CLOSED FORM SURFACE RAY TRACING ON OGIVAL SURFACES
R M 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
A new method based on t h e geodesic c o n s t a n t has been developed by t h e authors
f o r t h e s u r f a c e ray t r a c i n g on a n ogive. The 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
r e q u i r e d in t h e high f r e q u e n c y EM c a l c u l a t i o n s a r e o b t a i n e d in e x p l i c i t
a n a l y t i c a l f o r m . T h e o g i v e i s t r e a t e d a s a c o o r d i n a t e s u r f a c e of t h e
bispherical c o o r d i n a t e s y s t e m which is a geodesic c o o r d i n a t e system. All t h e
ray g e o m e t r i c p a r a m e t e r s a r e obtained in o n e - p a r a m e t e r form in t e r m s of t h e
First Geodesic C o n s t a n t h. These ray p a r a m e t e r s m a y b e used in t h e a n t e n n a
radiation p a t t e r n , mutual coupling and radar cross section calculations.
Introduction
The i m p o r t a n c e of t h e ogival s u r f a c e s lies in t h e f a c t t h a t they, along
with t h e Eisenhart (quadric) cylinders and s u r f a c e s of revolution, describe
nearly all t h e gross e x t e r n a l shapes e n c o u n t e r e d in Aerospace Engineering [I].
The Eisenhart quadric cylinders and s u r f a c e s of revolution a r e easily described
in t h e well known e l e v e n second-degree c o o r d i n a t e s y s t e m s r e f e r r e d t o as t h e
Eisenhart C o o r d i n a t e System [2]. In c o n t r a s t , a n ogive is not a coordinate
s u r f a c e of a n Eisenhart C o o r d i n a t e s y s t e m and cannot b e readily p a r a m e t r i z e d a s
such.
In this paper, a n ogive is identified as t h e c o o r d i n a t e s u r f a c e of a NonEisenhart C o o r d i n a t e System, namely t h e Bispherical Coordinate system.
A
s u r f a c e r a y analysis is c a r r i e d out a n d all t h e ray g e o m e t r i c p a r a m e t e r s
required for t h e high frequency EM a n t e n n a analysis have been obtained in onep a r a m e t e r form, for t h e f i r s t t i m e in open l i t e r a t u r e .
Formulation
T h e p a r a m e t r i c equation of a n ogive in Bispherical C o o r d i n a t e System is
given a s [ 3 ]
R s i n a cos
x -
cosh n
-
b
cos a
R sin a sin b
Y =
cosh n
-
R sinh n
z=
cos a
a >X/2
(I)
cosh n - cos a
w h e r e a and R a r e t h e shaping p a r a m e t e r s of t h e
ogive. W e identify t h e
c o o r d i n a t e s (n,& with t h e geodesic c o o r d i n a t e s (u,v) so t h a t e q (1) is of t h e
form
x = f ( u ) cos v
y = f(u) sin v
CH26642/89/000&1294 $1.00@l989 IEEE
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z = g(u)
(2)
where
R sin a
f(u) =
R sinh
g(u) =
-~
cosh
U
- cos a
cosh
U
(3)
- cos a
U
and t h e f i r s t and second p a r t i a l derivatives of f(u) and g(u) a r e obtained as
R sinh
f
U
sin a
R(cosh
U
g, =
z
(cosh
U
- cos a)'
(cosh
I? s i n a ( I -sinhzu -cosh
I uu:
(cosh
U
U
R sinh
cos a )
- cos a)'
U
U
cos a - I )
( I +sin2a -cosh
guu=
(4)
- cos a ) 2
(cosh u
-
U
cos a )
cos a)'
The subscript U d e n o t e s t h e p a r t i a l derivatives w.r.t.
U.
I t has been
shown by t h e authors t h a t t h e Bispherical C o o r d i n a t e System is a n e x a m p l e of a
Geodesic Coordinate System [31. H e n c e t h e g e n e r a l ray analysis developed by
t h e s e authors for s u r f a c e ray analysis in t h e Geodesic C o o r d i n a t e System c a n b e
extended t o t h e ogive a s well. 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 (of t h e
d i f f e r e n t i a l f o r m ) f o r t h e o g i v e c a n now b e readily o b t a i n e d by m e r e l y
substituting t h e value of f,, ,g , f u u , and g u u in t h e a p p r o p r i a t e equations. In
particular, t h e right (i.e. anticyockwise) m-th o r d e r geodesic, which completely
encircles t h e ogive m-1 t i m e s is expressed a s 131
where hr, i,s t h e F i r s t Geodesic C o n s t a n t obtained by i n t e g r a t i n g of t h e second
order canonical differential equation of t h e geodesic. The positive (negative)
sign of hr depends on t h e increasing (decreasing) n a t u r e of v p a r a m e t e r , which
in g e n e r a r i s of t h e f o r m
and h i m c a n b e
h;m is t h e Second Geodesic C o n s t a n t of t h e geodesic. h,,
determined s i n c e t h e s o u r c e and observation point c o o r d i n a t e s a r e known a
priori. All 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 r e q u i r e d f o r t h e high
frequency methods a r e expressed analytically in t e r m s of hrm. F,or example, t h e
rectangular c o o r d i n a t e c o m p o n e n t s of t h e unit t a n g e n t v e c t o r t a r e expressed as
t = t x i + t j + t Z k
Y
where
f u [ f ( u ) 2 - h & J ' / 2 c o s v - (f:+g:)1/2hm
t
sin, v
=
f (U) ( f ;+g; ) / 2
f u [ f ( u ) 2 - h 9 ' / 2 s i n v - (f;+g;)1/2hm
t
=
Y
f (U) ( f ; + g ; V
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cos v
A
-
Similarly n and b of t h e F r e n e t - f r a m e field v e c t o r s c a n b e readily derived.
W e summarily p r e s e n t t h e expression for some of t h e key ray g e o m e t r i c p a r a m e t e r s
( o n t h e o g i v e ) r e q u i r e d in t h e r a y - t h e o r e t i c f o r m u l a t i o n . T h e r a d i u s of
c u r v a t u r e p ( u ) and torsion 't(u) a r e readily obtained a s
g
f 3 ( f E+g$3/2
(8)
g u ) =
- f 3guf uu+f 3f u g u u + h ~ f g uuu-h&,ff
f
uguu+hhf~gu+h2?~
T h e a r c length s r m
integrating t h e m e t r i c
of t h e r i g h t m-th o r d e r geodesic i s obtained by
trm
IS defined a s t h e integral of a
The generalized Fock p a r a m e t e r
function of t h e radius of c u r v a t u r e p along t h e geodesic with r e s p e c t t o t h e
g
metric, fromn S t o P r m
Fock
The g e n e r a k e d Fock paAameter Trm is used in computing t h e generalized
i n t e g r a l s U ( q r m ) and V (Erm), which is t h e c r u x of UTD c o m p u t a t i o n [41.
Discussion
T h e use of Eisenhart C o o r d i n a t e System has enabled t h e EM analysis on a
very wide c l a s s of commonly used quadric s u r f a c e s , namely t h e Eisenhart
surfaces, which a r e e x t r e m e l y i m p o r t a n t f r o m t h e s u r f a c e modeling point of view.
However, t h e r e a r e s u r f a c e s , equally i m p o r t a n t f r o m s t r u c t u r a l considerations in
a e r o s p a c e engineering which a r e not Eisenhart surfaces. Ogive IS o n e such
rotationally s y m m e t r i c Non-Eisenhart s u r f a c e w h e r e s u r f a c e ray analysis until
now has been e x t r e m e l y difficult.
All t h e r a y 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 analysis
h a s been obtained in this paper by a Geodesic C o n s t a n t Method (GCM) developed by
t h e authors. F u r t h e r m o r e , e q (2) implies t h a t t h e ray p a r a m e t e r s obtained h e r e
a r e applicable t o t h e c o m p l e t e c l a s s of rotationally s y m m e t r i c Non-Eisenhart
surfaces, in general. The d e m o n s t r a b l e e a s e of i t s application t o t h e s u r f a c e
analysis for relatively complex s u r f a c e s shows t h e power of t h e GCM.
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REFERENCES
[ I ] R.M. Jha: "Surface Ray Tracing on Convex Quadrics with Applications t o
Analysis of Antennas on Complex Aerospace Bodies", 39th I n t e r n a t i o n a l
Astronautical Congress of t h e International A s t r o n a u t i c a l F e d e r a t i o n ,
Bangalore, India, O c t . 8-15, 1988.
121 P. Moon and D.E. Spencer, Field Theory Handbook. Berlin: Springer-Verlag,
1971.
131 R.M. J h a , Surface Ray Tracing on
Convex Quadrics with Applications to
Mutual Coupling between Antennas on Aerospace 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.
[41 P.H. Pathak and N. Wang, "Ray analysis of m u t u a l coupling b e t w e e n a n t e n n a s
on a convex surface", IEEE Trans. Antennas & Propagat. (USA), vol. AP-29,
no. 6, pp. 911-922, Nov. 1981.
Fig. I
Ray analysis on a c i r c u l a r ogive.
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