Generalized antiorthotomics and their singularities

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1 ISTITUTE O PHYSICSPUBLISHIG Inverse Problems 18 (2002) IVERSE PROBLEMS PII: S (02) Generalized antiorthotomics and their singularities Alamo 1 and C Criado 2 1 Departamento de Algebra, Geometria y Topologia, Universidad de Malaga, Malaga, Spain 2 Departamento de isica Aplicada I, Universidad de Malaga, Malaga, Spain nieves@agt.cie.uma.es and c criado@uma.es Received 31 January 2002, in final form 26 April 2002 Published 14 May 2002 Onlineat stacks.iop.org/ip/18/881 Abstract In this paper we construct, for a given wavefront and a source point, an uniparametric family of mirrors with the characteristic property of producing the given wavefront, after reflecting on each mirror, from the source point. This family is made up of the envelopes of certain families of hyperquadrics of revolution, and includes the antiorthotomic of the wavefront with respect to the source point. We prove that the singularities of these generalized antiorthotomics sweep out the caustic of the wavefront. We also give a method to determine the generalized antiorthotomics starting from the associated caustic. 1. Introduction The orthotomic and antiorthotomic concepts arise when we study, in the context of geometrical optics, the following situation. Suppose that we have given a hypersurface of R n+1, M and a point not in M. In what follows M will be called the mirror and the point source of light. Suppose that is a hypersurface of R n+1 such that its normal lines coincide with the straight lines that emerge from after reflecting on M. The hypersurfaces are called wavefronts. A classical problem is to determine a wavefront from the data of the point source and the mirror M. A particular wavefront can be constructed as the locus of all points obtained by reflecting in the tangent hyperplane of M at each point x M. This wavefront is usually called the orthotomic of M with respect to (see [1]). It seems that Adolphe Quételet was the first to study the orthotomic notion, also known as the secondary caustic [2]. Consider now the inverse problem, that is, we have given the wavefront and the source point, and we want to construct a mirror M such that is the orthotomic of M with respect to. Such a hypersurface M is then called the antiorthotomic of with respect to. rom the construction of the orthotomic it is obvious that the antiorthotomic is the envelope of the family of hyperplanes orthogonal to the straight line joining to each point of, and passing through its middle point. Keeping this idea in mind, we will give a method based on the optical properties of conics, to obtain a family of mirrors M a parametrized by a non-negative /02/ $ IOP Publishing Ltd Printed in the UK 881

2 882 Alamo and C Criado z y 2a x Q h a x d igure 1. Illustration of the characteristic optical property of the hyperbola h a x. This hyperbola has foci and x ;2a is the distance between vertices and d is the directrix. real number a, with the property of producing lines normal to after reflecting the rays that emerge from on each mirror. We have called the mirror M a the a-antiorthotomic of the wavefront with respect to the point source, because for a = 0 we recover the usual antiorthotomic described above. The construction of the a-antiorthotomic of with respect to proceeds by taking the envelope of a family of hyperquadrics of revolution (hyperboloids and ellipsoids), instead of the envelopes of hyperplanes. Basically, the hyperquadrics are those with foci, a point x varying in and with eccentricity x /2a (see figure 1). In the limit case of a = 0the hyperquadrics become hyperplanes. We will prove in the next section that M a is a suitable mirror (theorem 2.1) and that, under very general conditions (theorem 2.2) is an immersed submanifold of R n+1, with singularities in the points corresponding to the centres of curvature of. Moreover, we prove in theorem 2.3 that the singularities of the family of a-antiorthotomics sweep out the caustic of the wavefront. In section 3 we analyse some illustrative examples. inally, in section 4 we relate the a-antiorthotomic of with respect to with the caustic of, that is the locus of its centres of curvature. Specifically, we describe a method to construct the mirror M a as the envelope of certain hyperquadrics of revolution which have foci and a point varying in the caustic of (theorem 4.1). 2. The generalized antiorthotomics 2.1. Construction We now generalize the notion of antiorthotomic by considering the envelopes of certain families of hyperquadrics of revolution instead of the envelopes of hyperplanes. Let be an orientable smooth hypersurface of R n+1 and let be a point of R n+1 not contained in. Let a > 0. or any point x in, consider the number e = d(x, )/2a. Let h a (x) be the hyperquadric of revolution which has focus and whose directrix d x is the closest to of the two orthogonal hyperplanes to the straight line passing through and x, and that is at a distance of a/e from the centre of the segment x (see figure 1). Recall that h a (x) is the locus of all points y R n+1 such that the ratio of the distance from y to and the distance from y to d x is a constant and equals the eccentricity e (see [3]). Depending on whether the eccentricity e is greater or less than 1, h a (x) is a hyperboloid or an ellipsoid of revolution, and in both cases has foci and x, and the distance between its vertices is equal to 2a. ore = 1, h a (x) degenerates to the straight line passing through and x. It is convenient to include the case a = 0bytakingh 0 (x) to be the hyperplane parallel to the tangent hyperplane to at x, and passing through the middle point of the segment x.

3 Generalized antiorthotomics and their singularities 883 We define the a-antiorthotomic of with respect to as the envelope M a of the family {h a (x)} x of hyperquadrics. Observe that M 0 corresponds to the usual antiorthotomic of with respect to (see [1]). We will give the equations of M a in order to prove that it is an immersed submanifold of R n+1. rom now on, we assume that is the origin of R n+1. Let Q be the point where the directrix d x intersects the straight line passing through and x. Thus, Q = x/2 (a/e)x/ x = x(1/2 2a 2 / x 2 ), so that the equation of the hyperquadric h a (x), (y Q) (x/ x )e± y =0, becomes ( x y x ) +2a 2 ± 2a y =0. (2.1) 2 Observe that this equation includes the case of a = 0. In this way, for any a 0, we define the family of hyperquadrics {h a (x)} x by means of the map f a : R n+1 R given by f a (x, y) = x (y x/2) +2a 2 ± 2a y, so that for any fixed x the zero set of the map f a,x : R n+1 R given by f a,x (y) = f a (x, y),isthe hyperquadric h a (x). The envelope M a of the family {h a (x)} x is defined as the set { y R n+1 : t U with f a (x(t), y) = f } a (x(t), y) = 0, i = 1,...,n (2.2) t i for t 1,...,t n, the standard coordinates in R n,andx : U R n R n+1, a parametrization of. We will prove in theorem 2.2 that under very general conditions M a is an immersed manifold. Let us now prove that the a-antiorthotomic mirror M a produces the wavefront after the rays emanating from are reflected on it. Theorem 2.1. Let, and M a be as defined above. Then the mirror M a reflects the rays emerging from into the rays normal to. Proof. Consider a parametrization of, x : U R n R n+1 and let t = t 1,...,t n be the standard coordinates in R n. Then we obtain the envelope M a by eliminating t between the following equations: ( f a (x(t), y) = x(t) y x(t) ) +2a 2 ± 2a y =0 (2.3) 2 f a (x(t), y) = (y x(t)) x = 0, i = 1,...,n. (2.4) t i t i On the other hand, the equation of the line normal to at x(t), r n (x), can be given by (2.4) as well, and thus it follows that M a = h a (x) r n (x). (2.5) x We claim that, for h a (x) non-degenerate, the line normal to at x intersects the envelope M a at the points y(x) where the envelope itself is tangent to the hyperquadric h a (x). In fact, from the optical properties of conics (see [3]) we know that the reflected ray at the point y h a (x) emanating from, coincides with the ray that, starting from x, passes through y (see figure 1). This means that the rays produced from the source point by reflection on M a are the rays normal to. This conclusion also follows by continuity for the degenerate case e = 1, since in this case y = x.

4 884 Alamo and C Criado Theorem 2.2. Suppose that has curvature different from 0, and that M a does not contain any curvature centre of. Then the a-antiorthotomic M a is an immersed submanifold of R n+1. Proof. We first give an explicit parametrization of the a-antiorthotomic M a. To this end, let n(t) be an unitary normal vector field to. According to (2.5), M a can be given by the parametric equation y = x + λ(x)n(x) (2.6) where λ(x) have to be determined by the condition that y h a (x),thatis,y has to verify (2.1). A straightforward calculus gives two λ values: λ a = 2a2 x 2 /2 2a + x n and λ a = 2a2 x 2 /2 2a + x n. (2.7) Then, for a choice of n(x), M a can be decomposed in two sheets, M a = L a L a, with L a and L a parametrized by (2.6) with λ = λ a and λ = λ a respectively. otice that for a = 0, L a = L a. We claim that y i : U R n R n+1 given by y i (t) = x(t) + λ i (x(t))n(x(t)) (i = a, a) is an immersion, so that M a is an immersed submanifold of R n+1. Consider, for example, the case of y = y a and λ = λ a. The other case is similar. We have to verify that the vectors y = x + λ n + λ n, j = 1,...,n, (2.8) t j t j t j t j are linearly independent. To see that, we assume that the vectors x t 1,..., x t n are unitary and pairwise orthogonals. This is possible by taking a suitable parametrization x(t). Then it suffices to showthat the inner product of the vectors y t j with the vectors x t i results in a n n-matrix whose rank equals n. A simple computation gives the matrix ( x y ) ( = δ ij + λ x n ) ( ) = δ ij λ 2 x n, (2.9) where for the second identity we have used that 0 = ( x t j t i n ) = x t i n t j + 2 x n. But this matrix is singular if and only if 1/λ is an eigenvalue of the matrix ( 2 x n ) which is the second fundamental form of at x(t) (see [4]). Therefore, its eigenvalues are the principal curvatures of at x(t). It follows that the matrix (2.9) is singular if and only if 1/λ is a principal curvature of at x(t), or equivalently, y(t) is a centre of curvature of at x(t). Since, by hypothesis, this is not the case, we conclude that the matrix (2.9) has rank n, andsom a is an immersed submanifold Singularities Let be a wavefront which has curvature different from 0, and consider the a-antiorthotomic M a of with respect to the source point. According to theorem (2.2), we have that the singularities of M a consist of the points y M a which are centres of curvature of. The geometrical locus, C, of the centres of curvature of or, equivalently, the envelope of the lines normal to, is usually called the caustic C of. The caustic is in general a hypersurface of R n+1 that has singularities. The study of these singularities is an important subject of research (see [5]).

5 Generalized antiorthotomics and their singularities 885 L a L a c 1 C r O r 2a (a) (b) igure 2. (a) Representation of a particular a-antiorthotomic M a for the case of the wavefront being the sphere of example 3.1. In this case L a is a hyperbola (h) andl a an ellipse (e), both with foci C and. (b) Illustration of all the possible cases for the different values of a and c = C. The pair L a, L a consists of (h, h) in region (1), (h, e) in region (2) and (e, e) in region (3). In what follows we only consider wavefronts whose caustics are non-degenerate in the sense that each point c of the caustic corresponds to an unique point x of the wavefront. It is known that upon movement of the wavefront its singularities slide along the caustic; see [6]. The next theorem shows that a similar result holds regarding the singularities of the a-antiorthotomics M a when a varies in R + {0}. Theorem 2.3. The singularities of the family {M a } a 0 sweep out the caustic C of twice, that is, for each c C there are two values of a such that c is a singularity of the corresponding M a. Proof. ix an unitary normal vector field to, n. or each c C,takex and a principal curvature radius ρ of at x, such that c can be expressed as c = x + ρn(x). Then c is a singular point of M a if and only if c M a,sothatρ = λ a or ρ = λ a.sinceλ a and λ a are given by (2.7), a simple computation gives 2a = ρ ± c for λ a,and2a = ρ ± c for λ a. Since a 0, for any singular point c of M a there are two admissible values of a 2a = ρ + c and 2a = ρ c. (2.10) This result is illustrated in figure 4, for an elliptical wavefront. 3. Examples 3.1. Sphere We first consider the case of the wavefront being a sphere of dimension n in R n+1 with centre C and radius r. Take the unitary normal vector of at x to be n(x) = (C x)/r. Thenwe obtain the following result. Proposition 3.1. Let M a = L a L a be the a-antiorthotomic of the sphere with respect to a point. Then, (i) L a consists of the hyperquadric of revolution (hyperboloid or ellipsoid) with foci and C and eccentricity C /(r +2a) in the case of C r +2a, and reduces to the point C when C =r +2a.

6 886 Alamo and C Criado C x y h a x L a c L a igure 3. Illustration of theorem 2.3 about the singularities of the a-antiorthotomic M a,for wavefront being the ellipse of example 3.2. L a and L a are the two sheets of the a-antiorthotomic for a = 1.2, and h a (x) is a hyperbola of the family for a particular x. It is also illustrated the characteristic property of the mirror L a and its relation with the caustic C. (ii) L a consists of the hyperquadric of revolution (hyperboloid or ellipsoid) with foci and C and eccentricity C / r 2a in the case of C r 2a, and reduces to the point C when C = r 2a. Proof. or each y L a we have y = x + λ a n(x) = C + (λ a r)n(x),whereλ a is given by (2.7). Straightforward calculus gives y C = (λ a r)n(x) = (2a + r)2 C 2 n(x). (3.1) 2(2a + r) +2C n We deduce that for C =r +2a, λ a = r and therefore L a ={C}, andfor C r +2a, a simple computation gives y C ± y = 2a + r, which is the equation of the hyperquadric of revolution (hyperboloid or ellipsoid) with foci = 0andC, and eccentricity C /(r +2a). This proves (i); from a similar calculation follows (ii). In figure2(a)wehaverepresentedthecircle given by the parametric equation x(t) = ( 7 +10cost, 10 sin t) and the a-antiorthotomic M a for a = 3. In this case, L a is a hyperbola and L a an ellipse, both with foci C and. We have also represented four elements of the family {h a (x)} for x = x(t) with t = 0,π/2,π,3π/2, and the points where they are tangent to M a. The diagram presented in figure 2(b) illustrates all the possible cases for the different values of a and c = C. Let us denote by h and e the hyperboloids and ellipsoids, respectively, considered in proposition (3.1). Then the pair L a, L a consists of (h, h) in region (1); (h, e) in region (2), and (e, e) in region (3). inally, in the border between regions (1) and (2), is (h, C), and between (2) and (3), is (C, e). Moreover,fora = 0, besides the above, L a = L a, and for c = r both collapse to C. inally, for c = 0 the ellipsoids are spheres, and for 2a = r the sphere L a reduces to C. Remark 3.2. otice that, in the non-degenerate cases, the family of the a-antiorthotomics M a = L a L a constitutes a homofocal family of hyperboloids and ellipsoids, as a varies in R Ellipse In this example we consider the case of the wavefront being an ellipse. This example will illustrate theorem (2.3) about the singularities of the a-antiorthotomics. Consider the ellipse R 2 given by the parametric equation x(t) = ( 6+10cost, 7sint), t R. (3.2)

7 Generalized antiorthotomics and their singularities 887 C M a igure 4. Illustration of how the singularities of the family of a-antiorthotomics M a sweep out the caustic C for the case of the wavefront being an ellipse. The caustic C of this ellipse is the astroid given by the parametric equation c(t) = 51 ( 1 10 cos3 t, 1 7 sin3 t ), t R. (3.3) We have represented both curves in figure 3 and, for a = 1.2, they are also represented the two sheets L a and L a of the a-antiorthotomic M a. In the figure the hyperbola h a (x) for x = x(π/3) with its asymptotes is also represented, as well as the two points where it is tangent to M a. We also illustrate the characteristic property of the a-antiorthotomic M a,i.e.a ray emanating from and reflecting on M a at a point y M a h a (x) gives the ray normal to at x, which passes through y. The point c of the caustic corresponding to x is in the normal line to at x and the distance between x and c is the curvature radius of at x. Observe that when y and c coincide, the point y is a singularity of M a. igure 4 graphically shows how the singularities of the family of a-antiorthotomics sweep out the caustic. In this case we have represented the a-antiorthotomics sheets L a for a = 9, 10, 11, and L a for a = 0.1, 1.2, 2.3, 3.4, Construction of the a-antiorthotomics from the caustic In this section we give a method to construct the a-antiorthotomic mirror M a of a given wavefront with respect to a point, from the data of its caustic C. As in section 2.2 we only consider wavefronts whose caustics are non-degenerate in the sense that each point c of the caustic corresponds to an unique point x of the wavefront. Let us introduce some notation for the following theorem, which contains the main result of this section. or each regular point c C corresponding to a point x and a principal curvature radius ρ, such that c can be expressed as c = x + ρn(x), sete = c /2a and e = c /2a,where2a = 2a + ρ and 2a = 2a ρ. Consider the following hyperquadrics of revolution. If e 1(e 1 respectively), let h a (c) (h a (c) respectively) denote the ellipsoid or hyperboloid with foci and c, and eccentricity e (e respectively), and if e = 1(e = 1 respectively) h a (c) (h a (c) respectively) degenerates to the straight line passing through and c. We will state in the next theorem that the a-antiorthotomic M a of a wavefront with respect to a source point can be obtained as an envelope of the above family of hyperquadrics {h a (c) h a (c)} c C. Specifically, recall that, for a choice of the unitary normal vector n(x), the a-antiorthotomic M a consist of two sheets, L a and L a, that correspond to the parametric equation (2.6) with λ = λ a or λ = λ a, respectively. Then we have the following theorem. Theorem 4.1. With the above notation, L a and L a are in the envelopes of the families of hyperquadrics of revolution {h a (c)} c C and {h a (c)} c C, respectively.

8 888 Alamo and C Criado C L a igure 5. Construction of the mirror L a from the caustic C for the wavefront of example 3.2. There are represented five elements of the family {h a (c)} c C whose envelope is L a. Proof. We first consider the family {h a (c)} c C. We shall show that the hyperquadrics h a (c) and h a (x) are tangent at the point y a = x +λ a n(x), which is the point where h a (x) is tangent to L a, so that both hyperquadrics (and L a ) have the same tangent hyperplane at that point and thus L a is in the envelope of {h a (c)} c C. In order to demonstrate this result it suffices to show that the intersection between h a (c) and h a (x) is only one point, precisely y a = x +λ a n(x),sinceh a (c) and h a (x) are hyperquadrics. By definition, the points y h a (c) verify c ( y c 2 ) (2a + ρ)2 ± (2a + ρ) y =0. (4.1) If we suppose that a point y belongs to h a (c) and h a (x) simultaneously, then y has to verify equations (4.1) and (2.1), which yields n (y x) +2a ± y =0, (4.2) and therefore y x cos α +2a ± y =0, (4.3) where α is the angle between y x and n. Taking into account that the points of h a (x) also verify y x ± y = 2a, we get that cos α =±1. Thus y must be in the normal line r n (x), i.e. y = x +λn(x). To determine if λ = λ a or λ = λ a,sety = c +µn(x), with µ = λ ρ and substitute in (4.1). As in the proof of theorem 2.2, we get two solutions µ a and µ a similar to those given by (2.7) with c and 2a instead of x and 2a, respectively. It is easy to check that µ a = λ a ρ is the only valid solution, so that y = x + λ a n(x) as was claimed. The proof is similar for the family {h a (c)} c C. In figure 5 we show, for the ellipse and the sheet L a of the example of 3.2, five elements of the family {h a (c)} c C whose envelope is L a. Observe that, according to proposition 3.1, for the non-degenerate cases, h a (c) is the sheet corresponding to λ a of the a-antiorthotomic of the sphere centred at c and with radius ρ, S(c),andh a (c) is the sheet corresponding to λ a of the a-antiorthotomic of the same sphere. Therefore we can enunciate the following corollary. Corollary 4.2. With the notation above, the a-antiorthotomic of from is in the envelope of the a-antiorthotomics of the spheres S(c), c C. Moreover, L a (L a respectively) is the envelope of the sheets corresponding to λ a (λ a respectively) of the considered a-antiorthotomics of the spheres S(c). Remark 4.3. otice that, basically, for each fixed a, h a (c) and h a (c) are those elements classified in the diagram of figure 2.

9 Generalized antiorthotomics and their singularities Conclusions We have given an original procedure to solve the following inverse problem: given a wavefront and a fixed source point, find a mirror which produces the given wavefront after reflecting on it from the source point. In fact, we have found a one-parameter family of such mirrors and they are obtained as the envelope of certain quadrics. We have solved the problem in a general space R n+1, and proved that the mirror is an immersed submanifold, with singularities in the points corresponding to the centres of curvature of the wavefront. Moreover, we have proved that the singularities of our family of mirrors sweep out the caustic associated with the wavefront. Another method to reconstruct the mirror from the caustic is given. Here again the mirror is obtained as the envelope of quadrics. The fact that each of these mirrors is constructed as an envelope of quadrics can be used to design it as a tessellation of smaller quadrics mirrors. This composed mirror can be a good approximation of the original one, and therefore could be useful in the construction of optical, radar or acoustical devices. Acknowledgments Partial financial support from DGICYT grants PB and PB is acknowledged. References [1] Bruce J W and Giblin P J 1984 Curves and Singularities (Cambridge: Cambridge University Press) [2] Quételet L A 1829 Mém. Acad. Belg. 5 [3] Berger M 1980 Geometry II (Berlin: Springer) [4] Milnor J W 1963 Morse theory Annals of Mathematics Studies vol 51 (Princeton, J: Princeton University Press) [5] Arnold V I, Gusein-Zade S M and Varchenko A 1985 Singularities of Differentiable Maps vol I (Boston, MA: Birkhäuser) [6] Arnold V I and Givental A B 1990 Symplectic geometry Dynamical Systems IV (Encyclopaedia Math. Sci. vol 4) ed V I Arnold and S P ovikov (Berlin: Springer) pp 1 13