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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 in the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 In presenting th i s thesis i n p a r t i a l fulf i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library s h a l l make i t f reely avai lable for reference and study. I further agree that permission for extensive copying of th i s thesis for scholar ly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or publ ica t ion of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada SUPERVISOR: PROFESSOR Z. A. MELZAK ABSTRACT i i . This thesis is concerned with the mathematics of the design of parallel-plate equivalents of optical systems, in 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 is constant throughout. Consider the mean surface M , which is the locus of midpoints of the double normals, and suppose that microwave radiation is fed into the region between the plates. If M is not excessively curved the rays travel along its 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 C t h u s t h e system is equivalent to a parabolic mirror). It is assumed that M consists of two flats connected by a focusing bend B , where B is part of a tubular surface with directrix A . The problem is to determine the curve A so that the geodesies on the tube generated by 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 is given in terms of an implicit non-linear integro-differential equation in r(6) . This equation also proves too involved to solve exactly, but numerical approximations are calculated by two different schemes. One scheme is an analog of Euler's method, and the other is based on Galerkin's method of undetermined coefficients. The problem is so involved that analytical error analysis appears too difficult to handle. The best that can be achieved at this time is to i i i . calculate numerically the deviation of a beam from true parallelism. The results 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 for solving other optical problems are also taken up. i v . TABLE OF CONTENTS PAGE CHAPTER I PARALLEL-PLATE OPTICS 1 CHAPTER II MATHEMATICAL FORMULATION 8 1. The Mean Surface M 8 2. Geometrical Framework 10 3. Geodesies on B 14 4. Exact Formulation 18 CHAPTER I I I ANALYTICAL APPROXIMATIVE METHODS 20 1. Introduction: Osculating Tubes 20 2. Zeroth Order Approximation: Neglecting the Bend 22 3. F i r s t Order Approximation: the Osculating Cylinder 23 4. Second Order Approximation: the Osculating Torus 27 5. Higher Order Approximation 34 CHAPTER IV NUMERICAL APPROXIMATIVE METHODS 36 1. Introduction .36 2. Piecewise Osculating Polynomials 37 3. Galerkin's Method 42 CHAPTER V ERROR ANALYSIS 48 CHAPTER VI GENERAL SYNTHESIS OF PARALLEL-PLATE SYSTEMS 51 1. Simple Systems 51 2. Complex Systems 54 3. Three-Dimensional Systems 56 CHAPTER VII 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 author wishes to thank Dr. Z.A. Melzak for suggesting the topic of t h i s t hesis and for h i s excellent guidance during i t s preparation. Dr. F.G.R. Warren of the RCA research group i s also thanked f o r h i s thorough reading and constructive c r i t i c i s m of the manuscript. F i n a l l y , the f i n a n c i a l support of the Department of Mathematics i s appreciated. CHAPTER I PARALLEL - PLATE OPTICS This thesis i s concerned with the mathematics of the design of parallel-plate equivalents of optical systems, in particular with the parallel-plate equivalent of the parabolic mirror. A parallel-plate microwave system consists of a pair of metal plates S^ and S 2 , not necessarily plane, which are p a r a l l e l in the sense that they share common normals at every point, and the normal separation d i s constant throughout (see Fig. l a ) . To ensure a lower bound on the principal curvatures i t must be possible to move a sphere of radius d/2 freely between the plates so that i t always touches both plates, 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 (Fig. l b ) , and suppose that microwave radiation of wavelength X i s fed into the region between the plates. It i s assumed that d < A/2 so that only the TEM mode can propagate, with the same velocity as in free space and unit-index of refraction. Note that systems with d > X/2 , where other modes may propagate with non-unit indices, can also be designed, If the principal r a d i i of curvature of M exceed 1.5 to 2.0 times X , the radiation propagates along the geodesies of M and we are in the domain of geometrical optics, for the shape of M determines the focussing properties of the system. Parallel-plate systems are also known as geodesic lenses. FIGURE 1 3. A p a r a l l e l - p l a t e system i s sa i d to be equivalent 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 given system. For example, t h e p a r a l l e l - p l a t e equivalent of t h e p a r a b o l i c mirror must s a t i s f y the condition that a l l rays emanating from a f i x e d point Cthe focus) which enter the system, emerge i n a p a r a l l e l beam. An equivalent condition i s that the t o t a l o p t i c a l path length from the focus to a f i x e d l i n e Cor plane) be constant. A n a t u r a l problem a r i s e s now of designing the p a r a l l e l - p l a t e system, i n p a r t i c u l a r of shaping M , so as to achieve a pe r f e c t o p t i c a l system. Such a system s a t i s f i e s two conditions: 1) There i s a one - to - one correspondence between object points and image po i n t s . 2) C o l l i n e a r object points correspond to c o l l i n e a r image points, and conversely. In p r a c t i c e i t i s impossible to achieve a pe r f e c t system, f o r the image of a point i s a "patch" of f i n i t e s i z e which i s a combination of c e r t a i n aberrations. For any o p t i c a l system an aberration function i s defined and expressed i n a power s e r i e s , each term of which represents the magnitude of one of these aberrations (see [1]). In the p r a c t i c a l design of an o p t i c a l system we are guided by the d e s i r e to eliminate c e r t a i n aberrations. We s h a l l now examine how t h i s problem has been attacked i n the case of r e f l e c t i n g o p t i c a l systems, and how i t s s o l u t i o n can be applied to the design of p a r a l l e l - p l a t e systems. We consider r e f l e c t i n g o p t i c a l systems which are r o t a t i o n a l l y symmetric, noting that t h e i r p a r a l l e l - p l a t e equivalents w i l l be a x i a l l y symmetric but not rotationally symmetric. It can be shown that for such optical systems, the even-order terms in the aberration series must vanish. The f i r s t three odd-order terms, in order of decreasing magnitude are known as primary spherical aberration (1st order), primary coma (3rd order), and astigmatism (5th order). A spheroidal mirror suffers from a l l of these, primary spherical aberration.being most serious, for no parallel beam of rays focuses at a single point. A one-mirror system free of primary spherical aberration i s the paraboloid, which s t i l l suffers from primary coma and higher order aberrations; only one family of parallel rays exists which focuses at a single point. The problem of the design of reflecting systems free of both primary spherical aberration and coma was f i r s t solved by K. Schwarzchild in 1905 (reference [5]). His solution consists of two rotationally symmetric mirrors which are combined to give a system free of f i r s t and third order aberrations (see Fig. 2, where rays enter from the l e f t ) . Here there are i n f i n i t e l y many families of parallel beams, restricted within a f i n i t e angular range of the axis of symmetry, each of which focuses at a point on the focal sphere. 6. With t h i s s o l u t i o n i n mind we define a Schwarzchild System  of order k as a system of k m i r r o r s , the combination of which i s free of aberrations of orders 1 to 2k - 1 , i n c l u s i v e (see F i g . 3). A paraboloid i s a system of order 1, and Schwarzchild's o r i g i n a l system i s r o f order 2. Note that systems of order 3 or greater, being s i n g l e - l e v e l systems, are 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 are concerned because they are 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 eliminated for k = 2 by using a portion of the front mirror.as i n a Cassegnanian system). This i s not true 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 equivalents of these systems, f o r the mean surface of the Schwarzchild system of order k can be constructed as follows: Each mirror i s replaced at a d i f f e r e n t l e v e l by i t s p a r a l l e l -p l ate equivalent, and these are joined i n the proper sequence by planes (see F i g . 4 where k = 2). The major feature here i s that such systems, being m u l t i - l e v e l , are not s e l f - o b s t r u c t i n g . The Schwarzchild system of order k i s proposed here f o r two reasons. F i r s t , i t i s important i n i t s e l f , f o r i t provides the kth order approximation of a perfect o p t i c a l system, " t h e o r e t i c a l l y " f o r r e f l e c t i n g 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. Second, i t provides an example of how to attack the problem of designing p a r a l l e l -p l ate equivalents of "complex" o p t i c a l systems: i f the given o p t i c a l system can be r e a l i z e d as a combination of s i n g l e systems (sing l e mirrors or lenses), then the general problem i s reduced to the simpler one of designing the p a r a l l e l - p l a t e equivalents of simple systems. : Another important feature of p a r a l l e l - p l a t e systems i s that while 7. these systems have act u a l indices of r e f r a c t i o n of unity, by shaping M 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 of r e f r a c t i o n . Thus the problem of the design of p a r a l l e l - p l a t e equivalents of v a r i a b l e index systems i s concerned only with the problem of shaping the mean surface so as to achieve the equivalent e f f e c t i v e index of r e f r a c t i o n , while avoiding the problem of d i e l e c t r i c materials. The s t a r t i n g place, then, f o r the design of p a r a l l e l - p l a t e systems i s with the p a r a l l e l - p l a t e equivalent of the parabolic mirror, the subject of t h i s work. Not only i s t h i s system important i n i t s e l f , but the mathematical methods developed for i t s design can p o s s i b l y be employed i n the design of other systems. It may seem odd that a problem as simple as the p a r a l l e l - p l a t e equivalent of the parabolic mirror appears as yet unsolved. Although the problem i s simple, the s o l u t i o n i s involved, and even the exact mathematical formulation i s ghastly, because p a r a l l e l - p l a t e systems, with the exception of systems with r o t a t i o n a l symmetry Ce.g. the Tin Hat [2], [7 J ) are i n general d i f f i c u l t to handle mathematically. The mathematical f a c t behind t h i s i s that the only general c l a s s of surfaces whose geodesies can be expressed by e x p l i c i t formulae i s the c l a s s of L i o u v i l l e surfaces, which contains the c l a s s of surfaces of r e v o l u t i o n . CHAPTER II THE MATHEMATICAL FORMULATION 1. The Mean Surface M We now take up the problem of the design of the p a r a l l e l - p l a t e equivalent of the parabolic mirror, where we assume that the mean surface M c o n s i s t s of two p a r a l l e l f l a t s M^ and M.^ connected smoothly by a focussing bend B. An important consideration i s the choice of a family of surfaces f o r the focussing bend B , noting that surfaces of r e v o l u t i o n cannot be used (see [3]). There are three requirements which guide us i n our choice: One, the mechanical requirement of ease of construction of the p l a t e s S^ and (Fig. l a ) . Two, the electromagnetic requirement that the p r i n c i p a l r a d i i of curvature be bounded below-'so as to avoid severe r e f l e c t i o n s and standing waves. Three, since we expect to be unable to express the geodesies of B as e x p l i c i t formulae, we have the mathematical requirement that B be s u f f i c i e n t l y approximable by surfaces of r e v o l u t i o n , whose geodesies (solvable by e x p l i c i t formulae) w i l l approximate the geodesies of B. These requirements are s a t i s f i e d exactly by the family of tubular surfaces. A tubular surface, also known as a canal surface of constant radius, i s the envelope of the family of a l l spheres of constant radius a , whose centres l i e on a d i r e c t r i x curve A . For r e g u l a r i t y and to s a t i s f y requirement two above, we assume that the radius of curvature of A exceeds a at every point. In our case A i s a plane curve and B i s a p o r t i o n of the tube obtained by taking spheres of radius a centred on A , while the 9. bends of and are s i m i l a r l y obtained by taking spheres of r a d i i a^ and a.^ » r e s p e c t i v e l y . The normal separation i s d = a^ - a^ , (a + a ) and a = 1 2 . Our problem then, i s to determine the curve A so that a l l geodesies on M which o r i g i n a t e at F i n and pass over B , e x i t p a r a l l e l to each other i n . 1 0 . 2. Geometrical Framework Let the mean surface M l i e i n the rectangular Cartesian coordinate system as shown i n F i g . 5a, With the o r i g i n midway between the leading edges of and . The d i r e c t r i x A , which i s c l e a r l y symmetric, l i e s i n the x - y plane with the y-axis as axis of symmetry and as polar a x i s , where r = r(6) i s the polar equation of .A (see F i g . 5b which i s a top view of M). If P i s any point on B then the plane normal to A which contains P cuts B i n a s e m i - c i r c l e £ of radius a whose centre C l i e s on A (see F i g . 5c). The point C has polar coordinates (0, r(0)) and P has l a t i t u d e $ on £ so that the p a i r (0,<j>) determine P uniquely and w i l l be used as our coordinate system on B . The curves 0 = constant are s e m i — c i r c l e s and the curves <j> = constant are plane curves p a r a l l e l to A . In p a r t i c u l a r the curves A^ and determined by <j> = -ir/2 and cj> = TT/2 , the lower and upper rims of B , are the copies of A which terminate the f l a t s and so that A l i e s half-way between these. One reason for choosing t h i s coordinate system i s so that the two f a m i l i e s of coordinate curves are orthogonal, which s i m p l i f i e s somewhat the analysis to follow. A geodesic T s t a r t i n g at F with polar angle 0^ i s shown i n F i g . 6 , where n^ and TI^ are normals to A^ and . T follows a s t r a i g h t l i n e i n , cuts A^ at P^ with angle of incidence a , e x i t s from B c u t t i n g at with angle of e x i t 3 , and leaves the system v i a a s t r a i g h t l i n e i n M„ . F I G U R E 5 12. There are two equivalent focussing conditions f o r a p a r a l l e l beam, angle focussing condition denoted by AFC, and length focussing  condition, denoted by LFC. AFC i s given by (1) Q1 + y = a + 3 > > where y i s the angle between n^ and n^ . LFC i s given by (2) k = r ( 9 1 ) + r C 6 2 ) c o s ( 6 2 ) + s where i s the polar angle of P 2 , s i s the arc length of T on B(from r = - ir/2 to TT/2) , and k i s a constant. LFC expresses the condi t i o n that the t o t a l o p t i c a l path length 'from F to the upper edge of M 2 i s constant. For 0^ = 0 we have 9 2 = 0 so that k =• 2r(P) + ira .' We note that i n general AFC i s more s e n s i t i v e to data perturbations than LFC , and thus provides a more accurate estimate of e r r o r . To see t h i s , r e f e r to F i g . 7 where E i s any o p t i c a l system, L 2 i s the o p t i c a l path length from E to the f o c a l plane, L^ i s the length from E to the f o c a l plane f o r a perturbation i n d i r e c t i o n , AI{J i s the corresponding angle increment and AL the length incriment (AL = |L^-L 2|) . Since 2 2 cos Ai|> = 1 - — 2 " — we have AL = L^ — 2 — . Thus a small error i n angle corresponds to a much smaller error i n length, 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 rectangular Cartesian coordinates of a point 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 s i n $ . A geodesic on B i s a curve T = (x,y,z) where 0 and <j> are r e l a t e d by the following d i f f e r e n t i a l equation (see [6]): d ^ 22 V" T / V 12 22' r 2 i i The..symmetric ..Chr.istof.fel .symbols T are functions of the i j c o e f f i c i e n t s E, F, G of the f i r s t fundamental form of B , d s 2 = E d<{>2 + 2 F d<f> d0 + G d0 2 where E = x 2 + y 2 + z 2 , G = x 2 + y 2 + z 2 <p <p <p O D D and F = 0 since the coordinate curves are orthogonal. To evaluate E and G we c a l c u l a t e the p a r t i a l d e r i v a t i v e s : x. = - a s i n <f> sin(0-a) x Q = r ' s i n 0 + r cos 0 + a cos <f> cos (0-a) (1-a') y, = - a s i n (f i cosC0-a) y Q = r' cos 9 - r s i n 0 - a cos r sin(G-a) (1-a') <p 0 z, = a cos <j> z. = 0 <J) D where prime (') denotes d i f f e r e n t i a t i o n with respect to 0 . E i s e a s i l y evaluated: 2 2 2 2 2 2 2 2 2 E = a s i n <f> s i n (0-a) + a s i n <j> cos (0-a) + a cos <j> = a 15. To evaluate G we note that tan a = r ' / r and the polar equation of the radius of curvature R, of A, i s . 2 . ,2,3/2 R = Cr + r ' ) ( r 2 + 2 r ' 2 - r r " ) We also f i n d that 2 2 1 - tt. • Cr + r' > R We have r 2 -u - 2 -•. 2 2 A ( r 2 + r ' 2 ) G = r + r ' + a cos cj> — -z R ( r 2 + r ' 2 ) 1 7 2 + 2 a r f [ s i n 0 cos <J> cos(0-a) - cos 0 cos <f> sin(0-a)] K ( r 2 + r ' 2 ) 1 / 2 + 2 a r - [ cos 8 cos <f> cosC0-a) + s i n 0 cos <j> sinC9-a)J K and the bracketed 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 . Further s i m p l i f i c a t i o n gives ,2 :os <j>; symbols (see [6]) and get 1 1 2 r = r = r = o n Li2 l n u ' „1 • x t 2 , ,2. (R + a cos <j>) T 2 2 = s i n <j) (r + r 1 ) - — ^ ~ a R „ , 2 , , 2 v ( R + a cos <f>) X T , „ , . , . , . , G = (r + r ' ) 2 ^ — . Next, we compute the C h r i s t o f f e l p2 _ - a^ s i n (j> 12 (R + a cos <{>) r 2 _ Crr 1 + r ' r") a cos <|> R' 22 , 2 , ,2. RCR + a cos <j>) Cr + r ' ) Y 1 6 . Thus f o r the geodesic r on B , C3) becomes C4) d 29 _ s i n 9 Cr 2 + r ' 2 ) (R + a cos 9) / d Q \ 3 2 2 d<j> a R (dj) + (a cos 9 R' ( r r ' + r ' r " j \ /djV RCR + a cos 9) " ^2 + r , 2 } J [d$] | 2 a s i n 9 (dj) (R + a cos 9) The complete course of T on B i s determined when $ traverses the f u l l range - TT/2 to IT/2 . The i n i t i a l conditions to which (4) i s subject are determined as follows: Since a ray of r a d i a t i o n making angle 0^ with the polar axis cuts the lower rim A^ we have (5) 0( - TT/2) = Q1 . The tangent t to T at t h i s point has d i r e c t i o n (s i n 0^ , cos 0^ , 0) . Hence t o ( s i n 0^ , cos 0^ , 0) = 1 . Now t = -rr I I I |-rr| |1 at 9 = - TT/2 where dr/dtf) i s c a l c u l a t e d to be (r* s i n 0 n 4r + r cos 0.. 4r + a sin(0., - a) , 1 d9 1 d9 1 1 & d 0 . . d0 , * . r cos -rr - r sxn 0 n - r — + a cos(0.. - a) , - a) . 1 d9 i d9 1 Upon performing the indicated dot product, equating to unity and s i m p l i f y i n g we f i n d ( 6 ) d0_ d<{> = a tan a 9 = - TT/2 . 2 , ,2,1/2 (r + r ' ) 17. where r = r ( 9 ^ ) etc . . Let 6 2 denote 9(jr/2) . Solving (4) subject to (5) and (6) determines as a function of 9 ^ and a functional of r = rC9): e 2 = f(rce) , e1) . 18. 4. Exact Formulation Referring to F i g . 6 we have y = y(Q^,6^) the angle between the normals n^ and ri , s = s(0^,0 2) t^ i e a r c l e n g t h of T on B from <j> = - TT/2 to TT/2 , and g = 6 ( 6 2 ) the angle of e x i t of T . The exact formulation of (1), AFC, then i s (7) e1 + y ( e 1 , e 2 ) = acep + e(e2) where 6 2 = f(r(0),8^) i s determined by (4) subject to C5) and (.6). S i m i l a r l y ( 2 ) , LFC, i s (8) 2 r ( 0 ) + T 7 a = rC©^ + rC© 2) cos 0 2 + s(3 tB2) . These can be rewritten i n the following way: Let Gr = 0 1 + Y ^ , ^ ) - a (8.^ - BC82> and Lr = riQ^ + r(0 2) cos 0 2 + sC8 1 }0 2) - 2r(0) - 77a . Then (7) and (8) become, r e s p e c t i v e l y , Gr = 0 and Lr = 0 . In p r i n c i p l e , s e t t i n g Gr or Lr equal to zero determines r = rC8) . This, though rigorous, i s not s u i t a b l e f o r sol v i n g f o r r(0) , due to the f a c t that r enters f i n a very complicated way, and that f i s a f u n c t i o n a l of r rather than a function of r . In p a r t i c u l a r , f does not have a l o c a l dependence on r and i t s d e r i v a t i v e s , but depends on the change i n r from <f> = - ir/2 to 7 7 / 2 ; t h i s change i n r , moreover, depends v i a the focussing conditions on the f u l l course of T over B, which i n turn depends, v i a C4) whose c o e f f i c i e n t s are unknowns, on r(8) . I t appears that Gr = 0 and Lr = 0 are i m p l i c i t f u n c t i o n a l d i f f e r e n t i a l equations of a type not previously studied ( t h i s aspect i s discussed i n Chapter V I I l . Lt i s therefore necessary to attack the 19. problem by approximative methods, the subjects of the next two chapters. Note that f o r convenience we set r(0) ='1 since the focussing properties are independent of s c a l e . CHAPTER I I I ANALYTICAL APPROXIMATIVE METHODS 1. Introduction: Osculating Tubes Due to the ghastly nature of the preceeding we are forced to develop approximative methods to determine r = r(6) . Our plan of attack i s as follows: approximate A l o c a l l y by a simpler curve ft so the tube T^ with d i r e c t r i x ft approximates B l o c a l l y . T^ i s the os c u l a t i n g  tube of B_ . This approximation s i m p l i f i e s the way i n which r enters f so that Gr and Lr become simpler. We are guided i n our choice of ft by the requirement that the geodesies of T^ be expressible by e x p l i c i t formulae, since these approximate the geodesies of B , so that f w i l l have a l o c a l dependence on r and i t s higher d e r i v a t i v e s . This scheme i s -analogous to that employed i n the l o c a l approximation of curves: a curve i s approximated l o c a l l y to the 0-th order by a point, to the 1st order by i t s tangent l i n e , to the 2nd order by i t s osculating c i r c l e and so on. The tubes with these as d i r e c t r i c e s provide us with corresponding 0-th, 1st, 2nd, ... order approximations to B : the 0-th order approximation to B , with a point as d i r e c t r i x , i s the osculating  sphere of B ; the 1st order approximation to B , with tangent l i n e of A as d i r e c t r i x i s the osculating c y l i n d e r of 13 ; and the 2nd order approximation to B , with the oscu l a t i n g c i r c l e of A a s . d i r e c t r i x i s the osc u l a t i n g torus of B_ . Geodesies on these three surfaces can be expressed as e x p l i c i t formulae (elementary functions or e l l i p t i c i n t e g r a l s ) , and as we s h a l l see, f has a l o c a l dependence on r . Such approximations to B give r i s e to simpler equations G^r = 0 and L^r = 0 , whose solutions approximate the solutions of Gr = 0 and Lr = 0 . 21. With the exception of the 0-th order case, however, we were unable to so l v e G.r or L r e x a c t l y . Numerical methods f o r approximating s o l u t i o n s A A to these approximations are developed i n Chapter IV, and references to the numerical r e s u l t s s h a l l be made throughout t h i s chapter. 22. 2 . Zeroth Order Approximation: Neglecting the Bend For zeroth- order approximation we should replace A l o c a l l y by a point, and B by the tube whose d i r e c t r i x i s that point, namely a sphere of radius a . However, i n t h i s case, the geodesies on the sphere are great c i r c l e s and i t i s easy to see that a ray w i l l simply be reversed i n d i r e c t i o n so that AFC , for example, reduces to G^r = 6^  = 0 . Hence there i s no s o l u t i o n i n t h i s case. Such an approximation, then, i s too poor to achieve any r e s u l t s . Instead, a zeroth order approximation can be obtained by simply neglecting the bend B (see F i g . 8). This amounts to r e f l e c t i o n , so that 2 (7 ) becomes 6. = 2 a whose s o l u t i o n i s r(0) = -7=—; > a n c * 1 (1 + cos 0) 2 + Tta (8) becomes r(0) ~ •+ c o s 9) • Both solutions are., of course, parabolas (see F i g . 10). 23. 3. F i r s t Order Approximation: The Osculating Cylinder F i r s t order approximation i s achieved by replacing A locally by i t s tangent line T and B by i t s osculating cylinder C of radius a . The geodesies of C are helices, which approximate the geodesies of B (see Fig. 9). Here Y = 0 , a = 3 and s = . cos a AFC in this case is G.r = 8, - 2a = 0 whose solution i s the A 1 2 parabola r(0) = -rz—; „.. The "fai l u r e " of AFC to provide a better r (1 + cos 6) solution than the zeroth order case is due to the fact that y is independent of a , so in particular we could take a = 0 which is tantamount to neglecting B . LFC does not f a i l since s depends on a . Referring to Fig. 9 where k and L are the indicated lengths, we find that the total path length from the focus to the edge of is r(0^) + r(8^) cos 0^  + s - k where k = £ sin (0^ - a) and Z = TT a tan a . Thus Lr is replaced by 7T3. L.r = r(0 1) (1 + cos 0 1) H ira tan a sin (0, - a) - 2 - na . A X -L COS Ot JL Sat ting L^r = 0 , LFC becomes -TTQ (9) 2 + ira = r(0) (1 + cos 0) + — - ira tan a sin (0-a) cos a where we have written' -Q for 0^ . Note that f has a local dependence on r since 6^ is expressible in terms of 0^  , r(0^) and k . Lr = 0 has been reduced to the nonlinear implicit f i r s t order ordinary d i f f e r e n t i a l equation L.r = 0 (recall that tan a = r ' ( 0 ) / r ( 0 ) ) . f 9 8 / FIGURE 8 /IV / / L I I .rt FIGURE 9 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 6 = 0 to TT/2 i s plotted in 2 Fig . 10 along with the parabolas r(9) -,-i—; ^\— anc* • • r CI + cos 0) (2 + ita) r(&) = - y i — : — — r r - . For this particular example we have chosen CI + cos 0) a = .25 , noting that as a increases so does the "spread" of the solution. 27. 4. Second Order A p p r o x i m a t i o n : The O s c u l a t i n g Torus F o r second o r d e r a p p r o x i m a t i o n t o B we r e p l a c e A l o c a l l y by i t s o s c u l a t i n g c i r c l e w h i c h has t h e same t a n g e n t and c u r v a t u r e v e c t o r s as A 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 o f £ i s R , t h e r a d i u s o f c u r v a t u r e o f A h e r e . T h i s o s c u l a t i n g c u r v e £ t h e n s e r v e s as a new l o c a l g e n e r a t o r f o r t h e t u b e T w h i c h i s a t o r u s . B i s r e p l a c e d l o c a l l y by T , whose m i n o r r a d i u s i s a , and t h e g e o d e s i e s o f B a r e a p p r o x i m a t e d by t h e g e o d e s i e s o f T ( s e e F i g . 1 1 ) . The g e o d e s i e s o f T a r e f o u n d by s e t t i n g R(9^) = r ( 9 ^ ) = p ; p c o n s t a n t , and s h i f t i n g T so t h a t i t s c e n t r e i s a t t h e o r i g i n . The d i f f e r e n t i a l e q u a t i o n (4) becomes 2 3 / • i n i d 9 _ s i n 9 (p + a cos 9 ) / d8\ 2 a s i n 9 /d8\ L 1 2 a \d$) (p + a cos 9 ) \d$J ' = id*) To s o l v e t h i s s e t y = l - r r l and t = p + a cos 9" so t h a t (10) becomes a u d y _ * x j | t _ _ ^ _ d t = 0 m . fr a -4 M u l t i p l y i n g (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 t we f i n d t h a t ' 4 1 -2 i t becomes d ( y / t ) - d ( t ) = 0 so t h a t by t h e d e f i n i t i o n o f a t t h e s o l u t i o n o f (10) i s 28. where c i s a constant i n d i c a t i n g the angle of incidence of the geodesic. To f i n d c note that from (6) d6| d<H<j> and from CI2) a tan a TT/2 c a — I = p/p - c Equating, we f i n d that c = p s i n a , or since p = R , c = R s i n a . For the geodesic on T , by symmetry a = $ , and the qua n t i t i e s Y and s are given by Y = TT/2 -TT/2 d9_ d<|> d<j> = a R s i n a TT/2 d<j> —TT 12 CR + a cos <$>) /CR + a cos <J>) - R s i n a s = TT/2 -TT/2 ds d<j> d<|> = a TT/2 C& + a cos <j>) d$ -TT/2 / R + 2 2 2 r/2 CR a cos <}>) - R s i n a 2 2 2 2 2 where ds = a d<j> + (R + a cos <j>) d6 i s the f i r s t fundamental form of T . Note that under the i n t e g r a l s R and a are independent of <J) 3 0 . AFC then is G.r = 0 + a R sin a A TT/2 ,, d± - 2 a / R T - 2 ~ 2 - 1 -ir/2 CR + a cos <j>) (  + a cos $) - R sin a and equating G^r = 0 gives C13) 6 + a R sin a d j . = 2 a / R T - 2 " 2 • 2 CR + a cos <j>) C  + a cos <t>) - R sin a —TT/2 An expression for LFC , derived in a way similar to that for the cyl i n d r i c a l case (see Fig. 11), i s L A r = r C l + cos 9) + s - 2 R sin (Y/2) s inCy/2 + 8--<*) - '2 - v * . Equating L^r = 0 gives C14) 2 + ira = r C l + cos 9) + s - 2 R sin (y/2) s inCy/2 + 0 - a) . The equations Gr - 0 and Lr = 0 have been reduced to the implicit non-linear integro-differential equations G r = 0 and Pi. L^r = 0 , since f now has a local 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. One method requires the i n i t i a l values r (0 ) , r ' ( 0 ) and r " ( 0 ) . Since we previously set r C O ) = 1 , and since r ' C O ) = 0 by symmetry, we need only calculate r "C0) . This is accomplished by differentiating Q-3) w i t h respect to 0 and setting 9 = 0 , so that a = 0 and we obtain 3 1 . (15) 2(1 - k = 1 + a(R (1 - | l ) TT/2 d(J) 2 , (R + a cos <j>) -TT/2 Although i t was possible to express the i n t e g r a l i n t h i s equation i n terms of elementary functions, the r e s u l t i n g equation was too involved to solve exactly. I t was, however, easy to solve numerically; with a = .25 , r"(0) = 1 - R ^ Q N = 0.58415 to f i v e places. The c a l c u l a t e d approximations to (13) and (14) are p l o t t e d i n F i g . 12 along with the zeroth and f i r s t order s o l u t i o n s . In a d d i t i o n , the solutions to (13) with a = .25 and a = .50 are shown i n F i g . 13 for comparison. As expected, the spread of the d i r e c t r i x increases with a . FIGURE 13 3 4 . 5. Higher Order Approximations A natural problem now i s to continue with the above process: approximate A to the 3rd or higher order by a curve Q and approximate B by the tube with d i r e c t r i x . To examine t h i s problem we f i n d i t convenient to express (3) i n the form of two simpler equations (see[6]): a« $ " i 1 ( S ) ' + 2 r ^ + r l 1 ^ y - . <"> £ f + r Mi) 2 + -Mf)(f!) + r M£) 2 = ° 2 2 2 where ds = E d<j> + 2 F d<j> d9 + G d6 . In our case (16) and (17) are ,2. „ x ,1 _,2 d s ) ±4 ds + a s i n (j> ( r 2 + r ' 2 ) (R + a cos <{>) [de\ 2 = a 2 R 2 W (19) dT6 - 2 a s i n <j) ,2 (R + a cos ds ldQ\ /d£\ f ( r r ' + r ' r"] • ) \ d i ) i d s / l < r 2 + r . 2 > a cos <f> R (R + a cos  <!>)J |ds/ = 0 The equation (4) i s equivalent to (18) and (19) ( ( 4 ) i s obtained from them by eliminating ds ) . 35. Expressing (4) i n t h i s way allows us to see where the d i f f i c u l t i e s of s o l v i n g i t w i l l l i e , f o r higher order cases: l e t t = R + a cos 9 , so (19) i s (20) de d ds d9 ds 2 dt . d ( r 2 + r ' 2 ) ( r 2 + r ' 2 ) dR - dR ( 2 " R ) = 0 Each term of (20) can be integrated, except - — ', and i t i s t h i s term which renders (4) so d i f f i c u l t . For example, i n the t o r o i d a l case R i s constant so that t h i s term vanishes, rendering (20) and hence (4) easy to solve. Similar things happen with the c y l i n d r i c a l case a l s o , of course. To be more pr e c i s e , that term w i l l vanish only for d i r e c t r i c e s of constant curvature (giving c y l i n d e r s & t o r i ! ) , so since any 3rd or higher order approximation to A w i l l have non-constant curvature, the problem of solving (4) for such curves i s of the same order of d i f f i c u l t y as our exact formulation of the problem. We conclude then that higher order approximations are unfeasible at t h i s time. CHAPTER IV NUMERICAL APPROXIMATIVE METHODS 1. Introduction Since we were unable to solve (9), (13) or (14) exactly, numerical methods for approximating solutions to these approximations had to be developed. In what follows reference shall only be made to solving (13) since (9) and (14) are solved similarly, and as mentioned previously, AFC should yield the more accurate solution. Two distinct methods were developed, one a "local" method in 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 coefficients. Since error analysis for this problem appears too.„involved to tackle -analytically (see Chapter V), having two distinct methods provides some guide as to the accuracies of the numerical solutions. 37. 2. Piecewise Osculating Polynomials The l o c a l method consists of p a r t i t i o n i n g the domain of r(9) , [0, TT/2]., by the following mesh of N equally spaced arguments: 61 = 0 , 0 2 = TT/2(N - 1) , ... , 6 N = TT/2 . By the symmetry of A the domain [- TT/2, Oj need not be considered. Herein, r(9) denotes the exact s o l u t i o n to G^r = 0 ; remember that t h i s r i s an approximation of the exact s o l u t i o n to Gr = 0 . Let p ( 6 ) denote the numerical approximation of r ( 0 ) which we are about to c a l c u l a t e (p(6) i s defined only for 9 = 9^ ; i = 1 , ..., N) . -The i n i t i a l values r ( j ) (0) , (j = 0 , 1, 2) , are known, so define p ^ (0) = r ^ (0) . Assume i n d u c t i v e l y that p ^ (6 ) have been c a l c u l a t e d f o r l < i < K < N , ( j = 0 , l , 2 ) . To c a l c u l a t e P C J ) ( 6 K + ± ) , (j = 0 , 1, 2) , we extrapolate pC8 K + ±) and p ' ( 9 K + ± ) by the method described below, set r = p(9„ ..) and r ' = p ' ( 9 v • n ) K "r* X iv "T X i n (13), and solve (13) numerically f o r ^(9^. + -^ ) from which P"C6j, + }^ i s extracted. This "corrected" value of j > " - ) i s then used with K + X . p(8„ I T ) and p'(9 .. ) to proceed. Note that i n (9) only p(9 , , ) K * r l K + X K . + X i s extrapolated. To extrapolate p(9 , •,) and p ' ( 0 „ , ) an o s c u l a t i n g K . T X K + X polynomial P (0 ) of degree 2 i s c a l c u l a t e d which agrees with p ^ ^ ( 0 ) K. f o r j = 0, 1, 2, and p ( 0 v , ,) i s set equal to P(9 T, . , ) and K . + X K + 1 p ' ( e K + 1 ) i s set to P ' ( 0 K + 1 ) . 38. In passing we note that higher order osculating polynomials were t r i e d with no s u c c e s s , f o r values of p and p' were extrapolated which rendered (13) unsolvable. The extrapolation was improved by applying a form of Richardson's method of extrapolating to the l i m i t (see [7]). This was accomplished by p a r t i t i o n i n g [8 , 6 1 ] into 4 sub-intervals of length h/4 , where K K *r JL h = 8„ , , - 0„ . Improved approximations p(0„ , ,) and p'(9 ) were obtained from a sequence of piecewise osculating polynomials at 6V, Qv + h/4 , Qv + h/2 , 0., + 3h/4 , and 0„ L . i n the following way: Suppose that there e x i s t s a continuous l o c a l l y exact s o l u t i o n to (13) i n a neighborhood of &v which contains 8 , . . Denote t h i s s o l u t i o n by s(0i . We w i l l c a l c u l a t e approximations to s(8 .. ) and K. T i s'(0 ..) , and these w i l l be the p's here. K + J. A sequence of second degree piecewise osculating polynomials i s constructed as follows: Let P^(0) be the osculating polynomial which agrees with p ^ (0) at Qv f o r j = 0, 1, 2 . Set p. (0) = P. (0) , extrapolate p. ^ (9„ + h/2) for j = 0, 1, and solve (13) for p"(Bv + h/2) . Then I K I K c a l c u l a t e the polynomial P 2(8) which agrees with p ^ p ( 8 ) , for j = 0, 1, 2, at 0 K + h/2 . F i n a l l y , l e t p 2(8) = P 2(0) . Next, extrapolate pp^ (0 R + h/4) , for j = 0, 1, solve 39. (13) for P''C9TT + h/4) and calculate the polynomial P o(0) as before. Let P 3(6) = p 3 ( 6 ) a n d extrapolate p ^ C8R + h y / 2 ^ f o r J = °> 1 a n d solve for P-jt^ + h/2) . This determines P^CS) which i s used to calculate p (6 + 3h/4) and f ina l ly we obtain P^C^) which agrees with p 4 ( j ) C6) at 6 K + 3h/4 for j = 0, 1, 2 . Let p3C9) = P5C6> , and suppose that this process was continued at 8^ + h/8 etc. . In this way we obtain the sequences ^ 6 K + 1 > P3 C 6 K + 1> X l = P l ( 6 K + 1> X 2 = P 2 ( 9 K + ^ X 3 = P 3 C 6 K + 1> The assumption is made that x —^ sC9„ , ,) and x —^s C ^ , ) n K T I n . K. *r 1 worst asymptotially since this is known to be true for polynomials of degree one. This assumption is verif ied Cnumerically) la ter . What this means is that q l q l X i " a o + a l ( h / 2 i _ x ) + °< h ) • where a = s(9T. , .. ) , a, and q.. are unknown and independent of h. p K + 1 ' 1 ^1 r More p r e c i s e l y t h i s means that as 1 • Neglecting terms of o ( h q ) we have f o r i = 1, 2, 3 and . 6K + 1 " 9K h = X l Z S C 6 K + 1 } + 31 h q l X2 " s C-K + 1 } + a l ( h / 2 ) q l q l X 3 ~ s C 9 K + 1 } + a l 0 1 / 4 5 * This system i s solved f o r S ( 9 K + ^) by observing that i f X2 " X l q l Q = then Q = 2 and a q can be eliminated since X3 ~ X2 1  q l a 1 h x ± sCeR + x) x2 " s C e k + i } = —q = Q-; Q — s o t h a t S ^ K + i } Q x - x Q' xl - x' ^ X 1 __• •! 1__ _ T fl\ \ - J -_ j y and s i m i l a r l y s'C9 K + ^ CQ' - 1) 41. where Q' = * 3 - * 2 We then set p(6K + j) = S0K + ^) and p ' (9 K + ± ) = s' (6 ) , which are improved extrapolated values. The numerical jus t i f ica t ion of the above assumption was obtained by comparing the Q's at different arguments. From the sample l i s t i n g below i t can be observed that the Q's are nearly constant so we conclude that a and q. are (practically) independent of h : 6 Q 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 is contained in the Appendix. 42. 3. Galerkln's Method The previous method i s l o c a l i n nature, for we march from point to point, and use only knowledge of one previous argument. Since the exact problem Gr = 0 i s a global one we should l i k e a method of solving (13) which i s i t s e l f g lobal, even though i t may not provide a successful method of s o l v i n g the exact problem; furthermore we w i l l have something to compare the f i r s t s o l u t i o n to, from which conclusions can be made concerning the accuracies of numerical r e s u l t s . Galerkln's method of undetermined c o e f f i c i e n t s , f o r s o l v i n g ordinary and p a r t i a l d i f f e r e n t i a l equations, i s a global method which proves quite s u c c e s s f u l . This 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 Fy = 0 , i s roughly as follows: Suppose a s o l u t i o n y (x) of Fy = 0 i s sought, on a domain D . We suppose that (H, <, >) i s a H i l b e r t space of functions whose domains contain D , such that y e H ; l e t {g } be a basis for H . ot A Galerkin approximation of y consists of the p a r t i a l sum n oo y (x) = £ a. g.(x) where y(x) = £ a. g.(x) , the g's being basis n i = l 1 1 1=1 1 1 elements. The a^ , for i = 1, ... , n , are determined as follows: For y(x) to s a t i s f y Fy = 0 i t i s necessary that <Fy, g±> = 0 for i = 1, ... oo . 43. In p a r t i c u l a r <Fy , g.> - 0 for i = 1, . . . , n . n l This gives, then, a system of n non-linear equations i n the n unknowns a^, i = 1, . . . , n . In p r a c t i c e the g^' s a r e chosen i n advance so that n y^ (x) = £ a ± &± ^ s a t i s f i e s any i n i t i a l or boundary conditions auto-m a t i c a l l y . Note that i n case t h i s were not true, one of the equations, say <^'yn» = 0 > must be replaced by an equation which takes care of these conditions. In our case Fy = 0 i s the equation G.r = 0 , where nil D = [0, TT/2] , ,and <G^r, g ±> = j G^-Ce) • g^O) -d-9 . 0 n The r e s u l t i n g system of equations, when r (6) = £ a, g.(6) , i s n i = l 1 1 fir/2 G.r (6) • g.(9) d0 = 0 for i = 1 n . J o By the d e f i n i t i o n of G.r the i th equation i s rir/2 [9 + a R s i n a n n TT/2 d<J> -TT/2 5 f^r' Z 2 (R + a cos <j>) /(R + a cos <j>; - R s i n a ~ 2 ^ S i C 0 ) d9 = 0 44. where a and R are, r e s p e c t i v e l y , the angle of incidence and radius n n of curvature f o r the d i r e c t r i x with polar equation r n ( ^ ) • The numerical s o l u t i o n of t h i s system of non-linear equations was accomplished by Newton's method (see Appendix f or reference). In t h i s method the choice of the coordinate functions {g^} i s of paramount importance. They must, of course, be l i n e a r l y independent on [0, TT/2] , but not ne c e s s a r i l y orthogonal, since we only require them to be orthogonal to the operator G^r , and since we r e a l l y don't know what t h i s operator looks l i k e , a b i t of t r i a l and error i s involved. We t r i e d four d i f f e r e n t f a m i l i e s : _2 i ) g-^6) = cos (9/2) (the parabola) g ±(9) = s i n 2 ^ " 1 5(0/2) i = 2, ... , n . i i ) The Fourier cosine s e r i e s on [0, TT/2]: g ±(9) = c o s ( ( i - 1) 9) i = 1, ... , n . i i i ) The even order Legendre polynomials on [0, TT/2] . iv ) g ; L ( 9 ) = c o s " 2 (9/2) g.(9) = 9 2 ( 1 " 1 ) i = 2, ... , n . In cases i i ) and i i i ) , the 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 n's , implying that these 8^ ' s a r e u n s a t i s f a c t o r y . The d i f f i c u l t y here i s probably that, although the g 's are an orthogonal set, they are not 45. orthogonal to G^r . Case i ) worked reasonably w e l l , where, since a^ i s obviously 1, we only had to c a l c u l a t e a^, ... » a n • F o r n = 4 , the c o e f f i c i e n t s are a 2 = .21605 a 3 = -.195489 a. = 1.380398 . 4 It was observed that r^(9) agreed both with r^(.Q) and with the tabulated values obtained by the previous method to 5 places, the tolerance imposed on the equation s o l v i n g routine. Case iv,) .seemed..to work the,,bes.t,,...due to .it s s i m p l i c i t y (again a^ = 1) i n coding and to the f a c t that the a_^  seem to tend to zero, as can be seen i n the following table: n a 2 a^ a^ •' a^ 3 .0113415 .0304101 4 .0485788 .0052103 .0053799 5 .0409563 .0150133 .0005350 .0007852 We stopped the c a l c u l a t i o n s at n = 5 since r^(6) and r^(9) agreed to better than 5 places on the e n t i r e domain ( t h i s i s the usual c r i t e r i o n i n t h i s method). The agreement of r^(8) with the other method's s o l u t i o n was also excellent (4 p l a c e s ) . 46. The d i s t i n c t advantages of the Galerkin method are that we have a formula for r(8) , and the computing time was i n general a f r a c t i o n of that i n the l o c a l method. The program for t h i s method i s contained i n the Appendix. A natural question i s whether or not Galerkin's method works fo r the exact problem Gr = 0 . Since at any stage i n t h i s method we have a formula f o r r(6) , equation (4) can c e r t a i n l y be integrated numerically, and Gr evaluated. This was a c t u a l l y attempted, but with only p a r t i a l success. What occured was that true convergence was not obtained. By t h i s we mean that the aj_'s converged (successive i t e r a t i o n s changed only i n the 12th or higher place) but the equations f T T / 2 J Gr (6) g. (6) d9 = 0 were not within the desired tolerance. Q n I Thus the a ^ ' s converged, but not to the correct values. It was observed, however, that the deviation of a beam from true p a r a l l e l i s m was smaller than that for the s o l u t i o n to (13). We discuss t h i s i n more d e t a i l i n Chapter V. Thus, we did at l e a s t get some improvement of the approximate s o l u t i o n s . The question of why t h i s method did not converge to the true s o l u t i o n i s extremely involved. Several p o s s i b l e reasons are: i ) The inner product which we use may not be the correct one. In f a c t , i t i s questionable, i n the exact problem, [-TT/2 whether Gr • g^ (6) de i s an inner product at a l l , 47. for r(8) is defined on [0, TT/2] and yet Gr(0) requires values of r(0) outside of the domain of n defini t ion. That i s , we define r(0) = £ a. g.(8) i=l 1 1 only for 8 e [0, TT/2] , but when we evaluate Gr(8) at 0^ = TT/2 we require the value of r(®2^ w n e r e ©2 > TT/2 . Thus we extrapolate quite a distance outside of this domain. We make the assumption that r(8) can be written as a linear sum of the . This may not be possible, for a non-linear sum may be required. F ina l ly , the whole scheme, in this case, may be extremely sensitive numerically, i . e . unstable. CHAPTER V ERROR ANALYSIS There are two l e v e l s of error associated with the numerical sol u t i o n s of (13). At l e v e l I i s the truncation error due to approximating B by i t s osculating torus. Level II consists of three e r r o r s . F i r s t , the truncation error due to approximating r(8) , the exact s o l u t i o n of (13), by e i t h e r the osculating polynomial or by the Galerkin approximation rn ( 6 ) . Second, the error incurred by d i s c r e t i z i n g the domain of r (t h i s error f o r the l o c a l method o n l y ) . And t h i r d , there i s the roundoff error present i n any numerical computation. Level I error, then, i s the error between the exact s o l u t i o n to Gr = 0 and the exact s o l u t i o n to G.r = 0 , while l e v e l II error i s the error between the numerical s o l u t i o n s A and the exact s o l u t i o n to (13). The combination of l e v e l I and l e v e l II gives the t o t a l error between the calculated solutions and the exact s o l u t i o n to Gr = 0 . The t o t a l error determines another error, the o p t i c a l e r r o r , which f o r a given ray i s the actual deviation of the ray from true p a r a l l e l i s m . From the p r a c t i c a l viewpoint, the o p t i c a l error i s the most important. We s h a l l l a t e r show how t h i s can be ca l c u l a t e d numerically. Mathematically, the l e v e l I and l e v e l II errors are the most important, l e v e l I being of greatest i n t e r e s t . These er r o r s , while of such importance, seem too d i f f i c u l t 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 or numerically. As mentioned previously, the r e s u l t s of both numerical methods agreed very w e l l . Since these are such d i f f e r e n t methods, we can s a f e l y conclude that the l e v e l II errors are wit h i n reasonable tolerances. I t i s po s s i b l e , at l e a s t , to c a l c u l a t e the o p t i c a l error numerically. I f r (8) i s a Galerkin approximation, (4) can be integrated numerically, 49. and the tangent d i r e c t i o n of a geodesic can be calculated at i t s point Q£-.exit from B . This i s done as follows: Let t be the tangent vector of a geodesic T at the point of e x i t from the upper rim of B . True p a r a l l e l i s m for a ray (geodesic) s t a r t i n g from F at an angle 0^ i s achieved when t o (1, 0, 0) = 0 . We define the o p t i c a l error for t h i s ray as e(0^) = TT/2 - cos ^ (t (0^) o (1, 0, 0)) . This i s the angle between t and the axis of symmetry. The tangent t of Y i s given for 9 = TT/2 . Upon d i f f e r e n t i a t i n g and taking the dot product we f i n d that i / r ' s i n 0 O 4 r + r c o s eo TT ~ a s i n ( 0o ~ "<•>) eCe.) = TT/2 - c o s " 1 . 2 d * 2 d * 2- 2-1 2 d0 ^ ,2 dQZ ^ 2 A Runge-Kutta scheme was used to integrate (4) to c a l c u l a t e 62 a n d dT l 9 - ,12 - 62 • The 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 errors at 10 arguments i n [0, TT/2 ] for r^(9) and the " i n c o r r e c t " Galerkin approximation 2 ( i D to the exact problem (both with coordinate functions 0 ) are tabulated below, the 0's given i n radians and the errors i n degrees. 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 in the toroidal case i s smooth enough for us to infer that i t s maximum occurs at 0 = TT/2 . In the exact case, the error is not smooth, and attempts at smoothing i t out, by performing more iterations to solve the non-linear system, failed to improve matters. Thus we do not know where the maximum error occurs, nor how large i t i s . What we do know is that the integral of the error function remains too large for true convergence in the Galerkin scheme. CHAPTER VI  THE GENERAL SYNTHESIS OF PARALLEL-PLATE  EQUIVALENTS OF OPTICAL SYSTEMS 1. Simple Systems o We now take up the problem of the design of p a r a l l e l - p l a t e equivalents of simple o p t i c a l systems. By simple systems we mean either s i n g l e r e f l e c t i n g surfaces or s i n g l e continuous-index r e f r a c t i n g structures; the p a r a l l e l - p l a t e equivalents w i l l be s i n g l e - l e v e l systems. The o p t i c a l c h a r a c t e r i s t i c s of such systems are assumed to be given i n terms of a focussing function ^(6) where, r e f e r r i n g to F i g . 14, 6 i s the polar angle of a ray s t a r t i n g from some f i x e d point F , and ty(Q) i s the angle at which the ray cuts the normal to the x-axis as shown. In the case of r e f r a c t i n g structures we r e f l e c t the rays as shown i n F i g . 14b. For r e f l e c t i n g systems, ^(8) can be determined by elementary vector algebra, and f o r r e f r a c t i n g structures, from the index of r e f r a c t i o n . The focussing condition f o r the p a r a l l e l - p l a t e equivalent of a simple system whose focussing function i s ^(8) becomes Gr = 0 where Gr = Q1 + *C6 1) + yC'^, 0 2) - a t e p - B(6 2) (see F i g . 15), with 6 2 = f ( r ( 0 ) , 8 ) and the angles a, 3, y as before. Setting Gr = 0 we have (21) 0 1 + ^ ( 6 1 ) + Y(8 1, 0 2) = a(0 1) + 3 ( 0 ^ . The a n a l y t i c a l and numerical methods developed previously can be modified to render (21) simpler. For example the t o r o i d a l approximation gives TT/2 (22) 0 + i|)(8) + a R s i n a _dj_ (|R| + a cos <|>)/(|R| + a cos <|>)2-R2 s i n ^ x -TT/2 FIGURE 14 53. The sign of R depends on the concavity of A ; i t i s p o s i t i v e i f A i s concave down, negative i f concave up. At a point of i n f l e c t i o n R i s i n f i n i t e , and the i n t e g r a l ( i . e . y) i s zero. The absolute value of R i s needed under the i n t e g r a l since the r a d i i of a torus must be p o s i t i v e . The numerical methods appear a p p l i c a b l e also, with minor m o d i f i -cations. P o t e n t i a l troubles with the l o c a l method occur at points of i n f l e c t i o n and places where A i s nearly s t r a i g h t (R i s extremely l a r g e ) . 2 2 At a point of i n f l e c t i o n we merely set r + 2r' - r r " equal to zero to determine r " , but approaching such points may lead to d i f f i c u l t i e s . Another advantage of Galerkin's method i s that i t probably handles such problems with l i t t l e d i f f i c u l t y . 54. 2. Complex Systems We consider complex systems which can be r e a l i z e d as combinations of simple systems. The p a r a l l e l - p l a t e equivalent 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 equivalents of each simple system joined i n the proper sequence. The problem of complex systems i s s l i g h t l y more involved than i t f i r s t appears. As an i l l u s t r a t i o n , consider the problem of the design of the p a r a l l e l - p l a t e equivalent of Schwarzchild's system of order 2. Referring to F i g . 16, i s the bend corresponding to the front mirror ( d i r e c t r i x A^ has polar equation r ^ = r^(0)) , and B 2 i s the bend corresponding to the back mirror (with' = ^ ( 6 ) ) . A ray of r a d i a t i o n s t a r t i n g from F t r a v e l s over , cuts the x-axis with focussing angle if-^ » passes over and again cuts a l i n e p a r a l l e l to the x-axis with focussing angle i j ^ . The methods of s e c t i o n 1 are d i r e c t l y a p p l i c a b l e for c a l c u l a t i n g r ^ = r^(6) . A d i f f i c u l t y a r i s e s with r^ifi) , for we may not n e c e s s a r i l y apply these methods with (see the Figure) , for i n general =j= since the angle of incidence i s not equal to the angle of e x i t i n general. Since = when i s replaced l o c a l l y by i t s os c u l a t i n g torus, we see that r ^ can be used as with simple systems, but more error i s incurred. Presumably, for smooth enough bends t h i s error i s within reasonable l i m i t s , but the problem of the design of complex systems warrants further study. 55. FIGURE 1'6 56. 3. Three Dimensional Systems The p a r a l l e l - p l a t e systems which we have been considering are two-dimensional, i n that 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 better d e s c r i p t i o n might have been "... equivalent to a s l i c e through ..." rather than "... equivalent to ... ." We now examine the problem of the design of p a r a l l e l - p l a t e systems whose tubular part i s generated by a skew d i r e c t r i x ; we c a l l such systems three-dimensional. These systems may not be of p r a c t i c a l use, but they are c e r t a i n l y of t h e o r e t i c a l i n t e r e s t . We are only concerned here with the a n a l y t i c a l approximative analog of the osculating torus. Since A i s not a plane curve, we seek a skew curve V which approximates A , with the proviso that the geodesies on the tube generated by V be expressible i n e x p l i c i t formulae. The. obvious choice f o r V i s the c i r c u l a r h e l i x , which i s the skew analog of the c i r c l e , since i t has constant curvature. Indeed, the osculating h e l i x of A i s that h e l i x which has the same tangent, curvature vector, and t o r s i o n as A at the point of o s c u l a t i o n . In case A i s a plane curve, the t o r s i o n vanishes, and the h e l i x i s a c i r c l e , the oscu l a t i n g c i r c l e of A . We now show that the tubular surface with 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 surface, and that we can express the geodesies as e l l i p t i c i n t e g r a l s . Parameterize the h e l i x as y(v) = (a cos v, a s i n v, b v) , where a and b are p o s i t i v e constants, and 0 < v < 2TT . The tube with d i r e c t r i x y i s T(v, u) = y (v) + A cos u N(v) + A s i n u B(v) . N and B are the normal and binormal vectors of y at v , A i s the radius of the tube (A was a i n previous d i s c u s s i o n s ) , and 0 < u < 2 TT . 5 7 . We f i n d that the c o e f f i c i e n t s of the f i r s t fundamental form of T are E - A 2 dv G = P(u) T (a constant) i s the t o r s i o n of y , t i s arclength on y , and P(u) = T 2 A 2 + (1-A cos u ) 2 . We now transform the c u r v i l i n e a r coordinates so the coordinate curves are orthogonal: Let dv- = dv + A „T f U and du, = du where 1 G(u) 1 G(u) = 1 + A 2 x 2 + A 2 k 2 c o s 2 u . Thus v = v + A2 x . GluT d u , A 2 x 1 _ -l(A + B* tan u = v + — t a n I — s  A ^77 I (1 + B ) 2 2 * A k * 2 2 where B = —=- and A = 1 + A x are constants. 1 + A X The transformed c o e f f i e e n t s of the f i r s t fundamental form are now 2 A 4 x 2 F = A - — — E l A GC U l) F l = o G1 = GCu x) 58. 2 (A2 - x i 2 2 2 2 so that ds = I GTu~T/ ^ U l + G ^ u l ^ ' L e t , K U 1 ) = 1 + k cos ty 2 } 2 2 then ds = A 1 du + G(u 1) dv. which can be shown to be w r i t t e n 2 2 as (U^ + V^) (du^ + dv^ ) which v e r i f i e s the a s s e r t i o n that the tube i s a L i o u v i l l e surface. To c a l c u l a t e the equation of the geodesies on the tube we f i n d the I \ j to be: 1 = d E (u ) 2 1 2 = ' l l rrS r- ' l l *12 '22 2du x E (u^) o d G(u ) r 12 2 d u 1 GO^) 2 r22 = " d G < V w h e r e EC«i) " A " < i ; ( U l ) • 2du x E ( U ; L ) G C u l } Thus the d i f f e r e n t i a l equation (3) which determines a geodesic i s d 2 v, - d G Mv,\ 3 + d E d G d V l " 1 f d v l \ = 2 D U L E 2 E G du, du. du, 1 \ 1/ 1 1 ^T2 L e t t i n g «z. = j \ we have 59. Noting that we a r r i v e at ^ - d (E G" 2 z) G which y i e l d s -1 -2 C - G = E G z , C constant, o o Thus -2 1 d(E G z ) = — - ( G z d E - 2 E z d G + G E d z ) G J C G - G o 3 2 C G - G o 2 2 2 A (1 + k cos u^) SG-2 2 1 A (1 + k cos u^) Now dv = dv + A j_ du and u = u so that i n terms of our o r i g i n a l 1 G(u) 1 coordinates we have du r!CQ G ( u ) 3 - G ( u ) 3 A 2 (1 + k 2 cos 2 u 1 - A if—2—2-r, 11 + k cos (uj dv where G(u) = 1 + A 2 T 2 + A 2 k 2 c o s 2 u . 60. Thus the geodesies can be expressed as e x p l i c i t formulae g i v i n g f ( r , 0) a l o c a l dependence on r . In theory then, we can solve three dimensional problems approximatively as we did i n the t o r o i d a l case. CHAPTER VII IMPLICIT FUNCTIONAL DIFFERENTIAL EQUATIONS In t h i s chapter we formulate what appears to be a new c l a s s of problems, since the problem of the p a r a l l e l - p l a t e equivalent of the parabolic mirror does not seem to f i t into any known category. The d i f f e r e n t i a l equation (4) and the focussing condition which determine a tube B and i t s geodesies are of the general form 6" = F(4>, 8 , 8 ' , r ( 8 ) , r " ' ( 6 ) ) , G r = 9 1 + Y - c t - 3 = 0 where <j> e£- TT/2, , 8 ( 9 ) ejb, TT/2^  , and the i n i t i a l conditions are expressed as 8 . ( - TT/2) = 8 ^ , where 6^ ranges over ^0, TT/2J , and 8 ' ( - TT/2) = 1 ( 8 ^ r C 8 1 ) , r ' ( 8 1 ) ) . Although 9 = 8(<j>) i s an unknown, we are p r i m a r i l y interested i n determining r = r ( 8 ) . We r e s t r i c t ourselves here to second order d i f f e r e n t i a l equations, f i r s t f o r s i m p l i c i t y , and second since these occur i n many p h y s i c a l problems. In general now, (4) i s replaced by 0" = F(<j), 8 , 8 ' , r ( 8 ) ) where <f> e [a, bj 6 ( 9 ) e [c,d] r ( 8 ) = ( r ( 8 ) , r ' ( 8 ) , r " ( 8 ) , r ( n ) (0)) , f o r n > 0 . The i n i t i a l conditions are 8 ( a ) = 6^, where 8 ^ ranges over £c, dj , 6 ' ( a ) = I ( 8 ( a ) , r ( 8 ( a ) ) . In a s i m i l a r fashion the focussing conditions are replaced, 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. where = (e(<j>), 6' (cf>), 9"(<J>)) . . The general problem i s to determine r = r(6) so that G = 0 when 6 = e(<j>) i s a s o l u t i o n to 6" = ¥(<$>, 6, 6', r ( 6 ) ) . The unknown r i s imbedded i m p l i c i t l y not only i n F but also i n G, and G i s a f u n c t i o n a l equation i n r . We c a l l such a problem an i m p l i c i t f u n c t i o n a l d i f f e r e n t i a l equation. The chief d i f f i c u l t y i n such problems i s that i n general 6 = 0(<l>) cannot be solved exactly ( i . e . by e x p l i c i t formulae). We saw that i n our problem, when we were able, as i n the c y l i n d r i c a l case, to replace the d i f f e r e n t i a l equation by one which could be solved exactly, G was reduced to an i m p l i c i t ordinary d i f f e r e n t i a l equation. Also, i n the t o r o i d a l case where we were able to integrate once, G was reduced to an i m p l i c i t i n t e g r o - d i f f e r e n t i a l equation. In order to i l l u s t r a t e the general problem we now give a very simple example, due to Dr. Jon Schnute, i n which an exact s o l u t i o n can be found: Suppose there i s a v e r t i c a l force f ( y ) i n the x - y plane, and we wish to determine t h i s force so that a l l p r o j e c t i l e s s t a r t i n g from the o r i g i n , with a r b i t r a r y i n i t i a l (non-zero) v e l o c i t i e s and (non-vertical) d i r e c t i o n s , w i l l , a f t e r 1 unit of 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 formulation i s as follows: I f the p o s i t i o n at time t i s ( x ( t ) , y ( t ) ) we have y = f ( y ) y(0) = tan i|> 63. y(o) = 0 c l x ( t ) = C t (C const, tan i> = -r— o o C o where i s const). We are to determine f so that y ( l ) = 0 . This f i t s into our general c l a s s of problems where y corresponds to 6 t corresponds to cf> and f corresponds to r . Also F ( t , y, y', f ( y ) ) = f(y) and G =0 i s y ( l ) = 0 . We observe that y = c^ si n ( u t) s a t i s f i e s the desired i n i t i a l and boundary condition, where to = TT/2 + n"TT. Thus we f i n d the force to be f (y(t)) = - a)2 y ( t ) . Obviously, we need not r e s t r i c t ourselves to second order equations, nor to functions of one v a r i a b l e . However, due to the complexity of the formulation of such problems, we s h a l l not pursue more general formulations i n t h i s work. I t must be noted that more general problems are under consideration at the present time, the following problem i n d i f f e r e n t i a l geometry being an example: Roughly speaking, the problem i s to determine that manifold M , with boundary , so that each element V of the tangent space of M at some f i x e d point, a f t e r p a r a l l e l transport along the geodesic with tangent V here, belongs to some prescribed vector f i e l d on . I t CO i s assumed that M i s a sub-manifold of a C n-manifold with Riemannian metric, and that the vector f i e l d on i s prescribed by some r e l a t i o n i n the metric. The exact formulation of t h i s problem turns out to give an 64. i m p l i c i t f u n c t i o n a l d i f f e r e n t i a l equation of the type formulated here (except that r i s a function of several v a r i a b l e s ) . Methods of solving these problems, and existence proofs f o r s o l u t i o n s , are also under -consideration. CONCLUSION 65. Previous 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 optics appear to have dealt only with, systems possessing r o t a t i o n a l symmetry. In t h i s work, however, we have formulated a problem i n which the system i s not r o t a t i o n a l l y symmetric. The exact mathematical formulation proved so involved that an approximate s o l u t i o n required the development and synthesis of both a n a l y t i c a l and numerical methods. These schemes seem app l i c a b l e to other o p t i c a l problems, and we have examined how t h i s might be done. Also, i t appears that t h i s i n v e s t i g a t i o n may have led to the formulation of a new cl a s s of equations. Important problems i n regard to p a r a l l e l - p l a t e systems, f o r future consideration, are error analysis ( e s p e c i a l l y l e v e l I ) , existance proofs, p o s s i b l e numerical schemes for the exact problem, and the design of the p a r a l l e l - p l a t e equivalent of Schwarzchild's system of order 2. BIBLIOGRAPHY 66. M. Born and E. Wolf, P r i n c i p l e s of Optics (Pergamon Press, New York, 1959). R. C. Johnson, ,"The Geodesic Luneburg Lens," Microwave Journal, Vol. 5, August 1962; pp. 76-85. S. B. Myers, " P a r a l l e l - P l a t e Optics For Rapid Scanning," Journal of  Applied Physics, V o l . 18, 1947; page 221. E. Isaacson and H. K e l l e r , Analysis of Numerical Methods (John Wiley & Sons, Inc., New York 1966). K. Schwarzchild, "Untersuchungen zur geometrischen Optik," Part II Ges. Wiss. Gottingen Math. Phys. Klasse IV, 1905. D. Strui k , D i f f e r e n t i a l Geometry (Addison Wesley Inc., Reading, Mass 1950). F. G. R. Warren and S. E. A. P i n n e l l , "The Mathematics of the T i n Hat Scanning Antenna," Technical Report No. 7, Contract No. DRBS - 2 - 1 44 - 4 - 3 , RCA V i c t o r Company, Ltd., Montreal, 28 Sept 1951. APPENDIX 67. COMPUTER PROGRAMS Included are l i s t ings and brief explanations of three F0RTRAN programs, the two programs which solve (13) and the optical error program. Several l ibrary programs from the University of Br i t i sh 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 ed , 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 G, Q and F. The function G evaluates G^r at a given argument (G^r =0 is (13)), 0 gives equation (15) ; and F i s the integrand of the e l l i p t i c integrals in (13) and (15). 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. Rice, Mathematical Software (Academic Press, 1971); programmed by Mr. G. Imerzeel. 68 &/QCK I Read NMAX = no. of mesh points, and tolerances. Calculate r"(0) and mesh s i z e DELTA a rock 2. Calculate A l , B l , CI the c o e f f i c i e n t s of p-^e) P r i n t i n i t i a l values. , NMAX i aUck i Extrapolate p ^ ^ x ^ ) j = 0, 1 store i n T l , Tip . Extrapolate P J ^ X N J_ + h/2), solve (13) f o r r " ( x ^ + h/2) c a l c . new poly P 2(6) extrapolate p 2 ^ ^ ( x ) Store i n T2, T2P . As i n Block 4 at the points •+ h/4 , + h/2 and x N_ + 3h/4 . Store f i n a l r e s u l t s i n T3, T3P . G>!ock Q C a l c u l a t e Q, Q* ( c a l l e d ppl & pp2), improved r ^ ( x N ) j = 0, 1 and f i n a l r ' ^ x ^ . P r i n t r e s u l t s END 1 IMPLICIT E£AL«U { A - II, K,C-Z) 2 E X T E 6 8 A L Q R/ O C K d-3 EXTERHAL C U ^ N « C0MH3:! /PAR/SALPH, A , K, A X , B X , C A L P I I 5 COSKO.I /GG/TCLX 6 R E A D ( S , 5 5 ) HKun .XA,XTOL.XTOt 7 5 5 FORKAT ( 4 I , 3 D 2 2 . 1 S ) 8 T O L X = X T O I . 9 X N U t i = H M B a ' 1 0 A=XA • -1 1 I>1^3.14 1 5 9 2 6 5 3 5 9 0 0 1 2 T O L = X T O U 1 3 SALPI1=0. CO 1 4 . I H = 2 . 1 D O 1 5 BAD]=ROOT (Q,X8, 5 0 , T O l ) 1 6 X1 = O . D 0 1 7 B = 1 . 0 0 ' 1 8 B P = 0 . D 0 1 9 BAD=RAD1 2 0 R P P = 1 . C O - 1 . D O / R A O 2 1 DELTA=PI/(2 . CO'XllU.1) -2 2 A1 = R-RP»X l*RI'P*X 1 * X 1 / 2 . D 0 n . O 2 3 . B1=BP-RPP*X1 ir)/^UT-'N- cA 2 4 C 1 = RPP/2.D0 W 2 5 T I T 0 = 0 . D 0 2 6 J=l 2 7 H R I T E ( 6 , 3 ) J , X 1 , R, R P , R P P 2 8 • V R I T 2 ( 6 . 15) R A D , T I T O , T I T O , T I T O 2 9 H3R=li;il3R» 1 J O DO20N=2, ;iQa — 3 1 X = X1 * D E L T A (\ J C\ ( W 3 2 T 1 = A1 + ai*X>Cl*X*X I J ^ W - - 1 \ 3 3 T1P=D1+2.C0*C1*X 3 4 XX= (X*X 1) * . 5 0 0 3 5 T=A1'DI*XX>C1*XX*XX _ I vV 3 6 TP=E1 *2. t ; 0'C1«XX 1] C N A K H 3 7 EALP1I = TP/CSCRT (T»«2*TP«»2) \J ^  v \ 3 8 CALPH=DSQRT(T**2+TP**2)/TP 3 9 K= ( D A T A ! i ( T F / T ) -XX/2. CO) ^ C A L P ! ! / i 4 0 RX=BAD 4 1 At1S=ROOT ( G , R X , 5 0 , T O L ) 4 2 BRIANS (|3 TPP=T*2.DO»T P*TP/T- ( (T »T *T F »T F) * DSQRT (T »T *TP * T P ) ) / ( T * B R ) 4 4 C 2 = T P P * . 5 C 0 4 5 B 2 = T P - T F P * X X 4 6 A2=T-TP*XXtTPP»XX*XX*.5D0 4 7 T2=A2*D2«X*C2»X*X 4 8 T2P=U2»2. C0»C2*X 4 9 X X 1 = (XX*X 1) *.50O 5 0 X X 2 = X X 5 1 x x 3 = (xxtx) » . 5 c o f) i r\ t- \ c 5 2 T = A 1 »t!l*XX 1 » C 1 » X X 1 « X X 1 \}L U L K ) 5 3 T P = D 1 » 2 . C 0 « C 1 * X X 1 W J 5 4 CALPII= E5QRT (T«*2 >TP* *2)/TP 5 5 SALPi; = TP/CSQnT (T»*2 »TPt»2) 5 6 K= (DATAli ( T P / T ) -XX 1/2. DO) ' C A L P l l / A 5 7 A!IS=HOOT (G,RX,50,TOL) 5 8 RR=AHS 5 9 TPP=T» 2 . D0»TF»TP/T- ( (T *T * T F*T P) • DSQRf (T *T *T P* T P ) ) / (T*B R ) 6 0 C2 = T P P * . 5 C 0 6 1 B 2 = T P - T P P « S X 1 6 2 A2=T-TP»XX HTPP*XX1»XX1 * .5D0 6 3 T=A2*D2«XX2»C2»XX2«XX2 6 4 T P = 82*2. C 0 « C 2 * X X 2 6 5 SALPII = T P / C S Q P T (T*«2 'TP**2) 6 6 C A L P ] : = C S Q 3 T ( T * * 2 * T P * *2}/TP 6 7 K= (DATA!) (TP/T)-XX2/2. CO) * C A L P I I / A 6 8 A)lS=ROOT (G,RX,50,TOL) 6 9 RR=ANS 7 0 TPP=T*2. C 0 « T P « T P / T - ( (T»T*TP*TP) « D S Q R r (7»T>TP*TP) ) / ( T * n n ) 7 1 C 2 = T P P « . 5 C 0 7 2 B 2 = T P - T P P * X X 2 7 3 A 2 = T-TP«JX2 + T P P»XX2*XX2 * . 5C0 7 4 T=A2*I:2«XX J*C2»XX3»XX3 7 5 TP=U2*2 .D0»C2*XX3 7 6 SALPII = TP/CSQnT(Tt'»2*TP»*2) 7 7 C A L P I I = D 3 Q U T {T«»2»TP«*2)/TP 7 8 K= ( D A T A N (TP/T) - X X 3/2. DO) *"CA L P 11/A 7 9 A!IS=HOOT (G , RX ,50 , T O L ) 0 0 RR=A;IS 0 1 T P P = T*2. 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 B 2 = T P - T P P * I X 3 70; 81 A 2 = T - T P « X X 3 t T P P * X X 3 * X X 3 » . 5 D O " ' 85 T 3 = A 2 » D 2 » X « C 2 * X * X -86 T 3 P = U2 *2. C0 -«C2«X " . 87 IF ( T 3 .-KQ.T2) U0T05 . • . 88 . PP1 = ( T 2 - T 1 ) / ( T 3 - T 2 ) A ) L 89 I F ( P P 1 . E Q . 1 . C O ) G 0 T 0 5 Y7 U 90 R= ( P P I * T 2 - T 1 ) / ( P P 1 - 1 . D O ) • ' « 91 C0T06 92 5 H=T2 93 6 I F ( T 3 P . E 0 . T 2 P ) G 0 T O 2 91 PP2= <?2P-T I P ) / ( T 3 P - T 2 P ) 9 5 I F ( P P 2 . E Q . 1 . C O ) G 0 T 0 2 . . • 96 R P = ( P P 2 » T 2 P - T 1 P ) / ( P P 2 - 1 . D 0 ) . 97 G0T04 98 2 RP=T2P 99 4 S A L P H = R P / C S Q R T ( R * « 2 + R P * * 2 ) • . ' 100 CALPH=CSQRT( B « * 2 *RP * » 2 ) /R P . . . . 101 K= (DATAti (RP/li) - X / 2 . DO) «CALFII/& 10.2 A ! (S-npOT ( G . UX, 50 , TOL) 103 RR=AUS ' . 104 RAD=RR 105 RPP=R 17. D O » B F * R P / R - ( (R*R*RP <S) *RP) *ESQRT ( R * R * R P * R P ) )/ (H*nH) 106 C 1 = R P P / 2 . C 0 107 B1=RP-RPP*X 108 M = R - R P « X « X / 2 . 0 0 " • 109 V R I T B ( 6 . 3 ) S , X , R , R P , R P P 110 3 FORMAT ( IX , 'N= ' , 13, IX, ' X = ' , D 2 2 . 15, I X , 'R=> , D 2 2 . 1 5 , I X , ' n P = ' , D 2 2 . • 111 * I X , ' n P P = « , 0 2 2 . 1 5 ) 112 WRITS (6, 15) RAD, S A L P H , P P 1 , P P 2 113 15 FORK AT ( IX ; •RAO= ' , 0 2 2 . 15, 1X, 'SALPII= D2 2 . 15, 1X, 'PP1 = » , C 2 2 . I S , " 1 in • ' P P 2 ^ ' , E 2 2 . 15) 115 20 X1 = X 116 STOP ' 117 ZN 0 11 a 1 J o • • 137 FlWCTIOtl RCOT ( C , Z G U E S S , H , I P S ) 138 I M P L I C I T R F A L « 8 ( A - H , X, 0 -Z) J 139 ZO^ZGUBSS " ' •• 110 tlrt=l 111 I F t A G = 0 . , 112 WO = G(ZO) - » -113 T E S T ^ Z O 111 IF (DAES (30) . L E . EPS) COT03 115 Z 1 = 2 . D 0 » Z 0 116 Z2 = Z 0 / 2 . C 0 117 5 V I - G ( Z I ) 118 T E S T ^ Z I 119 I F (DABS (K 1) . L E . EPS) G0TO3 150 » 2 = G (Z2) . ' ' ' 151 TEST=Z2 152 IF (DAtS (W2) . L E . EPS) GOT03 ' 153 IF (HI * K2 . G £ .0 . HO) G0T01 151 7 Z3 = Z 1 - H 1 » ( Z 2 - Z 1) / (M2-H 1) 155 1 LIK=LI.1*1 156 IF ( L I K . G T . N ) C0TO2 157 W3 = G ( Z 3 ) 158 TEST=Z3 • * ." 159 IF (DADS (H3) . L E . E P S ) G0TC3 160 I F ( W 1 * W 3 . C E . 0.D0) C0T06 161 S2=U3 162 Z2=Z3 163 G0107 164 6 Z1 = Z3 165 W1 = W3 166 G0T07 167 4 Z 1 = Z 1 » Z 1 168 Z2 = Z 2 / 2 . C 0 169 I F L A G ^ I F L A G *1 • 170 IF (I F L A G . G E . II) GO TO 2 1 171 G010S 172 2 WRITE ( 6 , 10) LIH,7.1, a 1, Z 2 , 8 2 173 10 F O R K A T ( I X , ' L I K ' , 1 3 , 4 D 1 4 . 7 ) 174 STOP 175 21 W R I t E ( 6 , 1 1 ) I F L A G , Z 1 , W 1 , Z 2 , 9 2 176 11 FOR HAT IIX , ' I FLAG ' , 13, 4 U 11 . 7 ) * 177 STOP 178 3 ROOT = T £ S T 179 RETURU 180 EHD 7 1 . Next i s the Galerkin program. Since i t i s quite simple, we s h a l l only define pertinent function and sub-routines, and give the l i s t i n g . In the order i n which they occur, the routines are: N0NLIN, FK, CADRE, G, AND H. 2 N0NLIN solves the system of non-linear equations fit/2 ( i . e . G.r • g. = 0 ) whose values are evaluated by FK. CADRE i s as A X o before, H i s the integrand i n (13), and G i s G A r « A l i b r a r y subroutine from the U n i v e r s i t y of B.C. Computing Centre, taken from C o l l e c t e d Algorithms of CACM, Algorithm 3/6, by K. M. Brown. Hit* 307 JBPIICIT BUI »8 (X-B.C-2) 300 LOGICAL LIST 309 D ) PC .-.'L 1 C > 1 1CII1 (6,6) , 13UE (6) ,COF{6,6) ,TI«P (6) ,PABT 16} ,1 |5| 310 iisr-.Tsuj. 311 K-5 312 ltXIIT»20 313 311 »unsic«5 J 1 5 X(1)-1.CC 316 X (2) 3 . 1 2£ 3 15-762167215C0 317 X 111-.71611S7 1 5630060 3 (4 318 I (0) »-. 13om;;n )9£5922CC 319 X |5) »1.05 2 3 IC i 10 1112000 CO 320 O I L XCKLIMa'I.IIAIIT.BU.ISIG, 15 IHG , I, I tOI IT , I £ C3, COI.T (.a P ,P»«t, 321 • M I T ) 322 If (ISJKG. ICO ) V8 IT! <6,3C) 323 If (f AIII. IC.f. )1I(ITE |6,1C) 320 uaiiE(6,2C) (i ( i ) , i , s) ,n»i t t , i s t i o 325 20 PCBr.il (1 ),• 1 =' ,0 (C25. 16, 3X)/IX , D25. 16, 210) 326 30 FORSAT III, 'XO CO •) 327 31 S10P 328 110 329 330 331 132 331 130 315 336 337 'CEROUTtXH III (9,1,1, >) 338 IMPLICIT BEAL«8(A-fl,0-Z) 339 EX1ERKAL G • . 300 OlflEKSICH 1(5) , 8(5) 311 corr.o»/cn/ E,PI,J»,II 3<)2 aS-1 303 tt-l 310 OOU-1,1 315 1 8(J)=1|J) 306 p I- 1. 570796 3167') 0096D0 107 1EG-CACBt |C,C.C0,PI,0. [0,1.C-6,t,K,J) 348 T-TEG 319 BE1C8I 150 tig 351 352 357 358 PCHCTICS G (T) J5S j s p i i c i ? r . i ' i i!|i'J,c-:| 360 c i n t u s i o 6(5) 361 corros/cFF/ E ,Fi,3i,»I 362 n c j a i L ii 363 corros/Tvc/ i . s i t - ' 360 IF (T. E9- C . CO) GOT02 365 n=cccsn) 366 TE»I.C0>1T 367 368 S£=CSIS (T) 369 3 70 TF:2*TB«TB 371 B1="2. CO/T f • E (1) • « ! 372 Bi=2.C0«S[/T!•2.D0»8 (1) «TB 371 B3 = 2. co» n n ;.co• SE»SE/TI)/T2>2. CO • t H I 370 Tt-TK2«TU2 175 l i - H ! 2 » T 376 1J-IH2 377 TX-1 378 0011 = 2,111 379 J1-2 • I 380 0»P(1) ' 181 BI-BMO'TB 182 B2-B2tir'U >TI 181 n3«a) « n • i i i - i ) «o»Ta 161 T' = -jtr:-2 385 H-TI»1H2 386 TE» TH »TII 2 387 1 1I-11C»TH2 168 £S= B 1 » 3 1« 3 2 » B 2 389 «AI»S2/SS 190 " n«ss«CSCRT(s;J/lssta2-» l'»3) 39 I II = .2 5EO<9'.'i>l»CICBE|H,-Il,PI,0.tO(1. t-6 , 1,2,J) 192 f EE-T" • (2 •KX - 1) 393 G=f EE* I) I »T-; . CO • C AT AX (B2/S 1) ) 191 BF1UBJI 395 3 FEE»TE.".P 196 GCTOO 397 2 c«o.co • 198 BETUB H 399 EID 500 001 002 401 000 005 006 007 rescues 11 IT) 008 JBP11CIT B IXIHI (1-II,C-1) 009 corriik/ric/ t ,rxt 0 10 1. 1)0/ (!t • . ;V.HIK05 (T)) «0SPB1 ( (IH . 25PC • OCOS (1) ) • >?.- (It •S.M.) »«2) Oil UF 11111 012 tao 73. The o p t i c a l error routine ERR0R reads i n the c o e f f i c i e n t s a^ , n of the Galerkin approximation r (8) = £ a. g.(6) . The subroutine i = l 3 DRKC integrates (A), where (4) i s given by FUNC. Lines 407 to 530 c a l c u l a t e and p r i n t the o p t i c a l error i n degrees and radians. Note that t h i s routine can be used i n place of G i n the Galerkin program f o r attempting the exact problem; the a.'s are provided by N0NLIN. A l i b r a r y routine at the U.B.C. Computing Centre, taken from "Numerical Solutions of Ordinary Simultaneous D i f f e r e n t i a l Equations of the F i r s t Order Using the Method of Automatic Step Change," Num. Math., V o l . 14 #9 (1970). 071 C C E P 0 U T E 1 E If E.OB(T,l,»,RlCS,CfCS,T0L) 74 070 i n r u c t r > a t " « | i - i , i j - ! ) 075 E1UB>»1 l i n e 076 C I M E K S I C I £ ( 1 0 ) 077 corros o , » i 078 C1Bt»nic» »(10) , i (2) ,r (2) ,r,( J) ,S(2)#I (2) 079 B»D:*O.EC oeo DECS-'O. co o s i IF t i . co . c. CO) BZICBI 082 PI"3.1« 1 5 9 : 6 5 3 5 8 9 7 7 0 0 083 E'PI/6».CC oeo I-.KIIOIVKO.CC 085 E«IOl 0 86 087 z o - . s c o m 088 Z 2 » - Z 1 089 T(1)M 090 Till*1.CC»ECCS(1) 091 El = 2. C0/T1J1, 092 TIllVTJUl'Tilt. 093 S2 = 2. COUSI J | T ) / U l l 090 7EEE"T 095 12>lEr.t>t!ta 096 TEHP=T2 097 CC2 I' 1,1 ' • 098 D-l(I) 099 B(I)>0 500 I»2»I 501 B1"8 HO'tEIIP 502 B2«82«I1»0»TCnB 503 m t = iffa>i2 500 2 TElf»7t.l?,I2 50 5 I ( ? ) * . 2 5 E C ' B ; / ( M > C S C B T I S 1 » 8 I « 1 2 « 8 2 ) ) 506 c m CBKC c : i,z2,i, P.n,Hnis,s,ruac.c,3,i) 507 11=1(1) 508 I1P-r(2) 509 DUO. c o « c e c : t x l ) 510 Sl=2. tO/II! 511 BIP'2. CO'CSII (I1)/(C[f <tll) 512 T2=11»Z1 513 IEEE-I1 510 TE8P-T2 515 CO ) !• \.t 516 EO»|l> 517 II»2«I 518 gl»»l«U«IEHP 519 B1.f»B.lf»II»C»JIH 520 TE.1E"TE.13«I2 521 3 IEEEMEirf«I2 522 It-OATAS (S tP/S Ij -'523 s*=21'*Iir 520 Sl?=Blp»XlF «25 EE«C5CaT(P1»H»B1P*SIP».C6i5C0) S26 C»3 1?« ISIS II 1) <3 OOCCS II IJ-. 2500»OSE I (I l - l t ) ' ' J 2 1 RHCE-.SLC •(I-CAFCOS (C/EE) 526 CEGS-RltS* 13C.C0/PE J 2 9 B E 1 C 8 I 510 E»0 518 ' ' 1 1 1 ' ' 539 500 501 J C E B O C T E I I I S J C ( ! I , T , r ) 502 InPEICIT BE>L«8(1-H,C-1) 503 ClIENSK) 1 |1) .1 (1) , E 110) 500 COEnOS E,»J •05 11'CCCS 11 (1)| «06 IE-1.C0HT 507 12ME«Tt 508 SE-CSES (1 ( I ) ) 509 T E - I ( I ) 550 TE2»TH»TH e;i B 1=2. C C / I E<B (I)'IM 552 S2»2. EtO'SE/T ; »2. CO»B 11) 1T8 «<} B3-2.CC« |11» 3.CO«fr«SE/TI)/T212.CO•t |1) 555 S7-2. C0>S(»|6.!:0«5EtSE/T;»6.D0«1V12-1-00)/72 J55 1 E ' 1 H 2 « T H 2 556 TI»TH2M|lj «57 10=1H2 S58 "'T(I) <>9 CC102.11 560 ll»2«t 561 C*t(I) 5£2 S1=8I»0»T8 5tJ S?=B2»ll>0»Tt 540 »3«»3tII« HE- 1) 'D"IJ «t5 >7*B7MI< 1) • (11-2) <0*T( 5J6 TJ»TJ«TU2 t-tt 1011MH2 568 1I>r»»tMj «{9 t 11»1K»TII2 570 SS=BOB I>B;»B2 «71 E1-!S<)1<IM1M1 572 KU-SS.ISCKT (E5)/St «7J fi-3.E0«B;«(11'Sll/SS-l2.t0»(MB2O. EO "B :»B 3-BI • t J)/ST 57» • f-E-BI t.25EC«ECOS (11) • 75 1-1(2) 576 rot" < 1 7 »'.sno«rsii ( i t ) M/F6 578 (ECOS (H) •. ?5(JC«B5/Sb-S7( (»1.»1)/SS) 179 C-«.l>0MSl> (I M •H>».'.T.'.T«»»»H/(5S«S51 5S0 r(2l"»«H-.C iti »r1CKI E»0 

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