12-2
GEODESIC SPLITTING ON GENERAL PARABOLOID OF REVOLUTION
AND ITS IMPLICATIONS TO THE SURFACE RAY ANALYSIS
RM J h a , SA Bokhari, V Sudhakar and P R M a h a p a t r a
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
The a u t h o r s have observed an interesting phenomenon of geodesic splitting in t h e
c a s e of s u r f a c e ray propagation over a g e n e r a l paraboloid of revolution. Since
even t h e primary geodesics a r e s p l i t in both clockwise and anticlockwise
directions, t h i s leads t o a doubling of t h e r a y p a t h s t o b e considered in t h e
a n t e n n a c h a r a c t e r i s t i c s computations. This work provides a n insight i n t o t h e
ray-splitting phenomenon on simplest (i.e. of lowest order) possible convex
surface.
Introductlon
I t is well known t h a t b e t w e e n any t w o arbitrarily l o c a t e d points on a cone
o r a cylinder, t h e r e e x i s t primary and higher-order (of multiple e n c i r c l e m e n t s )
g e o d e s i c s in b o t h a n t i c l o c k w i s e ( r i g h t ) a n d c l o c k w i s e ( l e f t ) directions.
However, t h e number of geodesics of a given order and sense does n o t exceed one.
In c o n t r a s t , a d e t a i l e d s t u d y of t h e s u r f a c e r a y p r o p a g a t i o n o n g e n e r a l
paraboloids of revolution (GPORs) by t h e authors has shown t h a t t h e geodesics of
a given order and sense (e.g. right primary geodesic) could b e split into t w o
(Fig. I).
T h e implication of such split geodesic i s a n i n c r e a s e in t h e number of
geodesics (and t h e r e b y t h e s u r f a c e r a y p a r a m e t e r s associated with t h e various
geodesic paths) t h a t must b e t r e a t e d f o r a c o m p l e t e r a y - t h e o r e t i c analysis.
Effects of Geodesic Splitting
S i n c e t h e g e o d e s i c d o e s n o t s p l i t i n t h e c a s e of t h e r i g h t c i r c u l a r
cylinder, c o n e and sphere, t h e GPOR a s a m e m b e r of t h e quadric s u r f a c e s family
r e p r e s e n t s t h e simplest e x a m p l e w h e r e t h e phenomenon of geodesic splitting may
b e observed.
In t h e case of t h e high frequency EM a n t e n n a computations [1,2], knowledge
of several ray g e o m e t r i c p a r a m e t e r s associated w i t h e a c h of t h e e x i s t e n t
geodesic is required a priori.
This in effect m e a n s a doubling of t h e ray
g e o m e t r i c p a r a m e t e r s t o be d e t e r m i n e d in t h e c a s e of t h e GPOR as compared t o
t h o s e in t h e c a s e of t h e right circular cylinder and c o n e for any given order of
g e o d e s i c s . T h u s , f o r e x a m p l e , e v e n if t h e p r i m a r y g e o d e s i c s a l o n e a r e
considered on a GPOR, an e l e m e n t a l point-to-point t r e a t m e n t would require t h e
d e t e r m i n a t i o n of four primary geodesics which m a y b e present.
CH2654-2/89/W00-0196
$1.00 01989 IEEE
196
Since t h e GPOR is a nondevelopable s u r f a c e , in g e n e r a l i t r e q u i r e s a
bivariate s e a r c h f o r t h e s u r f a c e r a y g e o m e t r i c p a r a m e t e r s which m a k e s t h e
problem computationally i n t r a c t a b l e f o r most of t h e p r a c t i c a l cases.
In
c o n t r a s t , t h e a u t h o r s h a v e developed a geodesic c o n s t a n t m e t h o d (GCM) which
requires only a simple univariate s e a r c h [3,4].
This univariate s e a r c h is f o r
t h e First Geodesic C o n s t a n t associated with t h e geodesics.
The phenomenon of split geodesics serves t o e m p h a s i z e t h e a d v a n t a g e of t h e
Geodesic C o n s t a n t Method (GCM) developed by t h e authors f o r ray analysis over
g e n e r a l convex surfaces. I t may however b e n o t e d t h a t t h e ray g e o m e t r i c
p a r a m e t e r s in t h e c a s e of t h e GPOR a r e e x t r e m e l y sensitive t o t h e value of h
which must b e determined accurately. I t has been observed by t h e a u t h o r s t h a t
it usually suffices t o c o m p u t e h a c c u r a t e u p t o 8 d e c i m a l places f o r a given
a r b i t r a r y s e t of s o u r c e and observation on t h e GPOR.
In general, t h e r e a r e now t w o First Geodesic C o n s t a n t s h r l l a n d h r I 2
associated with t h e two right primary geodesics. O n c e t h e s e geodesics c o n s t a n t s
f o r t h e t w o right primary and h
and h 1 2 f o r t h e t w o l e f t
( h s l I and hr
primary geodesics) a r e obtained t h e a r c lewgth,
generalized F o c k p a r a m e t e r
and t h e F r e n e t - f r a m e fields v e c t o r s (t,n,b) required in t h e r a y - t h e o r e t i c
methods like t h e UTD c a n b e readily determined.
tk'
Summary
T h e phenomenon of geodesic splitting has been observed in t h e case of t h e
general paraboloid of revolution. T h e GPOR r e p r e s e n t s t h e simplest s u r f a c e over
which geodesic splitting c a n b e studied.
In g e n e r a l t h e r a y t r a c i n g over a g e n e r a l paraboloid of revolution would
require a bivariate search. The ray splitting in t h e c a s e of t h e GPOR tends t o
f u r t h e r increase t h e c o m p u t e r t i m e required substantially f o r t h e d e t e r m i n a t i o n
of 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 .
T h e authors have developed a Geodesic C o n s t a n t Method (GCM) involving a n
a c c u r a t e s i m p l e univariate s e a r c h which h a s brought t h e EM field c o m p u t a t i o n s
within t h e a m b i t of t r a c t a b i l i t y .
REFERENCES
[ I ] P.H. Pathak, N. Wang, W.D. Burnside and R.G. Kouyoumjian, "A uniform GTD
solution f o r t h e radiation f r o m sources on a convex surface", IEEE Trans.
Antennas & Propagat. (USA), Vol. AP-29, no. 4, pp. 609-22, July 1981.
[2] S.W. Lee, "A review of GTD calculation of m u t u a l a d m i t t a n c e of Slot
conformal arrays", E l e c t r o m a g n e t i c s (USA), vol. 2, pp. 85-127, Dec. 1982.
[3] R.M. J h a , S u r f a c e R a y Tracing o n C o n v e x Q u a d r i c s w i t h Applications to
Mutual Coupling b e t w e e n Antennas o n A e r o s p a c e Bodies, Ph D Dissertation.
Submitted t o 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, India, Nov. 1988.
N. Balakrishnan, "Ray analysis of mutual coupling
between a n t e n n a s on a general paraboloid of revolution (GPOR)", Electronics
L e t t e r s (GB), vol. 23, pp. 583-584, May 1987.
[ 4 ] R.M. J h a , V. Sudhakar and
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Fig.
1
Splitting of the right p r i m a r y geodeslc on t h e GPOI:.
198