Geometric Theory of Ray Tracing

Received 12 June 1969 13.2; 11.7

Geometric Theory of Ray Tracing

EDWARD S. EBY

U.S. Navy Underwater Sound Laboratory, Fort Trumbull, New London, Connecticut 06320

A temporal metric tensor is defined by combining the sound-speed function with the spatial metric tensor for a Riemannian space. Fermat's principle implies that spatial rays are temporal geodesics. Ray equations generalized to Riemannian spaces are shown to be temporal geodesic equations expressed in spatial terms. This geometric derivation leads to the consideration of geodesic deviation and its relation to three-dimensional spreading loss. Previous results [-E. S. Eby, "Frenet Formulation of Three-Dimensional Ray Tracing," J. Acoust. Soc. Amer. 42, 1287-1297 (1967)• are generalized to Riemannian spaces, and tensor expressions are derived for ray curvature and torsion.

INTRODUCTION

Physical laws governing stress, strain, mass conserva- tion, and pressure-density relations lead to the wave equation? 5 If the several physical parameters involved do not change too rapidly, the eikonal approximation to the wave equation can be used to define wavefronts (solutions of the eikonal equation). The family of curves normal to the wavefront surfaces are rays, and they can be shown to satisfy ray equations. 6

The ray equations alternately can be derived geo- metrically in a form generalized to a Riemannian (rather than a Euclidian) space by using the sound speed (a function of position) and the spatial metric tensor to define a temporal metric tensor. Fermat's principle, then, implies that spatial rays are temporal geodesics. The ray equations are the temporal geodesic equations expressed in terms of spatial quantities.

The geometric derivation presented here leads to the consideration of geodesic deviation and its relation to spreading loss in a three-dimensional Riemannian space.

Previously published results 7 obtained for ray tracing

1 j. W. Strutt Lord Rayleigh, The Theory of Sound (Macmillan and Company Ltd., London, 1896), 2nd ed., Vol. 2, Chap. XI (1940 reprint).

•' H. Lamb, The Dynamical Theory of Sound (Edward Arnold and Company, London, 1931), 2nd ed., Chap. VII, pp. 203 if.

a A. B. Wood, A Textbook of Sound (The Macmillan Co., New York, 1941), 2nd ed., pp. 56 ff.

4 p. G. Bergmann and A. Yaspan, Eds., ]>hysics of Sound in the Sea: ]>art I, Transmission, Nat. Def. Res. Counc. Div. 6, Sum. Tech. Rep., Vol. 8.

5 C. B. Officer, Introduction to the Theory of Sound Transmission with Application to the Ocean (McGraw-Hill Book Co., New York, 1958), Chap. I.

6 Ref. 5, Chap. 2, pp. 36-42. 7 E. S. Eby, "Frenet Formulation of Three-Dimensional Ray

Tracing," J. Acoust. Soc. Amer. 42, 1287-1297 (1967).

in three EuclidJan dimensions, using the Frenet formulas, are generalized to Riemannian spaces, and tensor expressions are derived for curvature and torsion of a ray.

I. RAY PATHS AS GEODESICS

For a Riemannian space with the spatial metric tensor gmn, the differential of arc length 8 is

dsS= gm,,dxmdx •. ( 1 )

We assume that sound speed c is a function of spatial coordinates only and introduce an associated temporal metric tensor a•= g,•,•/c 2. Then,

dF= a,•,dx'•dx'•(= g,•,dx•dx'•/c •= ds2/c•). (2)

Note that even if the space characterized by g,• is a EuclidJan space (a flat Riemannian space), the associated metric a,• characterizes, in general, a curved space. Since no essential simplification of the present theory is gained by restricting attention to EuclidJan space, we consider the more general Riemannian space.

Fermat's principle of least time implies that ray paths in the space characterized by the spatial metric tensor g,• are geodesics with respect to the temporal metric tensor am,•.

s Throughout this paper, the usual tensor notation with range and summation conventions is used. See, for example, J. L. Synge and A. Schild, Tensor Calculus (The University of Toronto Press, Toronto, 1952), for a discussion of tensor notation and these conventions.

The Journal of the Acoustical Society of America 273

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E. S. EBY

The equations of the temporal geodesics are 9

•(dx r d2x • { r } dxmdx n •\-•-t)=-•t •q- m n a d t d t • •=0, (3)

where •/•t is the absolute derivative 1ø with respect to't, and

{r} mn a

are the Christoffel symbols of the second kind computed for the temporal metric. Since the temporal metric is known in terms of the spatial metric, the geodesic equations can be translated into spatial terminology to obtain ray equations.

The Christoffel symbols of the first kind n are given by

l(Oap,•_} Oamp Oamn• [mn,p']• = •\ Ox---- •- Ox--- • Ox---• / 1 1/0 logc

=-•[-mn'p']a--c• • Ox m g•',* 0 logc 0 logc

-1 t- gmp OX---•'gmn), OX n

and those of the second kind 12 by

f r ] --ctrp[-mn,p•a m• a

(4)

0 logc __grp gmn , OX p (5)

since am,*=gm,*/d' implies amn---½•'g mn. The Christoffel symbols subscripted a are computed for the temporal metric (am,,), and those subscripted g for the spatial metric (gm,,).

Since d/dt=cd/ds implies

d9' (dd• dlogcd ) --=c 9' -[ -- ß (6• dF ds •s ' by substitution of Eqs. 5 and 6 into Eq. 3, we obtain the ray equations:

d•x• { r } dxmdxn dlogcdx • 01ogc --+ ..... . (7) ds 9' mn o ds ds ds ds grp, OX p Introducing the ray tangent T•= dx•/ds, we can write

dTr { r } dxn dlogc 01ogc •-[- Y m . = rr--g rp. , (8) ds mn • ds ds Ox p

9 j. L. Synge and A. Schild, Tensor Calculus (The University of Toronto Press, Toronto, 1952), Eq. (2.424).

10 Ref. 9, Eq. (2.511). 11 Ref. 9, Eq. (2.421). •2 Ref. 9, Eq. (2.422).

or

/iT • d logc O logc --=•r•--g •p- , (9) $s ds Ox p

where 8/Ss is the absolute derivative with respect to s. Equation 9 is the Eby-Frenet form of the ray equation 7 for Riemannian spaces.

Since c is a scalar, d(logc)/ds=8(logc)/8s, and Eq. 9 may be rewritten as

•S•-/=grp•(1/½)' (10) OX p

or, introducing the index of refraction n (which is Co/C, where ½0 is a constant reference sound speed), as

(a/as) (nrt)= g 'p On/Ox p. (11)

Equation 1! is the usual form of the ray equation 5 generalized to a Riemannian space.

II. GEODESIC DEVIATION

Let x t- xr(t,v) be a family of geodesics forming a two- dimensional subspace of the temporal Riemannian space. (The curves with v constant are the temporal geodesics, and t is temporal arc length along the geodesic.) Next, define geodesic deviation as the infinitesimal vector

Then, it can be shown 1• that, along a geodesic, v t satis- fies the Equation of Geodesic Deviation:

lis • dx • dx n ---•'Ra r .... --?] m----O, (13) $t •dt dt

where dxn/dt is the unit tangent vector to the geodesic (in terms of the temporal metric), and Ra t .... is the (temporal) mixed curvature tensor 14'

Ra r. smn OX m sn a OX n sm a

sn • pm a sm • pn • (14)

By a straightforward (but laborious) computation, Eq. 13 may be written, in terms of the spatial mixed curvature tensor Ra t..•mn, as

•2?]r dx p dx,, dx '• •?]• ---[-Rgr. pmn--?] m---- (10gC),n-- (•s 9' ds ds ds •s

t•?] m dx • dx,, dx • -- 2 (10gC),m-- 2 (10gC) ,m,,?] m --

•s ds ds ds

'-[-2grP(logc),p(lOgC),m?]m'nt-grP (logc),pm?]m=O, (1.5)

15 Ref. 9, pp. 90 if. 14 Ref. 9, p. 83.

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GEOMETRIC THEORY OF RAY TRACING

where (logc),• and (1ogc),•n are, respectively, the first and second covariant derivatives •5 of logc computed with respect to the spatial metric tensor

III. SPREADING LOSS IN A THREE- DIMENSIONAL SPACE

Let rt r= (Oxr/Ov)dv and •r= (Oxr/Ou)du be independent solutions of the second-order linear differential

equation, Eq. 15 (or Eq. 13), for family of rays emanat- ing from a common point x'(0), and measure arc length s(or t) from x'(0). Then rt'=rt'(s) and ur=u'(s) vanish for s-0. The general spreading-loss expression of Eby and Einstein, •6 generalized to a Riemannian three-space, gives the intensity ratio

I•/I= (e•Sk,%JTk)/(eiSk,%JTk),=•, (16)

where T'=dx'/ds is the ray tangent, rt' and u' the independent solutions of the equation of geodesic deviation, and egjk the permutation symbol. •7

Since Eq. 15 (or Eq. 13) is linear, linear combinations of the independent solutions, rt' and u', are also solutions. Let A,'q-Bu' and C,'q-Du' be independent (AD--BC•O) linear combinations of r/' and u'. Since

eok (A v•-+- Bta

we see that the intensity ratio given by Eq. 16 does not depend on our choice of the independent solutions and

Note that, to compute this intensity ratio, it is neces- sary to solve the ray equation (Eq. 3, 9, or 11) for the rays x •= x•(s), to substitute the rays into the equation of geodesic deviation (Eq. 13 or 15), to solve for =r/r(s) and/•=t•(s), and to combine these results in Eq. 16.

IV. THE FRENET FORMULATION

In a Riemannian three-space, the Frenet formulas •8 are

and

• T•/8s = q- •N •, (17) •Nr/•s = -- •T•-¾ rB •,

8B•/8s = --rN •,

where Tr=dx•/ds is the (unit) tangent, N • the (unit) normal, Br(= e"•nT,•Nn) the (unit) binormal, • the curvature, and r the torsion of the ray.

We rewrite the ray equation (Eq. 9) as

= r•r , ,• (•8) •X •' (logc),• --g (logc),• OE

•N•= (logc),•(T•T•--g•). (19)

• A. J. McConnell, Applications of the Absolute Differential Cal- culus (Blackie & Son Ltd., Glasgow, 1931), Chap. XII, pp. 140 ff.

•0 E. S. Eby and L. T. Einstein, "General Spreading-Loss Ex- pression," J. Acoust. Soc. Amer. 37, 933-934 (1965).

•* Ref. 9, Eq. (7.107). •s Ref. 15, p. 159.

Since gr•N•N•= 1 and g•TrT•= 1, Eq. 19 implies

= = g•, (logc),• (r • r •-- g•) (logc) ,, ( r n r s- gsn) -- (10gc),m(logc),ngr,

X (r'•rnT•r*--g"•rnr*--g*nr'•r•+g"•g *n) = (10gc),,,,(logc),n(g "*n-- Tmr").

(20)

Also, the expressions graTiNg=0, g•N•N •= 1, and Eq. 19 imply that the curvature can be expressed as

(21)

If the curvature does not vanish, Eq. 19 can be solved for the normal

N •= (1/•)(logc),,•(T"•T•--g"•). (22)

Substitution of

•;n = gnN

=

into the definition of the binormal yields

(23)

Differentiating the ray equation (Eq. 18) absolutely with respect to arc length s and simplifying, using the Frenet formulas and the relations (•i/as) (logc).• = (logc),mnr n and (•/•s)g•v=O, we obtain

--N•q-g(--•T•q-rB •) = (logc),mnTnTmT•-+ - (logc).,,

X (•N•)Trq - (logc).,•T•(•N • ) --g'v(logc),•,,•r •.

Computing the inner product of both sides of this expression with the binormal yields

•r= --grng'P(logc).,•vT'•B n

= -- (logc),,•nT'•B n.

Hence, we find the torsion is given by

r= -- (1/•)(logc) T'•B n (24)

Equations 18 through 24 generalize the corresponding results of Eby. •"

x• Ref. 7, Eqs. (44)-(49).

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Geometric Theory of Ray Tracing