CHR. GEORGIOU, TH. HASANIS AND D. K O U T R O U F I O T I S

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Consider an ovaloid M" in Euclidean E "+ 1 (n >t 1), the smooth and positively curved frontier of a bounded open convex set I(M"), that we shall call the interior of M". We pick a point O ~ I(M") and regard O as a light-source and M" as a mirror in vacuum. Thus, light rays emanating from O are reflected on M" in accordance with Snell's law of geometrical optics; this means that, reflection taking place at some P ~ M", the reflected ray lies in the same half-space with regard to the tangent hyperplane TpM" as O, and in the 2-plane spanned by the incident ray and the normal N(P) of M" at P; further, the angle between incident ray and N(P) is the same as the angle between reflected ray and N(P). The envelope of the family of reflected rays is called the caustic of the optical system {M", O}; it is a collection of submanifolds that may exhibit various types of degeneracy, being the set of focal points of a certain smooth hypersurface associated with our pair {M", O}, the so-called orthotomic. For example, an extreme case of degeneracy occurs when M z is an ellipsoid of revolution and O is one focus F1, in which case the caustic reduces to the other focus F 2. It is not hard to see that, for our choice of M" and O, at least some part of the caustic always lies in I(M"). The problem we shall consider in this article is to find sufficient conditions on Mn and O ensuring that the entire caustic lies in I(M"). Thus, our objective is two-fold: first to find a class of ovaloids M" for which there exist points O ~ I(M") with the property that the corresponding caustic lies entirely in I(M"), and then to locate such points O. It is noteworthy that not every ovaloid possesses interior points with the desired property; namely, we shall give an example of an oval M ~ for which any choice of 0 ~ I(M 1) yields a caustic that leaves the closure of I(M 1). In this example the curvature varies widely. The other extreme is the case of the circle where the curvature is constant and, for points O sufficiently close to the center, the corresponding caustic lies wholly in the interior, and is in fact arbitrarily close to the center. It is therefore reasonable to seek our ovaloids M" among those whose overall shape does not differ too much from that of a hypersphere, in some sense, and then try as light-sources points in I(M") which are not too close to the boundary M". A measure of roundness that turns out to be appropriate to our problem is the ratio of maximal to minimal principal Geometriae Dedicata 28 (1988), 153-169. © 1988 by Kluwer Academic Publishers.

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curvature; using this, we define explicitly a class of (mirror, source)-pairs that meets our objectives. T H E O R E M . Let M be an n-dimensional ovaloid in En+ l(n >>, 1), oriented so that the principal curvatures satisfy k I >1 ... >t k. > O. Suppose maxMk < (4/3)minMk i f n = 1, and maxMk 1 < (--1 +x~)minMk~ ifn > 1. There exists an open set U in the interior of M with the property that, for any 0 ~ U, the corresponding caustic lies entirely in the interior of M. The set U is the interior of the parallel ovaloid to M along the interior normal at a distance 3/(4minMk ) if n = 1, and 1/((-1 + x/~)minukn) if n > 1.

In proving this theorem, we shall use the description of the caustic of {M, O} as the focal set of the orthotomic. Since the orthotomic is obtained from the pedal of M with respect to O by a similarity with center O and factor 2, we shall briefly develop in the next two sections the theory of the pedal to the extent needed. In the final section this theory will be applied to caustics by reflection, and to the proof of the theorem in particular. 1. THE PEDAL If the proof of an assertion or formula in this section merely involves a computation, it will not be given. The interested reader will find complete proofs in [4] for n = 2, and in [3] for a general n. This holds in paticular for Propositions 1.1 and 1.2. Let M be a C°°-differentiable n-dimensional manifold (n >~ 1), which is also connected and oriented. We consider an immersion ~0: M ---,E" ÷ 1, pull back onto M the standard metric E "+ 1, and make the usual local identifications of M and q~(M). We choose an origin O ~ E "+ 1, denote by x the position vector of M, and set Ixl = r for the corresponding distance function. Let N be the unit normal vector field of M induced by the orientation. The support function f of M with respect to O is defined as f = - ( x , N ) . Setting x~ = t3x/t3u ~ and N i = ~N/~u ~in a chart (u 1. . . . , un), we have g~j = (x~, x j ) for the components of the metric tensor of M, and b~j = - ( x ~ , N j ) , n ~ i = ( N ~ , N j ) for the components of the second and third fundamental forms. For the principal curvatures of our hypersurface we adopt the numbering k 1 I> ... f> k n. The function K = kl ... kn is the Gauss-Kronecker curvature of M. Suppose there exists a point O that lies on no tangent hyperplane of M; we call such a point an admissible origin for M. Assuming we have chosen an admissible O as origin, the corresponding support function clearly never vanishes. Thus, because of connectivity, either f > 0 or f < 0 throughout. It follows that we can always choose an orientation of M which m a k e s f > 0. The

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corresponding vectorfield N at every P • M points in the half-space, determined by TpM, which contains the origin. In the sequel we shall always impose on M this preferred orientation, without mentioning this choice each time. D E F I N I T I O N . The pedal M~ with respect to O of the oriented hypersurface M, is the hypersurface with position vector x ~ = - f N with respect to O, and orientation the one inherited from M through the pedal transformation n: M --* M~. Here the point n(P) is the foot of the perpendicular through O to the tangent hyperplane of M at P. Thus, the pedal is the pair (M, n o ~0), and regularity of M~ means that the differential dl,(n o ~o)is injective at every P • M; that is, the n vectors Ox'/Ou i are linearly independent everywhere. P R O P O S I T I O N 1.1. For n >~2, M~ is a regular hypersurface if and only if the following two statements hold: (i) The Gauss-Kronecker curvature K of M is different from zero everywhere. (ii) The origin 0 is admissible for M. For n = 1, M~ is a regular curve if and only if M has curvature different from zero, and 0 does not lie on M. Suppose the spherical-image mapping of M is one-to-one and M , is regular; then n is a diffeomorphism of M onto M~, and M~ is star-shaped with respect to the origin. To see this, first note that n(P) = n(Q) implies that the tangent hyperplanes of M at P and Q coincide, hence so do the normals N(P) and N(Q) since they point in the same half-space. The fact that M~ is star-shaped follows now immediately from the formula for its position vector. Since n is one-to-one and M~ is regular, n is a local diffeomorphism. If, in particular, M is an ovaloid (oval for n = 1), a compact imbedded hypersurface with definite second fundamental form that is, and O is chosen in the interior I(M) of M, then, by Hadamard's theorem, rc is a global diffeomorphism of M onto the compact star-shaped hypersurface M~. In this case M , is imbedded, hence it is an ovaloid if and only if all its principal curvatures are nonzero and of the same sign. Example. Take a circle S 1 of radius 1 and center ( - d, 0) in the (x 1, x2)-plane. By Proposition 1.1, the pedal of S 1 with respect to any point O ¢ S 1 is a regular curve. The equation of the pedal of S 1 with respect to (0, 0) is easily computed in polar coordinates (q~,p) round the origin: p = 1 - d cos tp; this curve is an oval for 0 ~< d < ½, and a non-convex simple closed curve for ½ < d < 1. For d = 1 we obtain the cardioid, with the cusp at q~ = 0. For d > 1 the pedal is a closed curve with one double point at ~o = 0. Now we think of our plane as

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lying in E "+ 1, and rotate the whole configuration round the xl-axis. Clearly, we obtain in this manner the pedals of the hypersphere for the various choices of origin. In fact, let S" be the unit hypersphere, and S"~/2 the concentric hypersphere of radius ½. We conclude from the above considerations that all points inside S]/2, and only those, furnish as origins pedals that are ovaloids. The two extreme choices are the center of S" as origin, in which case the pedal is the same S", and the origin on S]/2, in which case there appear for the first time pedals that have points where the Gauss-Kronecker curvature is zero. If the origin is chosen in the open annulus between S]/2 and S", the pedal bounds a nonconvex, star-shaped compact set. For origins outside S", the pedal has one singular point if n > 1. P R O P O S I T I O N 1.2. The unit normal N" of M~ at the point ~(P), is given by (1.1)

N ~ - signK(x

+

r

2iN),

where x, N, etc., are computed at P. I f Tz(P) ~ P, the vector N ~ lies in the plane determined by the vector x(P) and the point ~(P);furthermore, the carrier of N ~ passes through the midpoint of the segment OP. We can now easily compute the components of the fundamental forms of M s in the local parameters {ui}: (1.2)

g~ = f i f j +f2no,

b'~ - sign K (2f~fj - fbij + 2f2nij), r

where f~ = Of/Oui. There are no simple formulas for the curvature functions of the pedal in arbitrary dimension. Even in the case of surfaces, where such formulas can be obtained, they are rather complicated [4,p. 136]. There is, however, an indirect way of obtaining information about the curvatures of M s through a classical decomposition of the pedal transformation into two involutive transformations. This decomposition is useful in reducing the complexity of various problems, as we shall have occasion to ascertain further on. We describe these geometric transformations briefly. Suppose M is an orientable hypersurface with K ~ 0, and O an admissible origin for M. We give M the preferred orientation ( f > 0). The polar reciprocal Mp of M with respect to O is the hypersurface with position vector y - - N/f, and orientation the one inherited from M through the defining transformation p: M ~ Mp. Because K ~ 0, we may assume that, in the local parameters {u~} of M, the n vectors ON/Ou ~are an orthonormal set at one point P; this simplifies the calculation of the following exterior product at P:

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X

(1.3)

Yl ^ "" ^ Y, = - ( s i g n K ) f , + l ,

where y~ = ~?y/~u~. It follows that Mp is a regular hypersurface. Its first fundamental form is easily computed to be (1.4)

gf. J

f/fJ +fZnlJ f4

From (1.3) we obtain immediately the normal of Mo, X

N o = - (sign K ) - , r

and thence its second fundamental form: (1.5)

bei _ sign fr K blj.

Supposing now, as we may, that the orthonormal vectors t?N/~3u~ are, in addition, in principal directions at P, we can compute the Gauss-Kronecker curvature of Mp from the formulas K p = det(bej)/det(9~), and we find (1.6)

f n+ 2

K ° = (sign K)" r" +2 K"

Finally, the support function fP = - ( y, N p) of M p is (1.7)

f o _ sign K, r

and this implies that O is an admissible origin for Mp as well; in particular, O does not lie on M o. The second transformation we need is the well-known inversion. Let M be an oriented hypersurface, and O a point that does not lie on M. The inverse M , of M with respect to O is the hypersurface with position vector z = x/r 2 and orientation the one inherited from M through the defining transformation a: M --, M , . This M , is a regular hypersurface with fundamental forms g~.

glj = ~,

b~-

2fg~j

b~j

r4

r2.

From these formulas we infer that inversion preserves principal directions. Further, we deduce from them that the principal curvatures k7 of M~ are given by (1.8)

k~' = 2 f - r2kl,

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where k i is the principal curvature of M at the corresponding point and in the corresponding principal direction. We verify directly that the pedal transformation n for a given M and a fixed admissible origin, is the composition ~ = o"p.

It follows, using (1.7) and (1.8), that the principal curvatures k~ of M~ and k~' of Mp at corresponding points and corresponding principal directions, are related by (1.9)

2(signK) k~ - - F

kf f2"

We shall have occasion to use this formula later; we note here that it gives us the curvature of the pedal of a plane curve C with k ~ O, with respect to an admissible origin; indeed, using (1.6) for n = 1, we obtain from (1.9): (1.10)

k~ - sign r k (2 - r2~)'

which is in fact valid even if the origin lies on a tangent line to C, but not on C. 2.

CONVEXITY

OF THE PEDAL

Let M be an ovaloid in E" + 1. As we know, any point in I(M) is an admissible origin, and the corresponding M~ is a compact hypersurface, star-shaped with respect to the origin O. Even if O is chosen 'centrally', M~ need not be an ovaloid. For example, the general ellipsoid (xE/a z) + (y2/b2) + (z2/c 2) 1 has as pedal with respect to its center the so-called surface of elasticity of Fresnel (x 2 + yZ + zZ)Z = aZx z + b2y2 + eZz2 [7, p. 159], which is not an ovaloid if the ellipsoid has sufficiently elongated shape. One can easily show, however, that each focus of an ellipsoid of revolution possesses a neighborhood, all of whose points as origins furnish pedals that are ovaloids. Thus, we may ask whether there exist always origins inside the given ovaloid M, with ovaloids as corresponding M~; the answer is no. It is sufficient to construct an example in the case of plane curves. First, consider generally an oval C, oriented so that k > 0. We choose an origin O inside C, and we construct the pedal C~. Suppose now that C~ is an oval. Clearly, we have again k ~ > 0, and formula (1.10) gives us k > f/(2r2); this is a necessary and sufficient condition on O ~ I(C) in order that C~ be an oval. Now take a square of length 1, and an oval C inscribed in it as in Figure 1. Suppose k(A) = k(B) <~~. Pick an origin O, say above the horizontal axis of symmetry of the square. Thenf(B) >f ½. If C~ with respect to =

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A

B Fig. 1.

O were an oval, then we would have r 2(B) > f(B)/(2k(B)) >~2, which is absurd since r 2 < 2 patently. The same argument applies if 0 is chosen below the axis of symmetry. Thus, for no choice of O ~ I(C) is C, an oval in this example. In order to obtain a sufficient condition ensuring the existence of 0 e I(M) with M~ an ovaloid, we take a hint from the example of S": it suggests that if M is sufficiently spherical in shape, there exists in I(M) a large open set of points of the desired type. Before we formulate precisely what we mean, we derive some useful formulas; it is convenient at this point to use the method of connections. Let M be an arbitrary oriented hypersurface and 0 an arbitrary origin. We decompose the position vector x of M in a component normal to M, and a component x r tangent to M: X

=

X T

-fN.

Let V2be the standard connection of E n÷ 1, and V the induced connection on M. Let X be a tangent vector of M. Since r 2 = (x, x ) and (TxX = X, we have X(r 2) = 2 ( x r, X). Now X(r 2) = 2rXr = 2r (grad r, X), hence Xr grad r = - - . r

The equations of Gauss and Weingarten are, respectively,

V7x Y = V x Y+ ( A X , Y ) N ,

VTxN = - A X ,

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where X, Y are vector fields tangent to M; using them, we compute: X = g?xX = gTx(Xr - f N )

= gTxX r - ( X f ) N - f g T x N

= Vxx r + (AX, xT)N - (Xf)N +fAX. Taking the normal component of this equation, we obtain (2.1)

A x T = gradf.

We need a lemma that will be used repeatedly in the sequel: L E M M A 2.1. Let M be an oriented ovaloid in E"+ l (n >1 1), whose positive principal curvatures satisfy max M k I < • min M k, for some constant ct > 1. Then: (i) the parallel hypersurface at a distance 1/(e minM k.) along the interior normal of M is an ovaloid ffl which lies in the interior of M; (ii) if the origin is chosen inside ffl, at every point of M we have f > l/(~t k.); (iii) if the origin is chosen inside ffI, then minM(f/r ) > 1/x/~. Proof. (i) According to Blaschke's first rolling theorem for hypersurfaces [6], the hypersphere of radius (maxM kl)-1 rolls freely inside M; that is, if it touches M internally at some point, then it lies entirely in the closure of the interior of M. If x is the position vector of M with respect to some origin, the parallel hypersurface ~r has position vector ~ = x + (0~rain k , ) - l N . Since (max M kl )- 1 > (e minM k,)- 1, the compact set M lies in the interior of M. Suppose we choose parameters u ~ on M so that all the x~ are in principal direction at P e M; then N~ = -k~ x~ at P, hence (2.2)

xi = [1 - (~ m i n k , ) - 1 kl]x v

The expression in the brackets is positive, by assumption; therefore, the ff~ are linearly independent at P, and )~ is a regular compact hypersurface. The mapping M ~ M defining M preserves principal directions, as is well known. Let ~ denote the principal curvature of )~ that corresponds to k~; we have (~i)-1 = ( k i ) - l _ (~minMk,)-i > 0. A theorem by Chern and Lashof [2, p. 317] states that a compact immersed orientable hypersurface is imbedded as the boundary of a convex body if and only if the degree of its spherical-image mapping is _ 1 and the Gauss-Kronecker curvature does not change sign. For our M, the degree in question is _ 1 because M and M have the same normal up to sign at corresponding points; in fact, we infer from (2.2) that, if N is unit normal of ~ in the induced orientation at a point which corresponds to a point of M with unit normal N, then ~7 = N. Therefore, 2~ is an ovaloid. Of course, the information on the degree is needed only in the case of curves (n = 1).

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(ii) Suppose the origin O is chosen in the interior of M. Let P E M, R ~ the point corresponding to P, and Q the foot of the perpendicular from O to the tangent hyperplane of M at P; we have t i P ) = [OQI > [RP[ = (~ rain M k,)- ~ >i (a ki)- 1 for all i. (iii) If M is a hypersphere and O its center, then the assertion is trivially true; so we suppose that O is not the center of a hypersphere M. The function f /r attains on M an absolute maximum equal to 1 at every point whose position vector is normal to M. It follows that the absolute minimum off/r over M is strictly less than 1, and that the position vector of such a point Po of minimum is not normal to M, hence XT(Po)~ O. Let X ~ TpoM: we have X(f/r) = O, which can be written using (2.1) as (rAXT--(f/r)XT, X ) leo=O, so Ax r = (f/rZ)xr at Po. It follows that XT(Po) is an eigenvector of A, with corresponding principal curvature, say, k,(Po)= (f/rZ)(Po). According to statement (ii) of our lemma,

(f/r2)(Po) >1 k.(Po) > (~f(Po)) -1

so

(f2/r2)(Po) > a -1.

P R O P O S I T I O N 2.2. Let M be an ovaloid in E "+ 1 (n >~ 1), oriented so that the normal points in the interior. Suppose the positive principal curvatures satisfy maxM kl < 2 minu k,. Then the parallel hypersurface along the normal at a distance 1/(2 min M k,) is an ovaloid ffl which lies in the interior of M; ifO is any point in the interior of ffi, then M s with respect to 0 as origin is an ovaloid. Proof First we derive a sufficient condition for M s to have definite second fundamental form. We may assume that at some P ~ M the third fundamental form (nij) of M is the unit matrix, and the second fundamental form (bij) is a diagonal matrix; then the diagonal elements of (b~j) are (k~)-1, and the expression (1.2) of b~ takes the form b~ = r - l[2f/fj + f ( 2 f - (ki)-l)bij] at e. Since f > 0 and (f~fj) positive-semidefinite, (b~) is positive-defnite if there exists a point O inside M, with respect to which as origin we h a v e f > (2kl)-1 for all i everywhere. Consider now the parallel hypersurface J~ along the interior normal at a distance (2 min M k,)-1: according to Lemma 2.1, it is an ovaloid in the interior of M; and if O is chosen in the interior of ~r, then f > (2k~)-1. Thus any origin inside M yields a compact imbedded M~ with k~ > O, an ovaloid that is.

3.

THE CAUSTIC

Let M be a connected and oriented hypersurface in E" + 1, and O an admissible origin for M. We think of M as a mirror, and of O as a light-source in vacuum. Light rays emanating from O hit M, and are reflected following Snell's law

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described in the Introduction. The mirror need not have K ~ 0, so the pedal of M may not be regular. If, however, we have K ~ 0, then the reflected ray at P ~ M is parallel to the normal of M~ at n(P). The usefulness of the pedal in geometrical optics is based on just this observation. In order to prove it, denote by po the point that is in symmetric position to O with respect to TeM, and draw the line l through pO and P; obviously, the ray reflected at P, is on 1. By Proposition 1.2, the normal N ~ at n(P) passes through the middle of the segment OP; consequently, by Thales' theorem, I is parallel to N ~. In fact, using simple vector calculus, we establish that the unit vector in the direction of the oriented reflected ray is

x + 2fN Y

whether the pedal of M exists as regular surface or not. When P ranges over M, the point pO describes a certain set M o which is called the orthotomic of M with respect to O I-5, p. 18]; we obtain it, clearly, by performing on M~ a similarity with center O and factor 2. Thus, the orthotomic is a regular hypersurface if and only if the pedal is a regular hypersurface. The envelope of the family of reflected rays is called classically the caustic of the optical system {M, O}. We give a description of the caustic in terms of the orthotomic Mo; this is more convenient in computations. Suppose Mo is regular. The line through pO and P is perpendicular to M 0 at pO. Hence, the caustic is the envelope of the normal lines of Mo; equivalently, it is the set of critical values of the mapping M o x R ~ E" + 1, {pO, t} ~ 2x ~ + tN". Therefore, the caustic is the collection of focal hypersurfaces of the orthotomic of M, assuming K ~ 0 on M. Of course, these n focal hypersurfaces may not be hypersurfaces at all; they may be lower-dimensional manifolds; some of them may coalesce, or vanish into infinity. Caustics have been investigated recently by Bruce, Giblin and Gibson in a series of papers, mainly from the point of view of differential topology and singularity theory; consult [1] and the references therein. Consider now an ovaloid M and an origin O in the interior of M. Does at least some part of the caustic of M with respect to O lie in the interior of M? This is a problem considered in [1, p. 193], where it is shown, using a theory of contact, that the answer is yes. In fact, ifPo ~ M is a point at which the distance function from O to M attains its maximum, these authors prove that all the points of the caustic corresponding to those in a sufficiently small neighborhood of Po, lie inside M. We can obtain this result easily using the polar reciprocal. Let k° = ~k~ 1 be the principal curvatures of the orthotomic M o of M, the orientation of M being the preferred one (k~ > 0). By continuity, it is sufficient

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to show that all the caustic points corresponding to Po lie in the interior of M. According to formula (1.9), the principal curvatures o f M o are related to those of Mp at corresponding points and corresponding principal directions by (3.1)

ko -

1

kf

r

2 f 2'

At the point Po, the position vector is perpendicular to M, h e n c e f has a critical value f(Po) = r(Po), and the fundamental forms of Mp assume the form 9~].j = nij/r 2, bfj = b J r 2 from (1.4) and (1.5); we deduce immediately from these that kf(Po) = 1/ki(Po). Further, the hypersphere of center O and radius r(Po) encloses M and is tangent to it at Po, since r(Po)= maxMr, therefore k~(Po) >1 1/r(Po). Putting all this together, we obtain at Po: 2r ~< k° =

1-

< -r ,

or r ( P o ) < 1/k ° (Po)<<-2r(Po), which clearly implies that all caustic points corresponding to Po lie in the interior of M. We ask now for a condition on M and O ensuring that the whole caustic lies in the interior of M. For this, it is necessary first that M o be an ovaloid. Proposition 2.2 furnishes sufficient conditions on M and O for M o to be an ovaloid, but we need a stronger pinching of the ratios of normal curvatures of M in order to be sure that the normal curvatures of Mo do not approach zero when O is picked close to .~t; this is seen most clearly in the case where M is a hypersphere of radius R, and ~t is the concentric hypersphere of radius R/2: if O s ~r, the pedal has points where K = 0. A measure of how 'round' M should be, and how 'centrally' O should be chosen, so that the caustic lie in I(M), is furnished by our theorem stated in the Introduction. Observe that this theorem is somewhat better for curves (n = 1, pinching constant 4/3) than for hypersurfaces (n > 1, pinching constant - 1 + x/~). In proving it, we shall use the following (compare [4, Lemma 5.3]): L E M M A 3.1. Let P be a point of the oriented hypersurface M, where K ¢0, and

0 an admissible origin for M. A tangent vector v ~ 0 at P is transformed under p into a principal vector of Mp at p(P) if and only ifthefollowin9 equation holds at P for a certain number kO(v): (3.2)

-f v - f ( x r , v ) x r = (sign K)k °(v)Av; r

this kP(v) is the normal curvature of Mp in the direction dp(v). Proof We introduce parameters (u 1. . . . . u") in a vicinity of P, and write

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v = E 2'(~3/~u ~) at P. We m a y suppose that the vectors x i = dx(c3/c3ui) are of length 1, and in principal directions at P, so that N~ = - k i x~ at P. The image of v under p is the nonzero vector dp(v) = dy(v), where y = - N/f; we c o m p u t e it: d y ( v ) = ~ 2idy

(3.3)

~

=~

2'fi N - f

2iyi=

~ 2iN i

1 1 1 = ~ 2 ( g r a d f , v ) N + ~ ~2ikixi=-f-i(Axr, v)N +~f ~ 2iAxi

= ~1( A X T , V)N + f A v . The unit n o r m a l of Mp is N p = - (sign K)x/r, therefore: (sign K) dN°(v) = (sign K) ~ 2 ~Nf = ~ i

(3.4)

=

1 ~(grad

1

)

)o~r~ x - -

1

r,v)x - -Vr = ~ ( x T / r ' v ) X

= -~ (Xr' V)XT - r

2 ~x~ .

1 -- -Vr

-- (xT' v)N,

whereby v is identified with Z 2 i x i. According to Rodrigues' theorem, dy(v) is in a principal corresponding n o r m a l curvature k"(v) if and only if dNP(v) + Substituting in this equation the expressions (3.3) and (3.4), n o r m a l and tangential components, we obtain (3.2) and the

f 3 (x T, v)

r

direction with k p (v) dy(v) = 0. and separating relation

= (sign K)kP(v)(AXT, v),

which is satisfied if (3.2) is, since we obtain it from (3.2) by scalar multiplication with x z and use of the identity x 2 = r 2 _ f 2 . Proof of the theorem. According to L e m m a 2.1, the parallel hypersurface to M along N at a distance 3/4 min u k for n = 1, and 1 / ( - 1 + x/~)minM k, for n > 1, is an ovaloid M inside M. Let O be a point in the interior of M. Set ~ for n = 1 and 0t = - 1 + x//5 for n > 1. Because ~ < 2, we conclude from Proposition 2.2 that O is an admissible origin for M, with respect to which M~, hence also the o r t h o t o m i c Mo, is an ovaloid. By our choice of orientation, sign K = + 1, so the n o r m a l of M~ is the interior one, and the principal curvatures k~ are all positive. The principal curvatures of M o are k ° = ~k~ 1 >0. Let pO ~ Mo ' and P the uniquely determined corresponding point of M. Consider the triangle OPP°; we have IP°PI = IPOI = r, and the line ppO is the

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165

carrier of the normal of M o at pO. Since the points of the caustic of M corresponding to P are the focal points o f M 0 corresponding to pO, we have to show first that 1/k ° > r for all i at P, and then find an appropriate bound for the 1/k ° from above. To this purpose, consider the hypersphere E1 with center x(P) + (1/max M kl)N(P) and radius 1/max M k l , and the hypersphere E z with center x(P) + (1/e min M k,)N(P) and radius 1/e min M kn; Y~I and Z 2 are both tangent to M at P, and in fact lg 2 lies in the interior of E 1, except for the point P, because max u k 1 < • minM kn. By Blaschke's first rolling theorem, Z 1 lies in the closed convex body bounded by M, hence so does E 2. Prolong now the segment pOp beyond P until it meets the point Q e 2 2 in the interior of M. p0

P

zr(p)

Fig. 2.

It is sufficient to show (3.5)

1

0 < ~ - r ~< LPQI.

We compute the quantities in these inequalities. Let T be the point of intersection of the normal line to M at P with E2, co the acute angle between N(P) and PQ. We have IPQI = IPTI cos co = (2/a min M k,) cos c~. Since co is also the angle between x(P) and N(P), cos co = fir; therefore [PQ[ = 2liar min M k,. In order to compute k °, we fix at re(P) a k7 and a corresponding principal direction w. Let v be the uniquely determined tangent direction at P, which is mapped under dTr into w, and k the corresponding normal curvature. Now we recall that inversion preserves principal directions and use formula (3.1), where kf is the principal curvature o f M o at p(P) in the uniquely determined principal

166

CHR. G E O R G I O U ET AL.

direction that is m a p p e d under d a into w. Since the tangent direction v at P is m a p p e d under dp into a principal direction, we m a y apply L e m m a 3.1 to c o m p u t e k~'. Suppose Ivl = 1, so that (Av, v) = k; we multiply E q u a t i o n (3.2) with v, and" solve for kP(v) = ke: (3.6)

i(

kf = ~

1

r2

j.

Combining (3.1) and (3.6), we obtain:

(3.7)

1 kO

r 2fr 2 k

r -

r 2 _ (x~,v) 2

1

We now prove the first inequality in (3.5). If ~o is the angle between x r and v, then r 2 -- (XT, V) 2 = r 2 - - I x r [ 2 c o s 2 cp

= r 2 _ (r 2 --f2)cos2 (p = r 2 sin 2 q~ + f 2 cos z ~o. Using L e m m a 2.1(ii) and ~ < 2, we have f2

2 f k > otfk > 1 >/sin 2 ~o + r~ cos 2 ~o,

hence 2 f r 2 k > r 2 _ ( x r , v) 2,

so that, by (3.7), (1/k °) - r > O. It remains to show r

2fr 2 k r2 _

(XT,

V) 2

1

~ < - 2ocm i n k .

f

r

or, equivalently, (3.8)

r2 - (xT' v)2 2fr 2

~ min k~ r E - ( x r, v) 2 + - ~k. 4

f2

In the case n = 1, ( x r , v) 2 = [xr[ 2, so f 2 = r 2 - ( x r , v)2 and the above inequality (3.8) becomes f ~ mink + ~k. 2r 2 4

ON THE CAUSTIC

OF A CONVEX

MIRROR

167

By our choice of O, we h a v e f > 1/a min M k and thus • min M k/2 >f/2r2; thus, it is sufficient to show in this case that

mink - - + 2

~mink ~
or

3~mink --~
which is satisfied for ~ - 4. In the case n > 1, it is clearly sufficient to prove, instead of (3.8), the following:

1 (3.9)

• mink.(r~ 2

~+

4

\~,) ~
Since f > 1/e minM k, by the choice of O, and f / r > 1/x/~ by using Lemma 3.1(iii), it is enough to show, instead of (3.9), that the inequality

is satisfied; this is indeed the case for ct = - 1 + v/5, the expression in the parentheses then becoming 1. This completes the proof. In view of this result, it is instructive to consider finally the case where M is the hypersphere S~ (n ~> 1), of radius R. Since S~ is a hypersurface of revolution with axis passing through the center and the origin O, we may apply our theorem for n = 1; it tells us that the caustic lies entirely inside S~ ifO is chosen in the interior of the concentric sphere of radius R/4. This condition on O is sufficient, but not necessary. In fact, we shall show now that the largest concentric sphere all of whose interior points as origins yield caustics inside S~, has radius R/3. Let O be a point in the interior of a circle S 1. We apply formula (1.10) with sign k = + 1, k = 1/R,k ° = k~/2, and obtain

1 k°

Rfr r

--

-

-

2r 2 - R f"

Let P e S~, pO the corresponding point on the orthotomic. We prolong the segment pop until it intersects S~ at the point Q. The caustic with respect to 0 lies entirely inside S~ if and only if, for every P e S~, (3.10)

0 <

Rfr

2r 2 -- R f

< IPQI.

If/5 is the antipode of P, then [PQ[ = IPPI cos e) = 2R f/r, where co is the acute angle between the position vector of P and N(P). Using this, the second

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CHR. G E O R G I O U ET AL.

inequality in (3.10) is equivalent to (3.11)

r 2 2R ~- > -~-.

Since the first inequality in (3.10) is equivalent to r 2 / f > R/2, we need only make certain that (3.11) is satisfied. Draw from O the perpendicular line to the position vector of P; it intersects the line through P and the center of the circle at a point T, and [PT[ cos 09 = r, s o r 2 / f = [PT]. Thus, O has the desired location if and only if 2R min IP T[ > - p 3 This minimum is obviously attained when P is the point of intersection of S~ and the ray starting at the center Z and passing through O, and for this P we have [PT] = R - IOZl, or equivalently IOZl < R/3. 4.

CONCLUDING

REMARKS

Our main theorem essentially says that a mirror close to a hypersphere ('close' with respect to a metric that takes account of ordinary distance in E n+ 1, but also of curvature) and a source close to the center of this hypersphere, generate a caustic located completely inside the mirror. N o w this is not too surprising, since the caustic of a hypersphere with respect to its center is that same center, and we have a certain continuous dependence of caustic on mirror and source. Therefore, the most important feature of the theorem is that it gives explicitly a class of (mirror, source)-pairs having the desired property, in terms of the ratio ct = max kl/min kn: only mirrors with ~ smaller than some given number are allowed. This class is certainly not the largest possible. In any dimension, the caustic of an ellipsoid of revolution with respect to a focus is the other focus. N o w the continuity mentioned above implies that every ovaloid sufficiently close in the above sense to an ellipsoid of revolution, has an internal caustic for sources near a focus of the ellipsoid. Thus, we find a class of solutions which is considerably larger than the one given in the main theorem: it contains the latter, but it also contains hypersurfaces with arbitrarily large ~. Does there exist yet another class of solutions, containing hypersurfaces not close to any ellipsoid of revolution? At the end of the preceding section we saw on the example of the sphere that the number ~ in our theorem is not always the best. We may thus ask whether a larger ct still works in the general case, or if there exist mirrors for which our is optimal, in some sense. Also, does the difference in our constant ~ for n = 1 and n > 1 have a geometrical significance?

ON THE CAUSTIC OF A CONVEX MIRROR

169

At the present, we have no answers, or even conjectures, to any of these questions. C o m p u t e r e x p e r i m e n t s m a y be helpful in this connection. W e t h a n k the referee for his close r e a d i n g of the first draft of o u r paper, which led to v a r i o u s i m p r o v e m e n t s . W e have i n c o r p o r a t e d his r e m a r k s a n d questions a l m o s t v e r b a t i m in this last section of the paper.

REFERENCES 1. Bruce, J. W., Giblin, P. J. and Gibson, C. G., 'On Caustics by Reflexion', Topology 21 (1982), 179-199. 2. Chern, S. S. and Lashof, R. K., 'On the Total Curvature of Immersed Manifolds', Amer. J. Math. 79 (1957), 306-318. 3. Georgiou, Chr., Hasanis, Th. and Koutroufiotis, D., 'The Pedal Revisited', Technical Report 100, University of Ioannina, Department of Mathematics, 1984. 4. Hasanis, Th. and Koutroufiotis, D., 'The Characteristic Mapping of a Reflector', J. Geom. 24 (1985), 131-167. 5. Lockwood, E. H., A Book of Curves, Cambridge Univ. Press, 1961. 6. Rauch, J., 'An Inclusion Theorem for Ovaloids with Comparable Second Fundamental Forms', J. Diff. Geom. 9 (1974), 501-505. 7. Salmon, G., A Treatise on the Analytic Geometry of Three Dimensions, Vol. II, Chelsea Publishing Company, New York (1965). Authors' addresses:

Ch. G e o r g i o u a n d Th. Hasanis, D e p a r t m e n t of M a t h e m a t i c s , U n i v e r s i t y of I o a n n i n a , 45110 Ioannina, Greece. D. K o u t r o u f i o t i s , D i m o h a r o u s Str. 29, 11521 Athens, Greece. (Received, March 11, 1986; revised version, March 8, 1988)