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3-D GEODESICS ON CONVEX QUADRICS FOR SURFACE RAY PROPAGATION :
A TURBO BASIC PACKAGE FOR COMPUTER-AIDED INSTRUCTION
K M J h a , SA Bokhari, V Sudhakar and PI? ivlahapatra
D e p a r t m e n t of Aerospace Engineering
Indian I n s t i t u t e of Science
Bangalore, 560 012 India
ABSTRACT
A c o m p u t e r package has been developed to d e m o n s t r a t e t h e various t y p e s of
geodesics on developable as well a s nondevelopable quadrics, with p a r t i c ~ i l a r
a p p l i c a t i o n to high f r e q u e n c y E M analysis. This user-friendly p a c k a g e ,
developed in TURBO BASIC, allows t h e user t o s i m u l a t e t h e various possible
geodesics b e t w e e n a n y t w o points l o c a t e d on a convex s u r f a c e .
Introduction
The ray-theoretic formulations such a s t h e C T D and UTD h a v e t h e common
f e a t u r e t h a t they require s u r f a c e ray geonietric p a r a n r e t e r s along t h e ray p a t h s
which a r e t h e geodesics of t h e convex surface. However, t h e d e t e r m i n a t i o n of
t h e geodesic ray p a t h s is n o t q u i t e easy. This is f u r t h e r hindered by t h e f a c t
t h a t q u i t e o f t e n t h e undergraduate s t u d e n t s (and s o m e t i m e s e v e n researchers!)
have preconceived notions about t h e geodesics which a r e o f t e n incomplete and in
several c a s e s e v e n wrong. For e x a n ~ p l e , one is o f t e n led to believe t h a t " t h e
geodesic is t h e s h o r t e s t p a t h (which is partially t r u e ) and consequently b e t w e e n
t w o points on a s u r f a c e t h e r e is only o n e geodesic (which is wrong)."
The a u t h o r s have developed a user-friendly package t o retilove s a m e of t h e s e
misconceptions of t h e students, a s well as t o convincingly d e m o n s t r a t e t h e
various types of geodesics on t h e various convex surfaces. W e have chosen t h e
faniily of quadrics for t h e demonstration of geodesics, s i n c e they f o r m t h e c l a s s
w h e r e most of t h e non-trivial properties of t h e geodesics c a n b e demonstrated.
Besides, t h e c l a s s of quadrics c o n s t i t u t e t h e Eisenhart C o o r d i n a t e S u r f a c e s [I],
which i s e x t r e m e l y i m p o r t a n t froni t h e Eh4 analysis point of view.
Description of the ceadesics
A s e t of c o m p u t e r codes has been developed in t h e TURBO BASIC which p e r m i t s
t h e user t o visualize t h e geodesics in a n i n t e r a c t i v e manner.
The most commonly held notion of a geodesic ( a s being of t h e s h o r t e s t path)
corresponds t o a primary geodesic. Figure I shows a primary geodesic on a r i g h t
circular cylinder. However, " t h e r e is only one geodesic" b e c o m e s a questionable
proposition, t h e m o m e n t a n o t h e r geodesic - t h e l e f t primary geodesic a p p e a r s in
t h e clockwise direction (Fig. 2).
Itealizing t h e geodesics on t h e developablrs like t h e right circular
cylinder and c o n e is relatively easy, s i n c e t h e development of t h e s e quadrics
i l l d l J 5 t h c geodesics o n t o t h e plane as (different) s t r a i g h t lines.
CH2864-2/89/W00-0223
$1.00 01988 IEEE
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On t h e o t h e r hand, t h e geodesics on t h e nondevelopable s u r i d c e s d r e q u i t e
cornplex. This is d e m o n s t r a t e d in Fig. 3, which shows t h e geodesics on a general
paraboloid of revolution (GPOK).
In such c a s e s t h e geodesic d i f f e r e n t i a l
e q u a t i o n expression in t h e Geodesic Coordinate System [2], r a t h e r t h a n t h e
s t m d i l r d cdriuriical gc:odesic equcltioii, is used t o d e s c r i b e tlw geodesic>. On il
s u r f a c e wliich is described in t w o geodesic coordinates U and v, t h e differential
equation of t h e geodesic t a k e s t h e f o r m
w h e r e E and C, t w o of t h e First Fundamental C o e f f i c i e n t s of t h e set (E,F,G),a r e
functions of U. lplplicit in this c h a r a c t e r i z a t i o n is t h e property t h a t t h e unit
principal normal n 05. t h e geodesic at any point on t h e c u r v e is equal t o t h e
unit s u r f a c e normal N a t t h a t point on t h e s u r f a c e on which t h e geodesic is
d e s c r i b e d . T h e s u b s c r i p t "rm" r e f e r s t o t h e m-th o r d e r r i g h t geodesic.
Similarly, several o r d e r s of l e f t geodesics may a l s o exist in general. T h e "+/" sign of h r m depends on whether t h e a r c length is a monotonically increasing o r
decreasing function of t h e geodesic p a r a m e t e r U a s shown in Figs. 4 and 5. In
m o r e complex cases, a geodesic may b e in p a r t s monotonically increasing a s well
a s d e c r e a s i n g function of t h e geodesic p a r a m e t e r U of which t h e right secondo r d e r geodesic in Fig. 3 is a n e x a m p l e 131. In such c a s e s t h e sign is also
t a k e n "+/-'I depending on t h e d e s c e n t / a s c e n t of t h e geodesic.
Application of t h e Geodesic C o n s t a n t Method
T h e authors have developed a Geodesic C o n s t a n t Method (GChl) where all t h e
s u r f a c e ray g e o m e t r i c p a r a m e t e r s required in t h e high frequency calculations [ 4 ]
a r e expressed explicitly in t e r m s o l t h e F i r s t Geodesic C o n s t a n t h.
This
analysis uses a g e n e r a l Geodesic C o o r d i n a t e System and is applicable t o a wide
c l a s s of quadrics and non-quadric s u r f a c e s .
The power of t h e GCM becomes a p p a r e n t with t h e e a s e of its application t o
t h e ogival surfaces, w h e r e t h e geodesics h a v e been h i t h e r t o described only by
computationally i n t r a c t a b l e t w o - p a r a m e t e r numerical methods [ 5 ] .
F u r t h e r analysis by t h e a u t h o r s [61 has shown t h a t c o n t r a r y t o t h e popular
belief, t h e number (i.e., o r d e r ) of geodesics on a circular c o n e is finite.
Finally, t h e a u t h o r s h a v e observed a splitting of t h e geodesics on t h e GPOR
(Fig. 6). The GPOR, being a quadric, is t h e simples1 sur-iace on which this
phenorrienon is observed [71. In t h e r a y - t h e o r e t i c approaches, this implies a
doubling of t h e e f f o r t in t e r m s of t h e r a y t r a c i n g of all t h e s u r f a c e ray
p a r a m e t e r s , required in t h e EM field computation.
REFERENCES
[ I ] P. Moon and D.E. Spencer, Field Theory Handbook. Berlin: Springer-Verlag,
1971.
[21 T.J. Willmore, An Introduction to D i f f e r e n t i a l Geometry. Oxford, UK:
Oxford University Press, 1959.
224
[31 R.M. J h a , V. Sudhakar and N. Ihlaltrishnan, "Ray analysis of niiitual
coupliiig L e t w c e l i rrriteiirias oii a gciieral paraboloid of r c v o l u t i o i i (Ll'Oit)'',
Electronics L e t t e r s (GB), vol. 23, pp. 583-584, May 1987.
[41 P.H. Pathak and N. Wang, "Ray analysis of mutual coupling b e t w e e n a n t e n n a s
on ii convex surface", ICEE Traiis. Antennas JC Propagut. (USA), vol. At'-29,
no. 6, pp. 911-922, Nov. 1981.
[51 R.M. J h a , "Surface Ray Tracing on Convex Quadrics with Applications to
Analysis of Antennas on Complex Aerospace h d i e s " , 39th International
Astronautical Congress of t h e International A ~ t r o n ~ i u r i c a l F e d e r a t i o n ,
Bangalore, India, Oct. 8-15, 198s.
[61 R.M. J h a , S.A. Bokhari, V. Sudhakar and N. Balakrishnan, "Closed-form
expressions for ray g e o m e t r i e s on a cone", IEE C o n f e r e n c e Proceedings,
F i f t h International C o n f e r e n c e on Antennas and Propagation,York, U.K.,
[CAP 87, vol. I , pp. 557-560, 30 March - 2nd April, 1987.
[71 R.M. J h a , Surface Ray Tracing on Convex Quadrics with Applications to
Mutual Coupling between Antennas on Aerospace Bodies, Ph D Dissertation.
S u b m i t t e d t o D e p a r t m e n t of Aerospace Engineering, Indiari I n s t i t u t e of
Science, Bangalore, India, Nov. 1988.
-Primary
-Right
Geodesic
Geodesic
Fig. I
2 Right and left primary geodesics
on a c i r c u l a r cylinder.
Primary (or 1st-order) geodesic
on a c i r c u l a r cylinder.
225
Fig. 3
Fig. 4
R i g h t 1st and 2nd-order geodesics
on a g e n e r a l paraboloid o f
revolution (GPOR).
R i g h t 1st-order geodesic on a GPOR
w h e r e a r c length nionotonically
i n c r e a s e s with u - p a r a m e t e r . I i e n c e
h,,,, is positive.
p i - -.---
.----S
Fig. 5
R i g h t 1st-qrder geodesic o n a C ; p o i <
w h e r e a r c length monotorilcally
d e c r e a s e s wlth t h e I n c r e a s e in up a r a m e t e r . I-lerlce h r r l r I S negative.
226
Fig- 6
Splitting of t h e right priniary
g e o d e s i c on a G P O R , among t h e
simplest s u r f a c e t o show this
phenomenon.