)  + r ' ) (R + a cos 9)  2  d<j>  the f u l l  B ,  on  a tan a . 2 , ,2,1/2 (r + r' )  simplifying  17.  where  r = r(9^) Let  determines  etc . .  6  2  denote  9(jr/2) .  as a function of e = f(rce) , e ) . 2  1  Solving (4) subject to (5) and (6)  9 ^ and a functional of  r = rC9):  18.  4.  Exact F o r m u l a t i o n R e f e r r i n g t o F i g . 6 we have  normals  n^  <j> = - TT/2  and  to The 1 +  where  6  2  ,  s = s(0^,0 ) g = 6(6 )  y(e ,e ) = 2  = f(r(0),8^)  i e a  r  l  c  e n  t h e a n g l e between t h e  g t h of  of (1),  (2),  LFC, i s  (8)  2r(0)  +T7a  on  B  from  T .  AFC, then i s  acep + e(e) 2  i s determined by (4) s u b j e c t  Similarly  T  the a n g l e o f e x i t o f  2  formulation 1  ^  t  2  TT/2 , and exact  e  (7)  ri  y = y(Q^,6^)  = r C © ^ + rC© ) cos 0 2  2  t o C5) and (.6).  + s(3 B ) t  .  2  These can be r e w r i t t e n i n t h e f o l l o w i n g way:  Let + r(0 ) 2  cos 0  respectively,  Gr = 0  2  1  + Y ^ , ^ ) - a (8.^ - BC8 > 2  + s C 8 0 ) - 2r(0) 1 }  Gr = 0  and  e q u a l to zero determines  Lr = 0 .  to the f a c t that  r  In p r i n c i p l e , s e t t i n g  enters r  i n a very  f  does n o t have a l o c a l dependence on from  rather  f  i s a f u n c t i o n a l of  r  complicated  than a f u n c t i o n of  < f > = - ir/2  r  to  r .  In p a r t i c u l a r ,  and i t s d e r i v a t i v e s , b u t depends 77/2  ; t h i s change i n  r ,  course of T  which i n t u r n depends, v i a C4) whose c o e f f i c i e n t s a r e unknowns,  B,  on r ( 8 )  or L r  way, and t h a t  moreover, depends v i a t h e f o c u s s i n g c o n d i t i o n s on t h e f u l l over  Gr  i s not s u i t a b l e f o r s o l v i n g f o r r(0) ,  f  on t h e change i n  Then (7) and (8) become,  r = rC8) .  T h i s , though r i g o r o u s , due  - 77a .  2  and L r = riQ^  .  I t appears t h a t  Gr = 0  and  Lr = 0  are i m p l i c i t  d i f f e r e n t i a l e q u a t i o n s o f a type n o t p r e v i o u s l y s t u d i e d discussed  i n Chapter V I I l .  functional  ( t h i s aspect i s  L t i s t h e r e f o r e n e c e s s a r y to a t t a c k t h e  19.  problem by a p p r o x i m a t i v e methods, t h e s u b j e c t s Note t h a t f o r convenience we p r o p e r t i e s a r e independent of s c a l e .  set  of the n e x t two  r ( 0 ) ='1  chapters.  s i n c e the f o c u s s i n g  CHAPTER I I I ANALYTICAL APPROXIMATIVE METHODS  1.  Introduction:  O s c u l a t i n g Tubes  Due to t h e g h a s t l y n a t u r e o f t h e p r e c e e d i n g we a r e f o r c e d t o develop a p p r o x i m a t i v e methods to determine i s as f o l l o w s : tube  T^  tube o f f  approximate  with d i r e c t r i x B_ .  so t h a t  A  ft  r = r(6)  and  ft by t h e requirement  Lr  approximates  become s i m p l e r .  B  locally.  T^  t h e way i n which  T^  so t h e  i s the o s c u l a t i n g r  enters  be e x p r e s s i b l e by e x p l i c i t  formulae, s i n c e these approximate the g e o d e s i e s o f r  ft  We a r e guided i n our c h o i c e o f  that the geodesies of  have a l o c a l dependence on  Our p l a n o f a t t a c k  l o c a l l y by a s i m p l e r c u r v e  T h i s approximation s i m p l i f i e s  Gr  .  B  , so t h a t  f  will  and i t s h i g h e r d e r i v a t i v e s .  T h i s scheme i s -analogous  to t h a t employed i n t h e l o c a l  a c u r v e i s approximated  of  curves:  to  t h e 1 s t o r d e r by i t s tangent l i n e , t o the 2nd o r d e r by i t s o s c u l a t i n g  c i r c l e and so on. corresponding  A  B , w i t h a p o i n t as d i r e c t r i x ,  B ; the 1st order approximation to  as d i r e c t r i x  the o s c u l a t i n g  o r d e r by a p o i n t ,  p r o v i d e us w i t h to  the 0-th  i s the o s c u l a t i n g  B , w i t h tangent l i n e o f  B , with the o s c u l a t i n g c i r c l e of  t o r u s of  and as we s h a l l s e e ,  B_ .  f  Geodesies  A  as.directrix  (elementary f u n c t i o n s o r e l l i p t i c  to  B  is  on these t h r e e s u r f a c e s can be  has a l o c a l dependence on  Such a p p r o x i m a t i o n s  integrals),  r.  g i v e r i s e to simpler equations  L ^ r = 0 , whose s o l u t i o n s approximate t h e s o l u t i o n s o f = 0 .  B :  i s t h e o s c u l a t i n g c y l i n d e r o f 13 ; and the 2nd o r d e r  e x p r e s s e d as e x p l i c i t formulae  Lr  0-th  0-th, 1 s t , 2nd, ... o r d e r a p p r o x i m a t i o n s  approximation to  and  to the  The tubes w i t h t h e s e as d i r e c t r i c e s  order approximation to sphere o f  locally  approximation  G^r = 0  Gr = 0 and  21. With the e x c e p t i o n of the solve  G.r A  or  L r  0-th  exactly.  o r d e r c a s e , however, we were u n a b l e t o  N u m e r i c a l methods f o r a p p r o x i m a t i n g  solutions  A  to t h e s e a p p r o x i m a t i o n s a r e d e v e l o p e d  i n Chapter I V , and r e f e r e n c e s t o  t h e n u m e r i c a l r e s u l t s s h a l l be made throughout t h i s  chapter.  22.  2.  Z e r o t h Order A p p r o x i m a t i o n :  N e g l e c t i n g the Bend  F o r zeroth- o r d e r a p p r o x i m a t i o n we by a p o i n t , and  B  sphere of r a d i u s  should r e p l a c e  locally  by the tube whose d i r e c t r i x i s t h a t p o i n t , namely a a .  However, i n t h i s case, the g e o d e s i e s  a r e g r e a t c i r c l e s and  i t i s easy t o see t h a t a r a y w i l l  in direction  AFC  so t h a t  , f o r example, reduces  Hence t h e r e i s no s o l u t i o n i n t h i s c a s e . is  A  too poor to a c h i e v e any  B  G^r = 6^ = 0 . then,  results.  (see F i g . 8).  can be o b t a i n e d by  ) becomes  6. = 2 a  whose s o l u t i o n i s  1  2  (8)  becomes  parabolas  r(0) ~  (see F i g .  +  •+ 10).  r ( 0 ) = -7=—  ;  (1 + cos 0)  >  a n c  *  Tta  c  o  s  9)  •  simply  T h i s amounts to r e f l e c t i o n , so t h a t  2 (7  sphere  s i m p l y be r e v e r s e d  Such an a p p r o x i m a t i o n ,  Instead, a zeroth order approximation n e g l e c t i n g the bend  to  on the  Both s o l u t i o n s are., of  course,  23.  3. F i r s t Order Approximation:  The Osculating Cylinder A  F i r s t order approximation i s achieved by replacing by i t s tangent l i n e a . of  T  and  The geodesies of B  (see F i g . 9).  C Here  B  by i t s osculating cylinder  are h e l i c e s , which approximate Y  =  0 , a = 3  and  C  locally of radius  the geodesies  s =  . cos a  AFC  i n t h i s case i s  G.r = 8, - 2a = 0 A 1  whose s o l u t i o n i s the  2 parabola  r(0) = -rz—;  „..  The " f a i l u r e " of  AFC  to provide a better  (1 + cos 6)  r  s o l u t i o n than the zeroth order case i s due to the fact that independent of  a , so i n p a r t i c u l a r we could take  tantamount to neglecting  where  k  LFC  does not f a i l since  and  L  k =  £ s i n (0^ - a)  1  X  which i s  s  depends on  a .  Referring to F i g . 9  are the indicated lengths, we f i n d that the t o t a l path  L.r = r ( 0 ) (1 + cos 0 ) H A  is  B .  length from the focus to the edge of where  a = 0  y  1  -L  Sat ting L^r = 0 , LFC  r(0^) + r(8^) cos 0^ + s - k  Z = TT a tan a .  and 7T3. COS  is  Thus  Lr  i s replaced by  ira tan a s i n (0, - a) - 2 - na . JL  Ot  becomes -TTQ  (9)  2 + ira = r(0) (1 + cos 0) + — cos a  where we have written' -Q on  r  since  6^  for  0^ .  -  Note that  i s expressible i n terms of  ira tan a s i n (0-a) f  has a l o c a l dependence  0^ , r(0^)  and  has been reduced to the nonlinear i m p l i c i t f i r s t order ordinary equation  L.r = 0  ( r e c a l l that  tan a = r ' ( 0 ) / r ( 0 ) ) .  k .  Lr = 0  differential  f9  8  /  FIGURE  8  /  /IV  L  FIGURE  9  /  I  I  .rt  25. Since we were unable to solve (9) exactly we had to approximate a solution numerically by the methods of Chapter IV. The resulting numerical approximation for the range Fig. 10 along with the parabolas • • r  (2 +  6 = 0 to  TT/2 i s plotted i n  2 r(9) -,-i—; ^\— CI + cos 0)  *  anc  ita)  ——rr. For this particular example we have chosen CI + cos 0) a = .25 , noting that as a increases so does the "spread" of the solution.  r(&)  =  -yi—:  27.  4. S e c o n d O r d e r A p p r o x i m a t i o n : For by  second order  approximation  i t s osculating circle  vectors  as  A  a s a new l o c a l  replaced B  l o c a l l y by  of  A  /•ini  equation  2  (10)  au  This  T T  curvature  osculating curve  £  then  T is  T  which i s a torus.  R ,  B  i s  a , and t h e g e o d e s i e s o f  (see F i g . 11).  a r e found by s e t t i n g so t h a t  locally  is  f o r the tube  i t s centre  R(9^)  =  r(9^) = p  i s at the o r i g i n .  ;  The  (4) becomes  a  To s o l v e  A  £  2 3 d 9 _ s i n 9 (p + a c o s 9 ) / d 8 \  1  we r e p l a c e  of  T , whose m i n o r r a d i u s  and s h i f t i n g  differential  L  here.  generator  The g e o d e s i e s o f constant,  B  w h i c h h a s t h e same t a n g e n t a n d  a r e approximated by t h e geodesies o f  p  to  Torus  a t t h e p o i n t o f o s c u l a t i o n , so t h e r a d i u s  the radius of curvature serves  The O s c u l a t i n g  this  \d$)  y = l-rrl  set  id*)  =  2 a sin9  /d8\  (p + a c o s  and  9)  \d$J  t = p + a cos  '  9"  so t h a t  becomes  d y  _*xj|t__  .  ^ _ d t  fr  =  0  m  a  -4 Multiplying it  ' becomes  (11) t h r o u g h by t h e i n t e g r a t i n g f a c t o r 4 d(y/t) -  1 a  t  t h e s o l u t i o n o f (10) i s  d (t  -2 ) = 0  t  we f i n d  that  so t h a t by t h e d e f i n i t i o n  of  28.  where  c  i s a constant i n d i c a t i n g  To f i n d  c  n o t e t h a t from  d6|  from  (6)  a tan a  d<H<j> and  TT/2  CI2)  —I  =  c  a  p/p E q u a t i n g , we f i n d  that  - c  c = p s i n a , or s i n c e  For the g e o d e s i c on Y  and  s  TT/2  Y =  s =  d9_ d<j> = a R s i n a d<|>  of  TT/2  —TT  TT/2  ds d<|> = a d<j>  -TT/2 r/2  -TT/2  where  T , by symmetry  p = R , c = R sin a . a = $ , and the q u a n t i t i e s  a r e g i v e n by  -TT/2  TT/2  t h e a n g l e of i n c i d e n c e of the g e o d e s i c .  2  ds = a  2  2  d<j>  12 CR + a cos <$>) /CR + a cos  <J>)  - R  sin a  C& + a cos <j>) d$ 2  2  //CR R + a cos <}>) - R  2  2  d<j> + (R + a cos <j>) d6  2  sin a  i s the f i r s t fundamental form  T .  Note t h a t under the i n t e g r a l s  R  and  a  a r e independent  of  <J)  30.  AFC  then i s  ,,  TT/2  G.r = 0 + a R s i n a A -ir/2  G^r = 0  and equating  C13)  - 2 a  d±  R T+ a cos $) CR + a cos <j>) //(R  ~ - R  2  2  s i -n  a  1  gives  dj  6 + a R sin a  .  =  / R T+ a cos < -t>) CR + a cos <j>) /CR 2  " - R  s i•n  2  a  2  —TT/2  An expression for the c y l i n d r i c a l L r A  LFC , derived i n a way s i m i l a r to that f o r  case (see F i g . 11), i s  = r C l + cos 9) +  s - 2 R s i n (Y/2) sinCy/2 + 8--<*) - '2 - v * .  Equating  L^r = 0  C14)  2 + ira = r C l +  gives  The equations i m p l i c i t non-linear  cos  Gr - 0  9)  +  and  s - 2 R s i n (y/2)  Lr = 0  s i n C y / 2 + 0 - a)  .  have been reduced to the G r = 0  i n t e g r o - d i f f e r e n t i a l equations  and  Pi.  L^r  = 0 , since  f  now has a l o c a l dependence on  r  and i t s f i r s t  two  derivatives. Again, unable to solve these approximate equations exactly, we calculated numerical approximations by the methods of Chapter IV. method requires the i n i t i a l values previously set r C O ) = 1 , and since calculate respect to  r"C0) 0  .  r(0)  , r'(0)  r'CO) =0  and  r"(0)  .  9 = 0 , so that  Since we  by symmetry, we need only  This i s accomplished by d i f f e r e n t i a t i n g Q-3)  and setting  One  a = 0 and we obtain  with  2 a  31.  (15)  2(1 -  k=  TT/2  1 + a(R (1 - | l ) ,  d(J) 2 (R + a cos <j>)  -TT/2  Although  i t was p o s s i b l e t o express  e q u a t i o n i n terms o f elementary i n v o l v e d to s o l v e e x a c t l y . with  R  Q N  = 0.58415  c a l c u l a t e d approximations  F i g . 12 a l o n g w i t h t h e z e r o t h and f i r s t s o l u t i o n s t o (13) w i t h comparison.  a = .25  As expected,  integral i n this  f u n c t i o n s , the r e s u l t i n g e q u a t i o n was too  I t was, however, easy  a = .25 , r " ( 0 ) = 1 - ^ The  the  and  to s o l v e n u m e r i c a l l y ;  to f i v e places.  to (13) and (14) a r e p l o t t e d i n order s o l u t i o n s .  a = .50  In a d d i t i o n , the  a r e shown i n F i g . 13 f o r  the spread of t h e d i r e c t r i x  increases with  a .  FIGURE 13  34.  5. Higher Order  Approximations  A n a t u r a l problem now approximate B  A  i s to c o n t i n u e w i t h the above p r o c e s s :  to t h e 3rd or h i g h e r o r d e r by  by the tube w i t h d i r e c t r i x  convenient  to express  .  Q  a curve  and  To examine t h i s problem we  (3) i n the form of two  a«  $ " i  <">  £ f M i ) + -Mf)(f!) M £ ) = °  where  1  +r  ds  2  +  r l  ±4  x  2  a  dT6 ,2 ds  -  2 a s i n <j) (R + a cos  a cos <f> R (R + a coss  The  equation  <!>)J  (4)  from them by e l i m i n a t i n g  _,2 + r' ) 2  2  R  |ds/  In our c a s e  (16) and  (17) a r e  2  W  2  • ) \di)ids/  (see[6]):  2  (R + a cos <{>) [de\  ldQ\ /d£\  find i t  ^ y - .  +r  „ ,1 a s i n (j> ( r  +  1  2  ds  (19)  +  2 2 = E d<j> + 2 F d<j> d9 + G d6 .  ,2.  ds)  ( S ) '  2 r ^  simpler equations  approximate  =  f ( r r ' + r ' r"] l  < r  2  +  r  .2  >  = 0  i s e q u i v a l e n t to (18) and ds ) .  (19)  ((4)  i s obtained  35.  Expressing of  solving i t w i l l  so  (19) i s  (4) i n t h i s way  a l l o w s us t o see where the  l i e , f o r higher order cases:  let  difficulties  t = R + a cos 9 ,  de d  (20)  2 dt  ds d9 ds  R  and hence (4) easy  2  2  r' )  +  r' )  2  (4) so d i f f i c u l t .  to s o l v e .  dR  "  ( 2  R  = 0  )  ', and  For example, i n the  S i m i l a r t h i n g s happen w i t h the  (20)  cylindrical  To be more p r e c i s e , t h a t term w i l l v a n i s h o n l y (giving cylinders & t o r i ! ) ,  3 r d or h i g h e r o r d e r a p p r o x i m a t i o n  c u r v a t u r e , the problem of s o l v i n g difficulty  -  2  d i r e c t r i c e s of c o n s t a n t c u r v a t u r e  s i n c e any  dR  i s c o n s t a n t so t h a t t h i s term v a n i s h e s , r e n d e r i n g  c a s e a l s o , of c o u r s e .  of  +  (20) can be i n t e g r a t e d , except - —  i s t h i s term which r e n d e r s  t o r o i d a l case  for  d(r (r  Each term of it  .  to  A  w i l l have  non-constant  (4) f o r such c u r v e s i s o f the same o r d e r  as our exact f o r m u l a t i o n of the problem.  t h a t h i g h e r o r d e r approximations  so  are u n f e a s i b l e at t h i s  We  conclude  time.  then  CHAPTER IV  NUMERICAL APPROXIMATIVE METHODS  1. Introduction Since we were unable to solve (9), (13) or (14) exactly, numerical methods f o r approximating solutions to these approximations had to be developed.  In what follows reference s h a l l only be made to solving (13)  since (9) and (14) are solved s i m i l a r l y , and as mentioned previously, AFC  should y i e l d the more accurate s o l u t i o n . Two d i s t i n c t methods were developed, one a " l o c a l " method i n  analogy to Euler's method for numerically solving ordinary d i f f e r e n t i a l equations, the other a "global" method based on Galerkin's method of undetermined c o e f f i c i e n t s .  Since error analysis f o r t h i s problem appears  too.„involved to tackle -analytically (see Chapter V), having two d i s t i n c t methods provides some guide as to the accuracies of the numerical s o l u t i o n s .  37. 2. P i e c e w i s e  Osculating  The  l o c a l method c o n s i s t s of p a r t i t i o n i n g  [0, T T / 2 ] . , by , 0  0  the f o l l o w i n g mesh of  6  =  By  the symmetry of  1  =  2  Polynomials  TT/2(N -  1)  A  ,  ...  , 6  N  the domain  denotes the exact  N =  TT/2  this  r  i s an a p p r o x i m a t i o n of the exact  p(6)  denote the n u m e r i c a l  define  p ^  been c a l c u l a t e d f o r  P  C J )  by  (6  K  +  ±  )  (0)  r .  (  j  )  G^r  r(0)  9 = 9^  (0)  , (j = 0 , 1,  2)  , we  solve  is  extracted.  pC8  r = p(9„  ..)  (13) n u m e r i c a l l y  for  This "corrected" value  ^(9^.  I T)  K * r l  is  and  p'(9  extrapolated.  .. )  +  X  To  extrapolate  p(9  polynomial  P(0)  are  about .-  p ^  (6 )  have  To c a l c u l a t e  )  K +  and  ±  p'(9  r' = p'(9  iv  -^)  from which  ) ^}  +  with  X .  +  and  p'(0„  X  K  ,) +  , ,) X  an o s c u l a t i n g  X  of degree 2 i s c a l c u l a t e d which agrees w i t h  p^^(0  ) K.  for  j = 0,  1,  2,  and  p(0  K.  p ' ( e  K  +  1  )  i s s e t to  , ,)  v  P ' ( 0  +  i s set equal  X  K  +  to  P(9 , T  K  1  ) .  )  X  P"C6j,  i s then used  ±  n  "T  K. +  T  K +  •  v  Note t h a t i n (9) o n l y p(9 , •,)  K.  Let  , a r e known, so  and  of j > " - )  to p r o c e e d .  K  we  .  X  +  K  p(8„  2)  2).  extrapolate  K "r*  (13), and  which  Assume i n d u c t i v e l y t h a t  the method d e s c r i b e d below, s e t  in  Gr = 0  ; i = 1 , ..., N)  l < i < K < N , ( j = 0 , l ,  , ( j = 0 , 1,  considered.  = 0 ; remember t h a t  s o l u t i o n to  a p p r o x i m a t i o n of  i n i t i a l values  need not be  s o l u t i o n to  i s defined only f o r  (0) = r ^  .  [- TT/2, Oj  r(9)  The  r(9) ,  e q u a l l y spaced arguments:  Herein,  to c a l c u l a t e (p(6)  the domain of  +  . ,) 1  and  38.  In p a s s i n g we  note t h a t h i g h e r o r d e r o s c u l a t i n g p o l y n o m i a l s were  w i t h no s u c c e s s , f o r v a l u e s o f  p  and  p'  tried  were e x t r a p o l a t e d which r e n d e r e d  (13) u n s o l v a b l e . The  e x t r a p o l a t i o n was  improved  method of e x t r a p o l a t i n g to the l i m i t partitioning h = 8„  [8 , 6  K  , , - 0„  .  K  1  ]  by a p p l y i n g a form of  (see [ 7 ] ) .  T h i s was  accomplished  i n t o 4 s u b - i n t e r v a l s of l e n g t h  *r JL  Improved a p p r o x i m a t i o n s  p(0„  , ,)  Richardson's  h/4  by  , where  and p'(9  )  were o b t a i n e d from a sequence of p i e c e w i s e o s c u l a t i n g p o l y n o m i a l s a t 6, V  Q  v  + h/4  , Q  v  + h/2  , 0., + 3h/4  , and  0„  L  .  i n the f o l l o w i n g  Suppose t h a t t h e r e e x i s t s a c o n t i n u o u s l o c a l l y to (13) i n a neighborhood s o l u t i o n by s'(0  s(0i .  of  &  v  which c o n t a i n s  8  , . .  We w i l l c a l c u l a t e a p p r o x i m a t i o n s  ..) , and t h e s e w i l l be the p's  exact  A sequence o f second degree p i e c e w i s e o s c u l a t i n g  Let  p ^  p. ^  I  K  (9„ + h/2)  j = 0, 1, 2 .  at  Set  f o r j = 0, 1, and  c a l c u l a t e the p o l y n o m i a l j = 0, 1, 2,  i  and  polynomials  be the o s c u l a t i n g p o l y n o m i a l which agrees  for  v  .. )  K. T  follows:  P^(0)  (0) a t Q  this  here.  K + J.  i s c o n s t r u c t e d as  solution  Denote  to s(8  way:  0  K  P (8) 2  + h/2  Next, e x t r a p o l a t e  .  s o l v e (13) f o r  (0  R  l e t p (8) 2  + h/4)  , extrapolate  p"(B  I K v  + h/2)  p^p(8),  which agrees w i t h  Finally,  pp^  p. (0) = P. (0)  = P (0)  , for  2  with  .  for  .  j = 0, 1,  solve  Then  39. (13) for Let  P''C9TT + h/4)  P (6) = ( ) p  3  solve for p p  6  a n d  3  ( j ) 4  extrapolate  P-jt^ + h/2) .  (6 + 3h/4) C6) at  P (0)  and calculate the polynomial p ^  C8 +  This determines  for  K  ^  f  o  J  r  °>  =  1  a n d  P^CS) which i s used to calculate P^C^) which agrees with  and f i n a l l y we obtain  6 + 3h/4  h y / 2  R  as before.  o  j = 0, 1, 2 .  Let p C9) = P C6> , 5  3  and suppose that this process was continued at 8 ^ + h/8  etc. .  In this way we obtain the sequences  l  X  =  P l K + 1> ( 6  ^ K 1> 6  +  X  2  =  P  2 K+  X  3  =  P  3 K + 1>  ( 9  ^  C 6  P3 K + 1> C6  The assumption i s made that  x —^ sC9„ , ,) n  K  T  and  x —^s C ^ , n.  I  K. *r  worst asymptotially since this i s known to be true for polynomials of degree one.  This assumption i s verified Cnumerically) l a t e r .  What this means is that q X  where  a  p  i " o a  +  a  l  ( h /  2i _ ) x  = s(9 . , .. ) , a, and K + 1 ' 1 T  l + °<  q h  l ) •  q.. are unknown and independent of ^1 r  h.  ) 1  More p r e c i s e l y t h i s means t h a t  as  1 •  Neglecting  terms o f  o(h )  we have f o r i = 1, 2, 3  q  and  . K + 1 " K h = 6  9  q  X  l  X  2  "  s C  3  ~  s C 9  Z  S  C  6  K+  1  }  +  3  1  l  h  l  q  -K + 1  }  +  K + 1  }  +  a  l  (  h  /  2  )  q  X  a  l  0  1  /  T h i s system i s s o l v e d  X  2 "  X  l  l Q = 2  then  and  (  ^)  9 K+  by o b s e r v i n g t h a t i f  q  can be e l i m i n a t e d  since  l  x  1  Q x  S  1  a h 2 "  a  X  s C e  l *  for  3 ~ 2 q  x  5  q  Q = X  4  k + i  - x _ j y ^ X  }=  —q  =  ±  sCe  R +  Q-;  x  Q —  ) s  o  Q' and s i m i•!l a r 1 l__ y 1 __•  C9 _sT'fl\  K +  ^\  t  h  a  t  S  ^  xl - x' J  -  CQ'  -  - 1)  K  +i  }  41. where  Q' = *3-*2 We then set s' (6  p(6  j) = 0 S  K  +  K  +  ^) and  p'(9  K  +  ±  ) =  ) , which are improved extrapolated values.  The numerical j u s t i f i c a t i o n of the above assumption was obtained by comparing the  Q's  at different arguments.  l i s t i n g below i t can be observed that the so we conclude that  6  a  and q.  Q  Q's  From the sample  are nearly constant  are (practically) independent of  h :  Q'  .10995  1.663  2.621  .12566  1.615  2.524  .14137  1.579  2.454  .15708  .1.551  2.399  The computer program for the above method i s contained i n the Appendix.  42. 3. G a l e r k l n ' s  Method  The  previous  to p o i n t , and exact  use  problem  method i s l o c a l i n n a t u r e , f o r we  o n l y knowledge of one  Gr = 0  (13) which i s i t s e l f  i s a g l o b a l one  concerning  we  argument.  should  g l o b a l , even though i t may  method o f s o l v i n g the exact to compare the f i r s t  previous  Since  l i k e a method o f not p r o v i d e  problem; f u r t h e r m o r e we  a  the solving  successful  w i l l have something  s o l u t i o n t o , from which c o n c l u s i o n s  the a c c u r a c i e s  march from p o i n t  can be made  of n u m e r i c a l r e s u l t s .  G a l e r k l n ' s method o f undetermined c o e f f i c i e n t s , f o r s o l v i n g o r d i n a r y and  partial differential  proves q u i t e s u c c e s s f u l . Fy = 0 , i s r o u g h l y  as  .  We  suppose t h a t  domains c o n t a i n  D  i s a g l o b a l method which  T h i s method, f o r s o l v i n g a d i f f e r e n t i a l  equation  follows:  Suppose a s o l u t i o n D  equations,  y  (H, <,  (x) of Fy = 0  >)  , such t h a t  i s sought, on a domain  i s a H i l b e r t space o f f u n c t i o n s whose y e H  ; let  {g  }  be a b a s i s f o r  H .  ot  A G a l e r k i n approximation of  y n  n (x) = £ a. g.(x) i=l 1  elements.  The  For <  Fy,  g> ±  = 0  y(x) for  c o n s i s t s o f the p a r t i a l  sum  oo  where  1  a^  y  , for  y(x) = £ 1=1 i = 1,  ...  to s a t i s f y i = 1,  ...  oo  a. g.(x) 1  , n  Fy = 0 .  , the g's  being  basis  1  , a r e determined as  i t i s necessary  that  follows:  43. In p a r t i c u l a r <Fy  n  , g.> l  -  0  i = 1,  for  T h i s g i v e s , then, a system of a^, i = 1,  . . . , n  .  n  . . . , n .  n o n - l i n e a r e q u a t i o n s i n the  In p r a c t i c e the  g^'  s  a  r  e  n  unknowns  chosen i n advance so t h a t  n y^  (x) = £  a  matically. say  <  ±  &±  ^  s a t i s f i e s any  initial  or boundary c o n d i t i o n s a u t o -  Note t h a t i n case t h i s were not t r u e , one of the  ^'y »  =  n  0  equations,  > must be r e p l a c e d by an e q u a t i o n which takes c a r e of  these c o n d i t i o n s . In our case  Fy = 0  i s the e q u a t i o n  G.r  = 0  , where  nil  <G^r, g > = j 0  D = [0, TT/2] , ,and  G^-Ce) • g^O) -d-9 .  ±  n The r e s u l t i n g system o f e q u a t i o n s , when  r (6)  =  n  £ a, g.(6) i=l 1  , is  1  fir/2 G.r  (6)  •  J  d0 = 0  g.(9)  for  i =  1  n .  o By the d e f i n i t i o n of  rir/2  G.r  the  + a R  n  sin a  d<J> n  5  -TT/2 (R  2  equation i s  TT/2  [9  ~  i th  ^ S i C ) d9 = 0 0  + a cos <j>) /(R  f^ ' r  + a cos <j>;  - R  2  Z sin  a  44. where  a  n  and  R  n  a r e , r e s p e c t i v e l y , t h e a n g l e o f i n c i d e n c e and r a d i u s  of c u r v a t u r e f o r t h e d i r e c t r i x w i t h p o l a r e q u a t i o n  r n  (^) •  The n u m e r i c a l  s o l u t i o n o f t h i s system o f n o n - l i n e a r equations was accomplished Newton's method  (see Appendix f o r r e f e r e n c e ) .  In t h i s method the c h o i c e o f t h e c o o r d i n a t e f u n c t i o n s i s o f paramount importance. on  by  They must, o f c o u r s e , be l i n e a r l y  {g^}  independent  [0, TT/2] , but not n e c e s s a r i l y o r t h o g o n a l , s i n c e we o n l y r e q u i r e them  to be o r t h o g o n a l to t h e o p e r a t o r  G^r , and s i n c e we r e a l l y don't  know  what t h i s o p e r a t o r l o o k s l i k e , a b i t o f t r i a l and e r r o r i s i n v o l v e d . We t r i e d f o u r d i f f e r e n t  families:  _2 i)  g-^6) = cos g (9) ±  ii)  ±  iv)  = sin ^  " (0/2)  2  1 5  = c o s ( ( i - 1) 9)  i = 2, ... , n .  [0, TT/2]:  i = 1, ... , n .  The even o r d e r Legendre p o l y n o m i a l s on  g ; L  (9) = cos"  g.(9) = 9  2 ( 1  2  , i m p l y i n g t h a t these  [0, TT/2] .  (9/2)  "  1  i = 2, ... , n .  )  I n cases i i ) and i i i ) ,  n's  (the parabola)  The F o u r i e r c o s i n e s e r i e s on  g (9)  iii)  (9/2)  8^'  s  i s probably that, although the  t h e r 's d i f f e r e d r a d i c a l l y f o r d i f f e r e n t n a  r  e  unsatisfactory.  The d i f f i c u l t y  here  g 's a r e an o r t h o g o n a l s e t , they a r e n o t  45.  o r t h o g o n a l to  G^r .  Case i ) worked r e a s o n a b l y w e l l , where, s i n c e 1,  we o n l y had t o  calculate  a^, ... »  a n  •  F  o  r  a^  i s obviously  n = 4 , the  c o e f f i c i e n t s are a  2  =  .21605  a  3  =  -.195489  a. = 4  1.380398 .  I t was observed  that  r ^ ( 9 ) agreed b o t h w i t h  r^(.Q) and w i t h the t a b u l a t e d  v a l u e s o b t a i n e d by t h e p r e v i o u s method t o 5 p l a c e s , the t o l e r a n c e imposed on the e q u a t i o n s o l v i n g r o u t i n e . Case iv,) .seemed..to work the,,bes.t,,...due a^ = 1)  i n c o d i n g and t o t h e f a c t t h a t the  to .its s i m p l i c i t y  a_^  (again  seem t o tend to z e r o ,  as can be seen i n t h e f o l l o w i n g t a b l e :  n  a  a^ •'  a^  3  .0113415  .0304101  4  .0485788  .0052103  .0053799  5  .0409563  .0150133  .0005350  We stopped r^(9)  a^  2  agreed  the c a l c u l a t i o n s at  t o b e t t e r than 5  u s u a l c r i t e r i o n i n t h i s method).  n = 5  since  .0007852  r ^ ( 6 ) and  p l a c e s on t h e e n t i r e domain ( t h i s i s t h e The agreement o f  method's s o l u t i o n was a l s o e x c e l l e n t  (4 p l a c e s ) .  r ^ ( 8 ) with the other  46. The  d i s t i n c t advantages of the G a l e r k i n method a r e t h a t  have a formula  for  r(8)  , and  of t h a t i n the l o c a l method.  the computing time was The  we  i n general a  fraction  program f o r t h i s method i s c o n t a i n e d  i n the Appendix. A n a t u r a l question f o r the exact  problem  have a formula  for  n u m e r i c a l l y , and  What occured  was  the 12th  r(6)  S i n c e a t any (4)  , equation  we  integrated  a c t u a l l y attempted, but w i t h o n l y p a r t i a l  converged  s  or h i g h e r  i n t h i s method  can c e r t a i n l y be  t h a t t r u e convergence was  a  stage  evaluated.  j_'  mean t h a t the  Gr = 0 .  Gr  T h i s was  i s whether or not G a l e r k i n ' s method works  p l a c e ) but  not o b t a i n e d .  By  success. this  we  ( s u c c e s s i v e i t e r a t i o n s changed o n l y i n the  equations  fTT/2  J  Gr  Q  Thus the  a  ^'  n  (6) g.  I  (6) d9 = 0  were not w i t h i n the d e s i r e d t o l e r a n c e .  converged, but not  s  I t was  to the c o r r e c t v a l u e s .  observed, however, t h a t the d e v i a t i o n of a beam from  t r u e p a r a l l e l i s m was  s m a l l e r than t h a t f o r the s o l u t i o n to  d i s c u s s t h i s i n more d e t a i l i n Chapter V.  Thus, we  (13).  We  did at least  get  some improvement of the approximate s o l u t i o n s . The  q u e s t i o n of why  t h i s method d i d not converge to the  s o l u t i o n i s extremely i n v o l v e d . i)  The one.  i n n e r product  Several p o s s i b l e reasons are: which we  use may  In f a c t , i t i s q u e s t i o n a b l e ,  whether  [-TT/2  Gr  true  • g^  (6) de  not be  the c o r r e c t  i n the exact  problem,  i s an i n n e r p r o d u c t  at a l l ,  47. for  r(8)  i s defined on [0, TT/2]  requires values of  and yet  Gr(0)  outside of the domain of n That i s , we define r(0) = £ a. g.(8) i=l  definition.  r(0)  1  only for at  0^  =  1  8 e [0, TT/2] , but when we evaluate TT/2  ©2 > TT/2 .  we require the value of  (®2^  r  Gr(8) w  n  e  r  e  Thus we extrapolate quite a distance outside  of this domain. We make the assumption that a linear sum of the  .  r(8)  can be written as  This may not be possible,  for a non-linear sum may be required. F i n a l l y , the whole scheme, i n this case, may be extremely sensitive numerically, i . e . unstable.  CHAPTER V ERROR ANALYSIS  There a r e two l e v e l s of e r r o r a s s o c i a t e d s o l u t i o n s of (13). At l e v e l I i s the t r u n c a t i o n B  by i t s o s c u l a t i n g  the  truncation  torus.  Level  r n  (6)  .  e r r o r due to a p p r o x i m a t i n g  I I c o n s i s t s of three e r r o r s .  e r r o r due to a p p r o x i m a t i n g  (13), by e i t h e r the o s c u l a t i n g  with the numerical  r ( 8 ) , the exact s o l u t i o n o f  p o l y n o m i a l or by the G a l e r k i n  Second, the e r r o r i n c u r r e d  approximation  by d i s c r e t i z i n g the domain of  ( t h i s e r r o r f o r the l o c a l method o n l y ) .  Gr = 0  Level  I e r r o r , then, i s the  and the exact s o l u t i o n t o  G.r = 0 , w h i l e l e v e l I I e r r o r i s the e r r o r between the n u m e r i c a l A and  t h e exact s o l u t i o n to ( 1 3 ) .  gives to  solutions  The combination of l e v e l I and l e v e l I I  the t o t a l e r r o r between the c a l c u l a t e d  Gr = 0 .  r  And t h i r d , t h e r e i s the r o u n d o f f  e r r o r p r e s e n t i n any n u m e r i c a l computation. e r r o r between the exact s o l u t i o n t o  First,  s o l u t i o n s and t h e exact  solution  The t o t a l e r r o r determines another e r r o r , the o p t i c a l e r r o r ,  which f o r a g i v e n r a y i s the a c t u a l d e v i a t i o n  of the r a y from t r u e p a r a l l e l i s m .  From the p r a c t i c a l v i e w p o i n t , the o p t i c a l e r r o r i s the most important.  We s h a l l l a t e r show how  t h i s can be c a l c u l a t e d  numerically.  M a t h e m a t i c a l l y , t h e l e v e l I and l e v e l I I e r r o r s a r e the most l e v e l I b e i n g of g r e a t e s t seem too d i f f i c u l t  interest.  agreed v e r y w e l l . conclude that  These e r r o r s , w h i l e o f such importance,  to c a l c u l a t e , e i t h e r a n a l y t i c a l l y o r  As mentioned p r e v i o u s l y ,  important,  numerically.  the r e s u l t s of b o t h n u m e r i c a l methods  S i n c e these a r e such d i f f e r e n t methods, we can s a f e l y  the l e v e l I I e r r o r s a r e w i t h i n r e a s o n a b l e  tolerances.  I t i s p o s s i b l e , a t l e a s t , to c a l c u l a t e t h e o p t i c a l e r r o r If  r (8)  i s a Galerkin  approximation,  (4) can be i n t e g r a t e d  numerically.  numerically,  49. and  the tangent d i r e c t i o n of a g e o d e s i c  Q£-.exit from  B  Let  .  T h i s i s done as  t  be  B  s t a r t i n g from  0^  a t an a n g l e  follows:  .  T  i s achieved  d e f i n e the o p t i c a l e r r o r f o r t h i s r a y as  (t  (0^)  0))  .  a t the  point  True p a r a l l e l i s m f o r a r a y  We  o (1, 0,  c a l c u l a t e d at i t s point  the tangent v e c t o r of a g e o d e s i c  of e x i t from the upper r i m of F  can be  when  e(0^)  t o (1, 0,  = TT/2 - cos  T h i s i s the a n g l e between  (geodesic)  t  and  0) = 0 .  ^  the a x i s of  symmetry. The  for  9  =  TT/2  tangent  t  Y  of  i s given  .  Upon d i f f e r e n t i a t i n g and  eCe.)  =  i  - cos"  TT/2  1  t a k i n g the dot p r o d u c t we  / r' sin 0 4r . * 1 2 d0 ^ ,2 dQ +  r  c  O  2  d  Z  A Runge-Kutta scheme was 6  2  a n d  dT l 9  The in  [0, TT/2 ]  ,12  -  6  2  o  s  o TT ~ * ^ 2 e  2  a  i  used to i n t e g r a t e (4)  r^(9)  problem  t a b u l a t e d below, the  -  "<•>) 2  to c a l c u l a t e  •  and  the " i n c o r r e c t " G a l e r k i n  (both w i t h 0's  ~  0  2  r e s u l t s of c a l c u l a t i n g the o p t i c a l e r r o r s a t 10 for  that  (o  n  d  given  coordinate  functions  i n r a d i a n s and  0  arguments  approximation 2 (i  to the exact  s  find  D )  are  the e r r o r s i n d e g r e e s .  50.  e  Torus Error  Exact Error  .015  -.004  -.005  .173  -.033  -.040  .329  -.012  -.040  .487  .063  -.008  .644  .167  + .018  .801  .276  +.017  .958  .385  -.002  1.11  .501  -.013  1.27  .611  -.003  1.43  .673  + .011  The error i n the t o r o i d a l case i s smooth enough f o r us to i n f e r that i t s maximum occurs at  0 = TT/2 .  In the exact case, the error i s not  smooth, and attempts at smoothing i t out, by performing more i t e r a t i o n s to solve the non-linear  system, f a i l e d to improve matters.  Thus we do  not know where the maximum error occurs, nor how large i t i s . What we do know i s that the i n t e g r a l of the error function remains too large f o r true convergence i n the Galerkin scheme.  CHAPTER V I THE GENERAL SYNTHESIS OF PARALLEL-PLATE EQUIVALENTS OF OPTICAL SYSTEMS  1. Simple Systems  o  We now take up the problem o f t h e d e s i g n o f p a r a l l e l - p l a t e e q u i v a l e n t s o f s i m p l e o p t i c a l systems.  By simple systems we mean  s i n g l e r e f l e c t i n g s u r f a c e s or s i n g l e c o n t i n u o u s - i n d e x r e f r a c t i n g the  p a r a l l e l - p l a t e e q u i v a l e n t s w i l l be s i n g l e - l e v e l The  optical characteristics  structures;  systems.  o f such systems a r e assumed t o be  g i v e n i n terms of a f o c u s s i n g f u n c t i o n 6  either  ^(6)  where, r e f e r r i n g  i s the p o l a r a n g l e of a r a y s t a r t i n g from some f i x e d p o i n t  t o F i g . 14, F , and  ty(Q) i s t h e a n g l e a t which t h e r a y c u t s t h e normal t o t h e x - a x i s as shown.  In the case o f r e f r a c t i n g  i n F i g . 14b.  structures  F o r r e f l e c t i n g systems,  v e c t o r a l g e b r a , and f o r r e f r a c t i n g The  focussing condition  ^(8)  = Q  with  6  Setting (21)  1  2  2  = f(r(0), Gr = 0 0  1  The  8 )  from the index o f r e f r a c t i o n .  f o r the p a r a l l e l - p l a t e equivalent of a  + * C 6 ) + yC'^, 0 ) - a t e p  1  t h e r a y s as shown  can be determined by elementary  structures,  s i m p l e system whose f o c u s s i n g f u n c t i o n Gr  we r e f l e c t  is  ^(8)  becomes  - B(6 ) a, 3, y  where  (see F i g . 15),  2  and the a n g l e s  Gr = 0  as b e f o r e .  we have  + ^ ( 6 ) + Y(8 , 0 ) = a ( 0 ) + 3 ( 0 ^ . 1  1  2  1  a n a l y t i c a l and n u m e r i c a l methods developed p r e v i o u s l y can  be m o d i f i e d t o r e n d e r  (21) s i m p l e r .  F o r example t h e t o r o i d a l  approximation  gives TT/2  (22)  _dj_  0 + i|)(8) + a R s i n a  (|R| + a cos -TT/2  <|>)/(|R|  + a cos <|>) -R s i n ^ x 2  2  FIGURE  14  53. The if  A  sign of  i s concave  R  depends on the c o n c a v i t y o f  down, n e g a t i v e i f concave up.  and t h e i n t e g r a l  A ; i t i s positive  At a point of i n f l e c t i o n  R  i s infinite,  ( i . e . y) i s z e r o .  The a b s o l u t e v a l u e o f  R  i s needed under the i n t e g r a l s i n c e the r a d i i of a t o r u s must be p o s i t i v e . The n u m e r i c a l methods appear a p p l i c a b l e a l s o , w i t h minor m o d i f i -  cations.  P o t e n t i a l t r o u b l e s w i t h the l o c a l method occur a t p o i n t s o f  i n f l e c t i o n and p l a c e s where  A  i s nearly straight  2 A t a p o i n t of i n f l e c t i o n we merely determine Another  r " , but approaching  set  r  (R  i s extremely  2 + 2r'  - rr"  equal to zero to  such p o i n t s may l e a d to d i f f i c u l t i e s .  advantage o f G a l e r k i n ' s method i s t h a t i t p r o b a b l y h a n d l e s  problems w i t h l i t t l e  difficulty.  large).  such  54. 2. Complex Systems We  c o n s i d e r complex systems which can be r e a l i z e d as  of s i m p l e systems.  The  combinations  p a r a l l e l - p l a t e e q u i v a l e n t of a complex system i s  a m u l t i - l e v e l system c o n s i s t i n g of the e q u i v a l e n t s of each s i m p l e joined  i n the proper  system  sequence.  The problem of complex systems i s s l i g h t l y more i n v o l v e d than it  first  appears.  As an i l l u s t r a t i o n , c o n s i d e r the problem of the d e s i g n  of the p a r a l l e l - p l a t e e q u i v a l e n t of S c h w a r z c h i l d ' s R e f e r r i n g to F i g . 16, (directrix  A^  i s the bend c o r r e s p o n d i n g  has p o l a r e q u a t i o n  r ^ = r^(0)) ,  c o r r e s p o n d i n g to the back m i r r o r ( w i t h '  parallel  to the x - a x i s w i t h f o c u s s i n g a n g l e  with  i s the bend  2  .  and ij^  , cuts the  again cuts a  . The methods of  r ^ = r^(6)  .  A  line  section  difficulty  r^ifi) , f o r we may not n e c e s s a r i l y a p p l y t h e s e methods w i t h  (see the F i g u r e ) , f o r i n g e n e r a l i s not e q u a l to the a n g l e o f e x i t  =j=  i n general.  i s r e p l a c e d l o c a l l y by i t s o s c u l a t i n g t o r u s , we used  B  t r a v e l s over  if-^ » passes over  1 are d i r e c t l y a p p l i c a b l e f o r c a l c u l a t i n g arises  F  2.  to the f r o n t m i r r o r  and  = ^(6))  A r a y of r a d i a t i o n s t a r t i n g from x - a x i s with f o c u s s i n g angle  system of o r d e r  s i n c e the a n g l e o f i n c i d e n c e Since see t h a t  as w i t h s i m p l e systems, but more e r r o r i s i n c u r r e d .  = r  when ^  can  be  Presumably,  f o r smooth enough bends t h i s e r r o r i s w i t h i n r e a s o n a b l e l i m i t s , but problem o f the d e s i g n of complex systems warrants  further  study.  the  55.  FIGURE  1'6  56. 3. Three D i m e n s i o n a l  Systems  The p a r a l l e l - p l a t e systems which we have been c o n s i d e r i n g a r e two-dimensional,  i n t h a t the r a d i a t i o n e x i t s i n a beam which l i e s  i n a plane.  A b e t t e r d e s c r i p t i o n might have been "... e q u i v a l e n t t o a s l i c e through ..." r a t h e r than "... e q u i v a l e n t t o ... ."  We now examine t h e problem  of the  d e s i g n o f p a r a l l e l - p l a t e systems whose t u b u l a r p a r t i s generated by a skew d i r e c t r i x ; we c a l l  such systems t h r e e - d i m e n s i o n a l . These systems may  not be o f p r a c t i c a l use, but they a r e c e r t a i n l y o f t h e o r e t i c a l We a r e o n l y concerned analog o f the o s c u l a t i n g t o r u s . a skew c u r v e  V  Since  which approximates  on t h e tube generated by obvious c h o i c e f o r of t h e c i r c l e ,  here w i t h t h e a n a l y t i c a l  V  V  A  i s not a p l a n e c u r v e , we  circle,  seek  A , with the p r o v i s o that the geodesies  be e x p r e s s i b l e i n e x p l i c i t  formulae.  The.  i s the c i r c u l a r h e l i x , which i s t h e skew a n a l o g  the o s c u l a t i n g h e l i x o f  A  same t a n g e n t , c u r v a t u r e v e c t o r , and t o r s i o n as A  approximative  s i n c e i t has c o n s t a n t c u r v a t u r e .  Indeed,  In case  interest.  i s t h a t h e l i x which has t h e A  at the p o i n t of o s c u l a t i o n .  i s a p l a n e c u r v e , t h e t o r s i o n v a n i s h e s , and t h e h e l i x i s a  the osculating c i r c l e of  A .  We now show t h a t t h e t u b u l a r s u r f a c e w i t h h e l i c a l d i r e c t r i x i s a L i o u v i l l e s u r f a c e , and t h a t we can e x p r e s s t h e g e o d e s i e s as e l l i p t i c integrals. P a r a m e t e r i z e t h e h e l i x as where  and  b  a r e p o s i t i v e c o n s t a n t s , and  with d i r e c t r i x  y  is  N  and  a  y ( v ) = (a c o s v, a s i n v, b v) ,  B  < u <  2  a r e t h e normal and b i n o r m a l v e c t o r s o f  TT  .  The tube  T ( v , u) = y (v) + A c o s u N(v) + A s i n u B(v) .  i s t h e r a d i u s o f t h e tube  0  0 < v < 2TT .  (A was  a  y  at  v ,  i n p r e v i o u s d i s c u s s i o n s ) , and  A  5  We f i n d  T  of  that the c o e f f i c i e n t s of the f i r s t  7  .  fundamental form  are E-  A  2  dv  G = P(u) T  (a c o n s t a n t )  P(u) =  T  2  A  i s the t o r s i o n o f  + (1-A  2  cos u )  2  y , t  i s a r c l e n g t h on  y , and  .  We now t r a n s f o r m t h e c u r v i l i n e a r c o o r d i n a t e s so t h e c o o r d i n a t e curves a r e orthogonal: Let  G(u)  Thus  = 1 + A  v  dv- = dv + 1 x  2  + A  2  A  = v +  k  2  , A  *  A  =  2  1  The  cos  2  E  l  2  x  where  u .  2  1  t  2 k  a  + A  and  A  *  _  -l(A+ I —  B* t a n u  I (1 + B )  = 1 + A  2  s  x  2  are constants.  X  transformed  = A  n  ^77  —=-  2 - — A  c o e f f i e e n t s of the f i r s t  x—  4  2  GC )  A  Ul  F l  =o  G  = GCu )  1  du, = du 1  d u  now F  and  x .  2  A  B  U  GluT  = v + —  where  „T f G(u)  A  x  fundamental form a r e  58. 2  so that  ty  2  U  2  1  }  +  as  2 2 (U^ + V^) (du^ + dv^ )  u  + G ( u ) dv.  L  e  t  ,  K  U 1  ) = 1 + k  I\j  which v e r i f i e s  o f t h e g e o d e s i e s on t h e tube we  2 1 ' l l *12  2 '22  =  x  r  22  d G(u ) 2 du  "  =  d  G  GO^)  1  w h e r e  < V  2du  E(  x  U ; L  1  du,  =  2  D  U  L  1  f l\ Mv,\ d  E  \  v  \  i  +  1/  we have  d  ;  C  (  U  u  E  2 E  l  )  l  }  •  which d e t e r m i n e s a g e o d e s i c i s  d G  3  2  «z. = j  (3)  equation  ^T Letting  "  <  G  - d G  2  C«i) "  2 A  )  Thus t h e d i f f e r e n t i a l  v d " v,  E  2  t h e a s s e r t i o n t h a t t h e tube  t o be:  d E (u ) r r S r2 d u E (u^)  o r 12  2  cos  surface.  f i n d the  =  '  which can be shown t o be w r i t t e n  1  To c a l c u l a t e t h e e q u a t i o n  1 'll  2  ^ l^  G  2  ds  a Liouville  du  2  then  is  = A  x i 2 GTu~T/ ^ l  -  (A  = I  ds  d  V  G  l du, du.  1 1  59. -2  Noting that  d(E G  z  1 ) = — G  ( G z d E - 2 E z d G  +  G E d z )  J  we a r r i v e a t  ^  -  d (E G"  z)  2  G which y i e l d s -1  C - G o  -2  =EG  z , C  o  constant,  Thus  C A  o 2  G  C  - G  (1 + k  2  cos  2  u^)  A  o 2  G  3  - G  (1 + k  2 2  cos  1  u^)  SG-  Now  dv 1  = dv + A j_ du G(u)  and  u  = u  so t h a t i n terms o f our o r i g i n a l  1  c o o r d i n a t e s we have  !C  Q  du  rA 1  G(u)  (1 + k  2  -  - G(u)  3  2  cos  2  dv  if—2—2-r,  A  G(u) = 1 + A  u  2  11 + k where  3  T  2  + A  2  cos k  2  (uj  cos  2  u .  60. Thus the g e o d e s i e s can be expressed as e x p l i c i t f(r,  0)  a l o c a l dependence on  r .  In t h e o r y then,  formulae  giving  we can s o l v e  t h r e e d i m e n s i o n a l problems a p p r o x i m a t i v e l y as we d i d i n t h e t o r o i d a l  case.  CHAPTER V I I  IMPLICIT FUNCTIONAL DIFFERENTIAL EQUATIONS  In t h i s c h a p t e r  we f o r m u l a t e  what appears t o be a new c l a s s o f  problems, s i n c e t h e problem o f t h e p a r a l l e l - p l a t e e q u i v a l e n t mirror  does n o t seem t o f i t i n t o any known The  differential  determine a tube 6"  =  8  F(4>,  where  ,  8  -  ,  =  TT/2)  r ( 8 ) ,  (4) and t h e f o c u s s i n g  r " ' ( 6 ) )  ,G  =  r  , 8 ( 9 ) e j b , TT/2^ ,  8 . ( - TT/2) = 8 ^ , rC8 ),  1 ( 8 ^  category.  1  r ' ( 8  where  We r e s t r i c t  ourselves  +  1  ct-3  -  Y  Although  9  =  0  conditions are  ^ 0 , TT/2J ,  8(<j>)  r = r ( 8 )  h e r e t o second o r d e r  form =  and t h e i n i t i a l  6^ ranges over  ) ) .  1  9  we a r e p r i m a r i l y i n t e r e s t e d i n d e t e r m i n i n g  first  c o n d i t i o n which  and i t s g e o d e s i e s a r e o f t h e g e n e r a l  <j> e£- TT/2,  e x p r e s s e d as  8 ' (  '  B  equation  o f the p a r a b o l i c  and  i s an unknown,  . differential  equations,  f o r s i m p l i c i t y , and second s i n c e t h e s e o c c u r i n many p h y s i c a l problems.  In g e n e r a l  now,  (4) i s r e p l a c e d by  0" = F(<j), 8  ,8 ' , r ( 8 ) )  where  <f> e [ a , b j  The  6 ( 9 )  e [c,d]  r ( 8 )  =  initial  ( r ( 8 ) ,  r ' ( 8 ) ,  r " ( 8 ) ,  r  (  n  (0))  )  ,  for  n  >  0  .  conditions are  8(a)  6'(a)  = 6^, where  =  8  ^ ranges o v e r  I(8(a), r(8(a))  £ c , dj ,  .  In a s i m i l a r f a s h i o n t h e f o c u s s i n g c o n d i t i o n s a r e r e p l a c e d , i n general,  by a r e l a t i o n G = G(^(a), r ( 8 ( a ) ) ;  8(b),  r(8(b)))  =  0  62. = (e(<j>), 6' (cf>), 9"(<J>)) .  where  The when  6  g e n e r a l problem i s to determine  = e(<j>) The  in  G,  and  an i m p l i c i t  6 = 0(<l>)  i s a s o l u t i o n to  unknown G  r  6',  so t h a t  chief d i f f i c u l t y  in  r .  G = 0  r(6)) .  i s imbedded i m p l i c i t l y not o n l y i n  i s a f u n c t i o n a l equation  cannot be  r = r(6)  6" = ¥(<$>, 6,  functional differential  The  .  We  call  F  but  such a problem  equation.  i n such problems i s t h a t i n g e n e r a l  s o l v e d e x a c t l y ( i . e . by  explicit  formulae).  We  t h a t i n our problem, when we were a b l e , as i n the c y l i n d r i c a l c a s e , r e p l a c e the d i f f e r e n t i a l e q u a t i o n G  was  an i m p l i c i t  be  which c o u l d be  to Dr.  Jon  to  G  Also, i n  was  reduced  to  equation.  to i l l u s t r a t e the g e n e r a l problem we  example, due  saw  solved exactly,  were a b l e to i n t e g r a t e once,  integro-differential  In o r d e r simple  by one  reduced to an i m p l i c i t o r d i n a r y d i f f e r e n t i a l e q u a t i o n .  the t o r o i d a l case where we  also  now  Schnute, i n which an exact  give a very  s o l u t i o n can  found: Suppose t h e r e i s a v e r t i c a l  and we  force  f(y)  i n the  x - y  plane,  w i s h to determine t h i s f o r c e so t h a t a l l p r o j e c t i l e s s t a r t i n g  the o r i g i n , w i t h a r b i t r a r y i n i t i a l directions, w i l l ,  (non-zero) v e l o c i t i e s and  (non-vertical)  a f t e r 1 u n i t o f time, be t r a v e l l i n g h o r i z o n t a l l y .  The mathematical f o r m u l a t i o n I f the p o s i t i o n at time (x(t), y(t)) y = f(y) y ( 0 ) = tan i|>  we  have  t  i s as f o l l o w s : is  from  63.  y(o) = 0 l t a n i> = -r— C o c  x(t)  = C  o  t  (C c o n s t , o where  We a r e to determine  f  and Also  T h i s f i t s i n t o our g e n e r a l  y  c o r r e s p o n d s to  6  t  c o r r e s p o n d s to  c> f  f  c o r r e s p o n d s to  r .  =0  is  F ( t , y, y', f ( y ) ) = f ( y ) and We observe t h a t  and  y(l) = 0 .  so t h a t  c l a s s o f problems where  i s const).  G  y = c^ s i n ( u t )  boundary c o n d i t i o n , where  y(l) = 0 .  satisfies  to = TT/2 + n"TT.  the d e s i r e d  Thus we f i n d  initial  the f o r c e to  be  nor  f (y(t)) =  - a) y ( t ) .  Obviously,  we need n o t r e s t r i c t o u r s e l v e s  2  t o f u n c t i o n s o f one v a r i a b l e .  to second o r d e r  However, due to the c o m p l e x i t y  f o r m u l a t i o n o f such problems, we s h a l l not pursue more g e n e r a l in  t h i s work.  equations,  of t h e  formulations  I t must be noted t h a t more g e n e r a l problems a r e under  c o n s i d e r a t i o n a t the p r e s e n t  time, the f o l l o w i n g problem i n d i f f e r e n t i a l  geometry b e i n g an example: Roughly speaking, w i t h boundary at  the problem i s to determine t h a t m a n i f o l d  , so t h a t each element  some f i x e d p o i n t , a f t e r p a r a l l e l  tangent  V  here,  V  o f the tangent space o f  transport along  the g e o d e s i c  b e l o n g s to some p r e s c r i b e d v e c t o r f i e l d  on  M , M  with .  It  CO i s assumed t h a t  M  i s a sub-manifold  m e t r i c , and t h a t the v e c t o r f i e l d in  the m e t r i c .  The exact  on  of a  C  n-manifold  with  Riemannian  i s p r e s c r i b e d by some r e l a t i o n  f o r m u l a t i o n o f t h i s problem t u r n s out to g i v e an  64. implicit (except  functional differential that  r  e q u a t i o n of the type f o r m u l a t e d  here  i s a f u n c t i o n of s e v e r a l v a r i a b l e s ) .  Methods of s o l v i n g these problems, and s o l u t i o n s , a r e a l s o under  -consideration.  existence  proofs  for  65.  CONCLUSION  P r e v i o u s i n v e s t i g a t i o n s i n p a r a l l e l - p l a t e o p t i c s appear to have d e a l t o n l y with, systems p o s s e s s i n g r o t a t i o n a l symmetry.  I n t h i s work,  however, we have f o r m u l a t e d a problem i n which the system i s not r o t a t i o n a l l y symmetric. involved  The e x a c t m a t h e m a t i c a l f o r m u l a t i o n proved so  t h a t an approximate s o l u t i o n r e q u i r e d  the development  s y n t h e s i s o f b o t h a n a l y t i c a l and n u m e r i c a l methods.  These schemes seem  a p p l i c a b l e to o t h e r o p t i c a l problems, and we have examined be done.  A l s o , i t appears t h a t t h i s i n v e s t i g a t i o n may  f o r m u l a t i o n o f a new  and  how  this  might  have l e d t o the  c l a s s of e q u a t i o n s .  Important problems i n r e g a r d to p a r a l l e l - p l a t e systems, f o r f u t u r e consideration, are error analysis  (especially l e v e l I ) , existance proofs,  p o s s i b l e n u m e r i c a l schemes f o r the e x a c t problem, and  the d e s i g n o f the  p a r a l l e l - p l a t e e q u i v a l e n t o f S c h w a r z c h i l d ' s system o f o r d e r  2.  66.  BIBLIOGRAPHY  M. Born and E. Wolf, P r i n c i p l e s o f O p t i c s 1959).  (Pergamon P r e s s , New  R. C. Johnson, ,"The Geodesic Luneburg L e n s , " Microwave V o l . 5, August 1962; pp. 76-85.  York,  Journal,  S. B. Myers, " P a r a l l e l - P l a t e O p t i c s For Rapid S c a n n i n g , " J o u r n a l o f A p p l i e d P h y s i c s , V o l . 18, 1947; page 221. E. I s a a c s o n and H. K e l l e r , A n a l y s i s of N u m e r i c a l Methods (John W i l e y & Sons, I n c . , New York 1966). K. S c h w a r z c h i l d , "Untersuchungen zur g e o m e t r i s c h e n O p t i k , " P a r t I I Ges. Wiss. G o t t i n g e n Math. Phys. K l a s s e IV, 1905. D. S t r u i k , 1950).  D i f f e r e n t i a l Geometry (Addison Wesley  I n c . , Reading, Mass  F. G. R. Warren and S. E. A. P i n n e l l , "The Mathematics o f the T i n Hat Scanning Antenna," T e c h n i c a l Report No. 7, C o n t r a c t No. DRBS - 2 - 1 44 - 4 - 3 , RCA V i c t o r Company, L t d . , M o n t r e a l , 28 Sept 1951.  67.  APPENDIX COMPUTER PROGRAMS Included are l i s t i n g s and brief explanations of three  F0RTRAN programs, the two programs which solve (13) and the optical error program.  Several l i b r a r y programs from the University of B r i t i s h Columbia  Computing Centre were used, the references being given where applicable. T r i v i a l subroutines and function routines are not l i s t e d , but their purposes are mentioned. The osculating polynomial program consists of a main program, the function routine R00T which finds the root  R(6)  of (13), a numerical  integration routine CADRE"'', and three t r i v i a l function routines F. 0  The function  G evaluates  gives equation (15) and  in (13) and (15).  ;  G^r at a given argument (G^r = 0  G, Q and i s (13)),  F i s the integrand of the e l l i p t i c integrals  A flow chart of the main program, divided into 6 blocks,  i s given, followed by the l i s t i n g of the main program and  R00T.  Taken from J . R. R i c e , M a t h e m a t i c a l Software (Academic P r e s s , 1971); programmed by Mr. G. I m e r z e e l .  68  &/QCK I Read NMAX = no. o f mesh p o i n t s , and tolerances.  C a l c u l a t e r " ( 0 ) and  mesh s i z e DELTA  a rock 2. Calculate  Al,  of p-^e)  B l , CI the c o e f f i c i e n t s  Print  initial  values.  , NMAX  aUck i  i Extrapolate  p^^x^)  j = 0, 1  store i n T l , T i p .  PJ^ N  Extrapolate  (13) f o r r " ( x ^ poly  J_  +  X  P (6)  h/2),  + h/2)  solve  c a l c . new  extrapolate p ^ ^ ( x  2  2  )  S t o r e i n T2, T2P .  As  i n Block 4 a t the points + h/2 and x _ N  final  results  + 3h/4 .  •+ h/4 , Store  i n T3, T3P .  G>!ock Q Calculate  Q, Q* ( c a l l e d p p l & p p 2 ) ,  improved r ^ ( x r'^x^  . Print  N  )  j = 0, 1 and f i n a l  results  END  1  IMPLICIT E£AL«U { A - II, K,C-Z)  2  Q  EXTE68AL  3 « 5 6  55  7 8 9  TOLX=XTOI. XNUti=HMBa  '  10  A=XA  11 12  I>1^3.14 1 5 9 2 6 5 3 5 9 0 0 TOL=XTOU  13 14 15 16 17 18  19  20  21 23. 22  24  25 26 27 28  29 JO  '  BAD=RAD1 RPP=1.CO-1.DO/RAO  DELTA=PI/(2 . CO'XllU.1) A 1 = R-RP»X l*RI'P*X 1 * X 1 / 2 . D 0 B1=BP-RPP*X1  C 1 = RPP/2.D0  H3R=li;il3R» 1 DO20N=2, ;iQa  —  53  54  55 56 57  58 59 60 61 62  63  64 65 66 67 68  69  70 71 72 73  74  75  76 77 78  At1S=ROOT  J  C\  (  W  IJ^W--1\  *2. EALP1I = CALPH=DSQRT(T**2+TP**2)/TP K= ( D A T A ! i ( T F / T ) -XX/2. C O ) ^ C A L P ! ! / i  BRIANS  51 52  (\  =  RX=BAD  50  O  cA  T  W  TIT0=0.D0  41  44 89  n  H R I T E ( 6 , 3 ) J , X 1 , R, R P , R P P • V R I T 2 ( 6 . 15) RAD,TITO,TITO,TITO  40  47  .  ir)/^U -'N-  J=l  T=A1'DI*XX>C1*XX*XX T1P=D1+2.C0*C1*X TP=E1 t;0'C1«XX T P / C S C R T (T»«2*TP«»2)  46  -  -  =  X1 O.D0 B=1.00 BP=0.D0  33 35  (|3 44 45  d-  N  IH=2.1DO  X = X1 * D E L T A  42  ^  BAD]=ROOT (Q,X8, 5 0 , T O l )  TX 1 = (XA1 i*X >5C0l0* X * X X *X + a 1) *.  39  OCK  SALPI1=0. CO  .  31  38  U  •  33 42 36 37  R/  EXTERHAL C C0MH3:! /PAR/SALPH, A , K, A X , B X , C A L P I I COSKO.I /GG/TCLX R E A D ( S , 5 5 ) HKun.XA,XTOL.XTOt FORKAT ( 4 I , 3 D 2 2 . 1 S )  _  I  vV H  1] C N A K \J ^  \  v  (G,RX,50,TOL)  TPP=T*2.DO»T P*TP/T-  ( (T »T *T F »T F) * DSQRT (T »T *TP * T P )  ) / (T*B R )  C2=TPP*.5C0 B2=TP-TFP*XX  A2=T-TP*XXtTPP»XX*XX*.5D0  T2=A2*D2«X*C2»X*X  XT2P=U2»2. X 1 = (XX*X C0»C2*X 1) *.50O XX2=XX x x 3 = (xxtx) » . 5 c o T = A 1 »t!l*XX 1 » C 1 » X X 1 « X X 1 TP=D1»2.C0«C1*XX1 CALPII= E5QRT (T«*2 > T P * * 2 ) / T P SALPi; = TP/CSQnT (T»*2 »TPt»2) K= (DATAli ( T P / T ) -XX 1/2. D O ) ' C A L P l l / A A!IS=HOOT  RR=AHS  TPP=T»2.  f) i r\ t- \ c \}L U L K )  W  J  (G,RX,50,TOL)  D0»TF»TP/T- ( (T *T * T F*T P) • DSQRf (T *T *T P* T P )  ) /  (T*B R )  C2 = T P P * . 5 C 0 B2=TP-TPP«SX1  A2=T-TP»XX H T P P * X X 1 » X X 1 * . 5 D 0 T=A2*D2«XX2»C2»XX2«XX2  TP=  82*2.  C0«C2*XX2  SALPII = T P / C S Q P T (T*«2 'TP**2) C A L P ] : = C S Q 3 T ( T * * 2 * T P * *2}/TP K= (DATA!) (TP/T)-XX2/2. C O ) * C A L P I I / A A)lS=ROOT (G,RX,50,TOL) RR=ANS TPP=T*2. C 0 « T P « T P / T - ( (T»T*TP*TP) « D S Q R r  (7»T>TP*TP) )  /  (T*nn)  C 2= T P P « . 5 C 0 B2=TP-TPP*XX2  = T-TP«JX2 + T P P » X X 2 * X X 2 * . 5 C 0 T=A2*I:2«XX J*C2»XX3»XX3 A2  TP=U2*2.D0»C2*XX3  SALPII = TP/CSQnT(Tt'»2*TP»*2) C A L P I I = D 3 Q U T {T«»2»TP«*2)/TP K= ( D A T A N (TP/T) - X X 3/2. D O ) *"CA L P 11/A  79 00  A!IS=HOOT RR=A;IS  01  T P P = T*2.  (G , RX , 5 0 , T O L )  D O « T P * T P / T - | ( T * T *T F + T P) *[)!>'QRT ( T * T * T P » T P ) ) / ( T * R H )  04  83 81 85 86 87 88 89 90 91 92 93 91 95 96 97 98 99 100 101 10.2 103 104 105 106 107 108 109 110 111 112 113 1 in 115 116 117 11 a 1J o 137 138 139 110 111 112 113 111 115 116 117 118 119 150 151 152 153 151 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180  B2=TP-TPP*IX3 A2=T-TP«XX3tTPP*XX3*XX3».5DO T3=A2»D2»X«C2*X*X T 3 P = U2 *2. C 0 - « C 2 « X I F ( T 3 .-KQ.T2) U 0 T 0 5 .  5 6  2 4  3  15 20  PP1 = ( T 2 - T 1 ) / ( T 3 - T 2 ) I F ((PPPP1I.*ETQ2. -1T. 1 C )O/ () P GP 0 T10- 51 . D O ) R= C0T06 H=T2 IF(T3P.E0.T2P)G0TO2 PP2= <?2P-T I P ) / ( T 3 P - T 2 P ) IF(PP2.EQ.1.CO)G0T02 RP=(PP2»T2P-T1P)/(PP2-1.D0). G0T04 RP=T2P SALPH=RP/CSQRT(R*«2+RP**2) C A L P H = C S Q R T ( B « * 2 *RP * » 2 ) / R P K= (DATAti ( R P / l i ) - X / 2 . DO) « C A L F I I / & A!(S-npOT ( G . UX, 5 0 , TOL) RR=AUS RAD=RR  6 4  2 10 21 11 3  .  A Y7 «  ) U  .  • '  .  • .  L  .•  .  '  .  . .  '  .  RPP=R 17. D O » B F * R P / R - ( ( R * R * R P <S) *RP) * E S Q R T ( R * R * R P * R P ) )/ (H*nH) C1=RPP/2.C0 B1=RP-RPP*X M=R-RP«X«X/2.00 " • VRITB(6.3) S,X,R,RP,RPP FORMAT ( I X , 'N= ' , 1 3 , I X , ' X = ' , D 2 2 . 15, I X , 'R=> , D 2 2 . 1 5 , I X , ' n P = ' , D 2 2 . * IX,'nPP=«,022.15) WRITS (6, 15) RAD, S A L P H , P P 1 , P P 2 FORK AT ( I X ; •RAO= ' , 0 2 2 . 15, 1 X , ' S A L P I I = D2 2 . 15, 1X, ' P P 1 = » , C 2 2 . I S , " • ' P P 2 ^ ' , E 2 2 . 15) X1 = X STOP ' ZN 0 • J  tlrt=l  7 1  " . •  F l W C T I O t l RCOT (C,ZGUESS,H,IPS) I M P L I C I T R F A L « 8 ( A - H , X, 0 - Z ) ZO^ZGUBSS "  5  70;  "' -  IFtAG = 0 WO = G ( Z O ) TEST^ZO I F (DAES (30) Z1=2.D0»Z0 Z2 = Z 0 / 2 . C 0 VI-G(ZI) TEST^ZI I F (DABS (K 1) » 2 = G (Z2) TEST=Z2 I F ( D A t S (W2) I F (HI * K2 . G £  '  •  •  ••  . ,  -  »  . L E . EPS)  COT03  . L E . EPS) G0TO3 . . L E . EPS) GOT03 .0 . HO) G0T01  '  ' '  '  /  Z3 = Z 1 - H 1 » ( Z 2 - Z 1) (M2-H 1) LIK=LI.1*1 IF ( L I K . G T . N ) C0TO2 W3 = G ( Z 3 ) TEST=Z3 I F (DADS (H3) . L E . E P S ) G0TC3 I F ( W 1 * W 3 . C E . 0 . D 0 ) C0T06 S2=U3 Z2=Z3 G0107 Z1 = Z 3 W1 = W3 G0T07 Z1=Z1»Z1 Z2 = Z 2 / 2 . C 0 IFLAG^IFLAG*1 • I F (I F L A G . G E . II) GO TO 2 1 G010S WRITE ( 6 , 1 0 ) L I H , 7 . 1 , a 1, Z 2 , 8 2 FORKAT(IX,'LIK',13,4D14.7) STOP WRItE(6,11) IFLAG,Z1,W1,Z2,92 FOR HAT IIX , ' I FLAG ' , 13, 4 U 1 1 . 7 ) STOP ROOT = T £ S T RETURU EHD  •  *  ."  *  -  71. Next i s t h e G a l e r k i n program.  S i n c e i t i s q u i t e s i m p l e , we  s h a l l o n l y d e f i n e p e r t i n e n t f u n c t i o n and s u b - r o u t i n e s , and g i v e t h e l i s t i n g . In the o r d e r i n which they o c c u r ,  N0NLIN,  the r o u t i n e s a r e :  FK, CADRE, G, AND H. 2  N0NLIN  s o l v e s t h e system o f n o n - l i n e a r  equations  fit/2  (i.e.  o  G.r • g. = 0 ) A X  whose v a l u e s a r e e v a l u a t e d by FK.  b e f o r e , H i s t h e i n t e g r a n d i n ( 1 3 ) , and G  is  G  r A  CADRE i s as  «  A l i b r a r y s u b r o u t i n e from t h e U n i v e r s i t y o f B.C. Computing C e n t r e , from C o l l e c t e d A l g o r i t h m s o f CACM, A l g o r i t h m 3/6, by K. M. Brown.  taken  Hit* 307 300 309 310 311 312 313 311 J15 316 317 318 319 320 321 322 323 320 325 326 327 328 329 330 331 132 331 130 315 336 337 338 339 300 311  JBPIICIT B U I »8 (X-B.C-2) LOGICAL L I S T D ) PC .-.'L 1 C > 1 1CII1 (6,6) , 13UE (6) ,COF{6,6) ,TI«P (6) ,PABT 16} ,1 |5| iisr-.Tsuj. K-5 ltXIIT»20 »unsic«5 X(1)-1.CC X ( 2 ) . 1 2£ 3 15-762167215C0 X 111-.71611S7 1 5630060 3 ( 4 I (0) »-. 13om;;n )9£5922CC 3  X |5) »1.05 2 3 IC i 10 1112000 CO O I L XCKLIMa'I.IIAIIT.BU.ISIG, 15 IHG , I, I tOI I T , I £ C3, COI.T (.a P ,P»«t, • MIT) If (ISJKG. I C O ) V8 IT! <6,3C) If (f AIII. IC.f. )1I(ITE |6,1C) 20 30 31  310  'CEROUTtXH  corr.o»/cn/  aS-1 tt-l  1  III (9,1,1, >) •  .  E,PI,J»,II  OOU-1,1 8(J)=1|J) p I- 1. 570796 3167') 0096D0 1EG-CACBt |C,C.C0,PI,0. [0,1.C-6,t,K,J) T-TEG BE1C8I  tig  357 358 J5S 360 361 362 363 360 365 366 367 368 369  PCHCTICS G (T) jspiici? r.i'ii!|i'J,c-:| c i n t u s i o 6(5) corros/cFF/ E ,Fi,3i,»I n c j a i L ii corros/Tvc/ i . s i t IF (T. E9- C . CO) GOT02 n=cccsn) TE»I.C0>1T  -  '  S£=CSIS (T)  3 70 371  372 371 370 175 376 377 378 379 380 181 182 181 161 385 386 387 168 389 190 39 I 192 393 191 395 196 397 198 399 500 001 002 401  110  IMPLICIT BEAL«8(A-fl,0-Z) EX1ERKAL G OlflEKSICH 1(5) , 8(5)  3<)2 303 315 306 107 348 319 150 351 352  uaiiE(6,2C) (i ( i ) , i , s) ,n»i t t , i s t i o PCBr.il (1 ),• 1 =' ,0 (C25. 16, 3X)/IX , D25. 16, 210) FORSAT III, 'XO CO •) S10P  TF:2*TB«TB B1="2. CO/T f • E (1) • « ! Bi=2.C0«S[/T!•2.D0»8 (1) «TB B3 = 2. co» n n ;.co• S E » S E / T I ) / T 2 > 2 . CO • t H I Tt-TK2«TU2 li-H!2»T 1J-IH2 TX-1 0 0 1 1 = 2,111 J1-2 • I  1  0»P(1) BI-BMO'TB B2-B2tir'U >TI n3«a) « n • i i i - i ) «o»Ta T' = -jtr:-2 H-TI»1H2 TE» TH »TII 2  '  1I-11C»TH2 £S= B 1 » 3 1« 3 2 » B 2 «AI»S2/SS " n«ss«CSCRT(s;J/lssta2-» l'»3) II = .2 5EO<9'.'i>l»CICBE|H,-Il,PI,0.tO 1. t-6 , 1,2,J) f EE-T" • (2 •KX - 1) G=f EE* I) I »T-; . CO • C AT AX (B2/S 1) ) BF1UBJI FEE»TE.".P GCTOO c«o.co • BETUB H (  3 2  EID  0 00 005 006 007 008 009 0 10 Oil 012  r e s c u e s 11 IT) JBP11CIT B I X I H I (1-II,C-1) c o r r i i k / r i c / t ,rxt 1. 1)0/ (!t • . ;V.HIK05 ( T ) ) «0SPB1 ( (IH . 25PC • OCOS (1) UF 11111 tao  )  • >?.- (It •S.M.) »«2)  73. The o p t i c a l e r r o r r o u t i n e  of  the G a l e r k i n approximation  ERR0R  reads i n t h e c o e f f i c i e n t s  n r (8) = £ a. g.(6) . i=l  a^ ,  The s u b r o u t i n e  3 DRKC  integrates  ( A ) , where (4) i s g i v e n by FUNC.  c a l c u l a t e and p r i n t  the o p t i c a l  e r r o r i n degrees and r a d i a n s .  t h i s r o u t i n e can be used i n p l a c e o f the  e x a c t problem;  L i n e s 407 to 530  G  Note t h a t  i n the G a l e r k i n program f o r a t t e m p t i n g  t h e a.'s a r e p r o v i d e d by  N0NLIN.  A l i b r a r y r o u t i n e a t the U.B.C. Computing C e n t r e , taken from "Numerical S o l u t i o n s o f O r d i n a r y Simultaneous D i f f e r e n t i a l E q u a t i o n s o f the F i r s t Order U s i n g the Method o f Automatic Step Change," Num. Math., V o l . 14 #9 (1970).  071 070 075 076 077 078 079 oeo osi 082 083 oeo 085 0 86 087 088 089 090 091 092 093 090 095 096  097  098 099 500 501 502 503 500 50 5 506 507 508 509 510 511  512  513 510 515 516 517 518 519 520 521 522 -'523 520 «25 S26 J21 526 J29 510 518 539 500 501 502 503 500 •05 «06  #  zo-.scom Z2»-Z1 T(1)M Till*1.CC»ECCS(1) El = 2. C0/T1J1, TIllVTJUl'Tilt. S2 = 2. COUSI J | T ) / U l l 7EEE"T 12>lEr.t>t!ta TEHP=T2 CC2 I' 1,1 D-l(I) B(I)>0 I»2»I B1"8 HO'tEIIP B2«82«I1»0»TCnB m t = iffa>i2 2 TElf»7t.l? I2 I(?)*.25EC'B;/(M>CSCBTIS1»8I«12«82)) cmC B K C c : i , z 2 , i , P.n,Hnis,s,ruac.c,3,i) 11=1(1) I1P-r(2) DUO. c o « c e c : t x l ) Sl=2. tO/II! BIP'2. CO'CSII (I1)/(C[f < t l l )  «{9  570 «71 572 «7J 57» • 75 576 < 578 179 5S i 1 7  0  ti  ' •  ,  T2=11»Z1  3  IEEE-I1 TE8P-T2 CO ) !• \.t EO»|l> II»2«I gl»»l«U«IEHP B1.f»B.lf»II»C»JIH TE.1E"TE.13«I2 IEEEMEirf«I2 It-OATAS (S tP/S Ij s*=21'*Iir Sl?=Blp»XlF EE«C5CaT(P1»H»B1P*SIP».C6i5C0) C»3 1 ? « ISIS II 1) <3 OOCCS II IJ-. 2500»OSE I (I l - l t ) RHCE-.SLC •(I-CAFCOS (C/EE) CEGS-RltS* 13C.C0/PE BE1C8I E»0 ' ' 1  1  1  JCEBOCTEII ISJC(!I,T,r) InPEICIT BE>L«8(1-H,C-1) C l I E N S K ) 1 |1) .1 (1) , E 110) COEnOS E,»J 11'CCCS 11 (1)| IE-1.C0HT  12ME«Tt  507  508 509 550 e;i 552 «<} 555 J55 556 «57 S58 <>9 560 561 5£2 5tJ 540 «t5 5J6 t-tt 568  74  C C E P 0 U T E 1 E If E.OB(T,l,»,RlCS,CfCS,T0L) inructr >at"«|i-i,ij-!) E1UB>»1 l i n e CIMEKSICI £(10) corros o , » i C1Bt»nic» »(10) , i (2) ,r (2) ,r,( J) ,S(2) I (2) B»D:*O.EC DECS-'O. co IF t i . c o . c. CO) BZICBI PI"3.1« 1 5 9 : 6 5 3 5 8 9 7 7 0 0 E'PI/6».CC I-.KIIOIVKO.CC E«IOl  t  SE-CSES (1 ( I ) ) TE-I(I) TE2»TH»TH B 1=2. C C / I E<B (I)'IM S2»2. EtO'SE/T ; »2. CO»B 11) 1T8 B 3 - 2 . C C « |11» 3.CO«fr«SE/TI)/T212.CO•t |1) S7-2. C0>S(»|6.!:0«5EtSE/T;»6.D0«1V12-1-00)/72 1E'1H2«TH2 TI»TH2M|lj 10=1H2 "'T(I) CC102.11 ll»2«t C*t(I) S1=8I»0»T8 S?=B2»ll>0»Tt »3«»3tII« HE- 1) 'D"IJ >7*B7MI< 1) • (11-2) <0*T( TJ»TJ«TU2 1011MH2 1I>r»»tMj  11»1K»TII2  SS=BOB I>B;»B2 E1-!S<)1<IM1M1 KU-SS.ISCKT (E5)/St fi-3.E0«B;«(11'Sll/SS-l2.t0»(MB2O. EO "B :»B 3-BI • t J)/ST • f-E-BI t . 2 5 E C « E C O S (11) 1-1(2) rot" »'.sno«rsii ( i t ) M/F6 (ECOS (H) •. ?5(JC«B5/Sb-S7 (»1.»1)/SS) C-«.l>0MSl> (I M •H>».'.T.'.T«»»»H/(5S«S51 r(2l"»«H-.C »r1CKI E»0 (  ' '  '  '  

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