Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows

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1 American Journal of Computational Mathematics, 5, 5, Published Online June 5 in SciRes. Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows Rafik Aramyan, Russian-Armenian (Slavonic University, Yerevan, Armenia Institute of Mathematics Armenian Academy of Sciences, Yerevan, Armenia rafikaramyan@yahoo.com Received February 5; accepted May 5; published 5 May 5 Copyright 5 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY. Abstract In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article. Keywords Integral Geometry, Convex Body, Projection Curvature, Support Function. Introduction The problem of reconstruction of a convex body from the mean and Gauss curvatures of the boundary of the body goes back to Christoffel and Minkwoski []. Let F be a function defined on -dimensional unit sphere S. The following problems have been studied by E. B. Christoffel: what are necessary and sufficient conditions for F to be the mean curvature radius function for a convex body. The corresponding problem for Gauss curvature is considered by H. Minkovski []. W. Blaschke [] provides a formula for reconstruction of a convex body B from the mean curvatures of its boundary. The formula is written in terms of spherical harmonics. A. D. Aleksandrov and A. V. Pogorelov generalize these problems for a class of symmetric functions G R, R of principal radii of curvatures (see []-[5]. How to cite this paper: Aramyan, R. (5 Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows. American Journal of Computational Mathematics, 5,

2 n Let R ω,, Rn ω signify the principal radii of curvature of the boundary of B at the point with outer normal direction ω S n. In n-dimensional case, a Christoffel-Minkovski problem is posed and solved by Firay [6] and Berg [7] (see also [8]: what are necessary and sufficient conditions for a function F, defined on S n to be function Ri ( ω R i ω for a p convex body, where p n and the sum is extended over all increasing sequences i,, i of indices p chosen from the set i =,, n. R. Gardner and P. Milanfar [9] provide an algorithm for reconstruction of an origin-symmetric convex body K from the volumes of its projections. D. Ryabogin and A. Zvavich [] reconstruct a convex body of revolution from the areas of its shadows by giving a precise formula for the support function. In this paper, we consider a similar problem posed for the projection curvature radius function of convex bodies. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article. The solution of the system of differential equations is itself interesting. Let B R be a convex body with sufficiently smooth boundary and with positive Gaussian curvature at every point of the boundary B. We need some notations. S the unit sphere in R, S ω S the great circle with pole at ω S, B ( ω projection of B onto the plane containing the origin in R and orthogonal to ω, R ( ω, ϕ curvature radius of B ( ω at the point with outer normal direction ϕ S ω and call projection curvature radius of B. Let F be a positive continuously differentiable function defined on the space of flags = {( ωϕ, : ω S, ϕ S }. In this article, we consider: ω Problem. What are necessary and sufficient conditions for F to be the projection curvature radius function R ( ω, ϕ for a convex body? Problem. Reconstruction of that convex body by giving a precise formula for the support function. Note that one can lead the problem of reconstruction of a convex body by projection curvatures using representation of the support function in terms of mean curvature radius function (see [7]. The approach of the present article is useful for practical point of view, because one can calculate curvatures of projections from the shadows of a convex body. Let s note that it is impossible to calculate mean radius of curvature from the limited number of shadows of a convex body. Also let s note that this is a different approach for such problems, because in the present article we lead the problem to a differential equation of spatial type on the sphere and solve it using a new method (so called consistency method. The most useful analytic description of compact convex sets is by the support function (see []. The support function of B is defined as B R be a convex body with sufficiently smooth boundary and let = sup R H x yx,, x. y B Here, denotes the Euclidean scalar product in R. The support function of B is positively homogeneous and convex. Below, we consider the support function H of a convex body as a function on S (because of the positive homogeneity of H the values on S determine H completely. k ( S C denotes the space of k times continuously differentiable functions defined on S. A convex body B k is k-smooth if its support function H C ( S. Given a function H defined on S, by H ω ( ϕ, ϕ S ω we denote the restriction of H onto the circle S ω for ω S, and call the restriction function of H. Below, we show (Theorem that Problem. is equivalent to the problem of existence of a function H defined on satisfies the differential equation S such that H ω Hω ( ϕ + Hω ( ϕ = F( ωϕ,, for ϕ Sω ( ϕϕ for every ω S. Definition. If for a given F there exists H defined on S that satisfies Equation (, then H is called a solution of Equation (. In Equation (, H ω ( ϕ is a function defined on the space of an ordered pair orthogonal unit vectors, say e, e, (in integral geometry such a pair is a flag and the concept of a flag was first systematically employed by 87

3 R.V. Ambartzumian in []. There are two equivalent representations of an ordered pair orthogonal unit vectors e, e, dual each other: ( ωϕ, and ( Ω,, ( where ω S is the spatial direction of the first vector e, and ϕ is the planar direction in S ω coincides with the direction of e, while Ω S is the spatial direction of the second vector e, and is the planar direction in S Ω coincides with the direction of e. The second representation we will write by capital letters. g ωϕ,, we denote by g the image of g defined by Given a flag function where ( ωϕ, (, g g( ωϕ = Ω (dual each other. Let G be a function defined on. For every circle S ω. Definition. If G ( ω, is a solution of that equation for every Equation (. G ωϕ, satisfies Definition. If a flag solution G Ω, =,, ( ω S, Equation ( reduces to a differential equation on the (, G ω S, then G is called a flag solution of Ω = Ω (4 (no dependence on the variable, then G is called a consistent flag solution. There is an important principle: each consistent flag solution G of Equation ( produces a solution of Equation ( via the map G ωϕ, G Ω, = G Ω = H Ω, (5 and vice versa: the restriction functions of any solution of Equation ( onto the great circles is a consistent flag solution. Hence, the problem of finding a solution reduces to finding a consistent flag solution. To solve the latter problem, the present paper applies the consistency method first used in []-[5] in an integral equations context. We denote: e[ Ω, ] the plane containing the origin of R, direction Ω S, determines rotation of the plane around Ω, B [ Ω, ] projection of B onto the plane e[ Ω, ], R ( Ω, curvature radius of B Ω, at the point with outer normal direction Ω S. It is easy to see that [ ] where ( Ω, is dual to (, ( Ω = R R ω ϕ,,, ωϕ. Note that in the Problem. uniqueness (up to a translation follows from the classical uniqueness result on Christoffel problem, since R( Ω + R( Ω = R (, d. Ω (6 Equation ( has the following geometrical interpretation. It is known (see [] that times continuously differentiable homogeneous function H defined on convex if and only if where R, is Hω ( ϕ + Hω ( ϕ for every ω S and ϕ Sω, (7 ϕϕ H ω is the restriction of H onto S ω. So in case F >, it follows from (7, that if H is a solution of Equation ( then its homogeneous extension is convex. It is known from convexity theory that if a homogeneous function H is convex then there is a unique convex body B R with support function H and F ( ωϕ, is the projection curvature radius function of B (see []. The support function of each parallel shifts (translation of that body B will again be a solution of Equation (. By uniqueness, every two solutions of Equation ( differ by a summand a, defined on S, where 88

4 a R. Thus we have the following theorem. Theorem Let F be a positive function defined on. If Equation ( has a solution H then there exists a convex body B with projection curvature radius function F, whose support function is H. Every solution of Equation ( has the form H( + a,, where a R, being the support function of the convex body B + a. The converse statement is also true. The support function H of a -smooth convex body B satisfies Equation ( for F = R, where R is the projection curvature radius function of B (see [6]. The purpose of the present paper is to find a necessary and sufficient condition that ensures a positive answer to both Problems, and suggest an algorithm of construction of the body B by finding a representation of the support function in terms of projection curvature radius function. This happens to be a solution of Equation (. Throughout the paper (in particular, in Theorem that follows we use usual spherical coordinates τ, for points S based on a choice of a North Pole and a reference point τ = on the equator. The point with coordinates τ, we will denote by ( τ,, the points (,τ lie on the equator. On S ω we choose anticlockwise direction as positive. On the plane ω containing S ω we consider the Cartesian x and y-axes where the direction of the y-axis y is taken to be the projection of the North Pole onto ω. The direction of the x-axis x we take as the reference direction on S ω and call it the East direction. Now we describe the main result. Theorem Let B be a -smooth convex body with positive Gaussian curvature at every point of B and R is the projection curvature radius function of B. Then for Ω S chosen as the North pole H ( Ω = R (, τ, ϕ cosϕdϕ dτ 4 ( + R, τ, ϕ ( ϕ cosϕ sin ϕ dϕ dτ 8 + ( sin d d,, sin d τ R τ ϕ ϕ ϕ cos is a solution of Equation ( for F = R. On S ω we measure ϕ from the East direction. Remark, that the order of integration in the last integral of (8 cannot be changed. Obviously Theorem suggests a practical algorithm of reconstruction of convex body from projection curvature radius function R by calculation of support function H. We turn to Problem. Let R be the projection curvature radius function of a convex body B. Then F R necessarily satisfies the following conditions: a For every ω S and any reference point on S ω F( ωϕ, sin ϕ d ϕ = F( ωϕ, cos ϕ d ϕ =. (9 This follows from Equation (, see also [6]. b For every direction Ω S chosen as the North pole (( τ,, y d, ( F τ= = τ (Theorem 5. Let F be a positive times differentiable function defined on. Using (8, we construct a function F defined on S : where the function F is the image of F (see ( and y is the direction of the y-axis on (, F( Ω = F( (, τ, ϕ cosϕdϕ dτ 4 + F ((, τ, ϕ (( ϕ cosϕ sin ϕ dϕ dτ 8 + sin d d τ F (( τ,, ϕ sin ϕdϕ cos Note that the last integral converges if the condition ( is satisfied. (8 ( 89

5 Theorem A positive times differentiable function F defined on represents the projection curvature radius function of some convex body B if and only if F satisfies the conditions (9, ( and the extension (to R of the function F defined by ( is convex.. The Consistency Condition We fix ω S and try to solve Equation ( as a differential equation of second order on the circle S ω. We start with two results from [6]. a For any smooth convex domain D in the plane where normal direction at s, = ϕ R ( h ϕ ψ sin ϕ ψ d ψ, ( h ϕ is the support function of D with respect to a point s D. In ( we measure ϕ from the R ψ is the curvature radius of D at the point with normal direction ψ. b ( is a solution of the following differential equation ( ϕ ( ϕ ( ϕ. One can easy verify that (also it follows from ( and ( R = h + h ( = ϕ F ( G ωϕ, ωψ, sin ϕ ψ d ψ, (4 is a flag solution of Equation (. Theorem 4 Every flag solution of Equation ( has the form = ϕ ( + + g ωϕ, F ωψ, sin ϕ ψ dψ C ω cosϕ S ω sinϕ (5 where C n and S n are some real coefficients. Proof of Theorem 4. Every continuous flag solution of Equation ( is a sum of G+ g, where g is a flag solution of the corresponding homogeneous equation: for every H (6 ( ϕ + H ( ϕ =, ϕ S, ω ω ϕϕ ω ω S. We look for the general flag solution of Equation (6 in the form of a Fourier series = n + n g ωϕ, C ω cos nϕ S ω sin nϕ. n=,,, After substitution of (7 into (6 we obtain that g (, g = C + S ωϕ satisfies (6 if and only if ωϕ, ω cosϕ ω sin ϕ. Now we try to find functions C and S in (5 from the condition that g satisfies (4. We write (, dual coordinates i.e. g( ωϕ, = g ( Ω, and require that g (, Ω S, i.e. for every where (, Ω S ( g ( G C S (7 g ωϕ in Ω should not depend on for every Ω, = ωϕ, + ω cosϕ+ ω sinϕ =, (8 G ωϕ was defined in (4. denotes the derivative corresponding to right screw rotation around Ω. Differentiation Here and below with use of expressions (see [4] sin ϕ τ =, ϕ = tan sin ϕ, = cos ϕ, (9 after a natural grouping of the summands in (8, yields the Fourier series of ( G ( ωϕ, the Fourier coefficients. By uniqueness of 9

6 where ( ( S ( ω τ C ω + + tan C( ω = A( ωϕ, cos ϕdϕ ( ( ( S ( ω τ C ω tan C( ω = A( ωϕ, dϕ ( C ( ω τ S ω + tan S( ω = A( ωϕ, sin ϕd ϕ, ( ϕ A( ωϕ, = F( ωψ, sin ( ϕ ψ + F( ωψ, cos( ϕ ψ ϕ d ψ. (. Averaging Let H be a solution of Equation (, i.e. restriction of H onto the great circles is a consistent flag solution of Equation (. By Theorem there exists a convex body B with projection curvature radius function R = F, whose support function is H. To calculate H ( Ω for a Ω S we take Ω for the North Pole of S. Returning to the Formula (5 for every ω = (, τ SΩ we have H( Ω = R( ω, ψ sin ψ d ψ + S( ω, ( We integrate both sides of ( with respect to uniform angular measure dτ over [, to get Now the problem is to calculate ( Ω = ( τ ψ ψ ψ τ + ( τ τ ( H R,, cos d d S, d. (( τ τ = S S, d. (4 We are going to integrate both sides of ( and ( with respect to dτ over [,. For ω ( τ, where, and (, τ we denote S( =, S = τ, d τ, (5 ϕ A( = d τ R( ω, ψ sin ( ϕ ψ R( ω, ψ cos( ϕ ψ ϕ dψ sin ϕd ϕ. + (6 Integrating both sides of ( and ( and taking into account that for [, we get ( C ( τ, dτ= ( ( ( i.e. a differential equation for the unknown coefficient We have to find τ S + tan S = A, (7 S. S given by (4. It follows from (7 that ( A(. S = (8 9

7 Integrating both sides of (5. with respect to d over [, we obtain Now, we are going to calculate It follows from (5 that ( A( ( S S = cos d. (9 S. ϕ S ( = H ω ϕ R ω, ψ sin ϕ ψ dψ sinϕdϕdτ = H sin d d R(, (( cos sin d d. ω ϕ ϕ ϕ τ ω ψ ψ ψ + ψ ψ τ Let S ω, ω= τ,. As a point of spherical coordinates ut, with respect to Ω. By the sinus theorem of spherical geometry and ϕ be the direction that corresponds to ϕ [, for From (, we get ( S, let ϕ have sinϕ = sin u. ( = sin ϕ. ( u = Fixing τ and using ( we write a Taylor formula at a neighborhood of the point = : H ( ϕ = H, ((, ϕ+ τ + H ((, ϕ+ τ sin ϕ o. τ + Similarly, for ψ [, we get R ( τ,, ψ = R, τ, ψ+ τ + R, τ, ψ + τ sin ψ + o. Substituting ( and (4 into ( and taking into account the easily establish equalities we obtain S ( ( H, ϕ+ τ sinϕdϕdτ = R τ ψ + τ ( ( ψ ψ + ψ ψ τ = (5,, cos sin d d ( lim = H, sin d d cos + ϕ τ ϕ ϕ τ,, sin (( cos sin d d R τ ψ + τ ψ ψ ψ + ψ ψ τ = H ((, d R ((,, d. τ τ τ τ 4 y = Theorem 5 For every -smooth convex body B and any direction Ω S, we have ( (4 (6 9

8 (( τ,, y d, (7 R τ= = where y is the direction of the y-axis on ( τ,. Proof of Theorem 5. Using spherical geometry, one can prove that (see also ( (( τ,, y = (( τ, + ϕϕ (( τ, R H H = = = H( ( τ, + H ττ H tan cos = [ H ], ττ = where H is the support function of B. Integrating (8, we get ( y [ ] R H = ττ = = τ,, dτ = dτ =. 4. A Representation for Support Functions of Convex Bodies Let B be a convex body and Q R. By H we denote the support function of B with respect to Q. Q Theorem 6 Given a -smooth convex body B, there exists a point O R such that for every Ω S chosen as the North pole (( τ H, dτ =. (9 O = Proof of Theorem 6. For a given B and a point on S Clearly, K Q Q R, by ( Ω = HQ( ( τ = (8 K we denote the following function defined Q, d τ. K Q : K is a continuous odd function with maximum Q = max ( Ω. K Q K Q Ω S It is easy to see that K( Q for Q. Since which Let = K( Q K O K Q is continuous, so there is a point min. Ω be a direction of maximum now assumed to be unique, i.e. = ( Ω = ( Ω K O max K K. Ω S If K( O = the theorem is proved. For the case K( O a OO = ε Ω. It is easy to demonstrate that H H ε (, O O that K( O = a ε, contrary to the definition of O. So O O O for = > let O be the point for which Ω = Ω ΩΩ, hence for a small ε > we find K O =. For the case where there are two or more directions of maximum one can apply a similar argument. Now we take the point O of the convex body B for the origin of R. Below H, we will simply denote by O H. By Theorem 6 and Theorem 5, we have the boundary condition (see (6 Substituting (9 into ( we get S =. (4 9

9 A H( Ω = R, τ, ψ cosψdψdτ d ( d =,, cos d d R τ ψ ψ ψ τ ϕ R( ω, ψ sin ( ϕ ψ + R( ω, ψ cos( ϕ ψ ϕ dψ sin ϕdϕd τ. ( Using expressions (9 and integrating by dϕ yields ( Ω = R H, τ, ψ cosψdψdτ d + d τ R ( ω, ψ I R ( ω, ψ tanii d ψ, + where II = ( = ( ψ cosψ sinψ ( + sin ψ sin ϕcos ϕ ψ sinϕdϕ sin ψ, ψ and ( ψ cosψ sinψ ( + sin ψ I = sin ϕsin ( ϕ ψ cosϕd ϕ = +. ψ 4 4 Integrating by parts (4 we get sin sin ψ H( Ω = R, τ, ψ cosψdψdτ d d, d τ R ω ψ ψ cos dτ R ((, τ, ψ Idψ + lim d τ R( ( a, τ, ψ Id ψ. cos a a Using (4, Theorem 5 and taking into account that we get I dψ = sinsin ψ H ( Ω = R, τ, ψ cosψdψdτ d d, d τ R ω ψ ψ cos dτ R( (, τ, ψ Id ψ. From (44, using (9 we obtain (8. Theorem is proved. (4 (4 (4 (44 5. Proof of Theorem Necessity: if F is the projection curvature radius function of a convex body B, then it satisfies (9 (see [6], the condition ( (Theorem 5 and F defined by ( is convex since it is the support function of B (Theorem. Sufficiency: let F be a positive times differentiable function defined on satisfies the conditions (9, (. We construct the function F on S defined by (. There exists a convex body B with support function F since its extension is a convex function. Also Theorem implies that F is the projection curvature radius of B. Funding This work was partially supported by State Committee Science MES RA, in frame of the research project SCS 94

10 -A44. References [] Minkowski, H. (9 Theorie der konvexen Korper, insbesondere Begrundung ihresb Oberflachenbergriffs. Ges. Abh.,, Leipzig, Teubner, -9. [] Blaschke, W. (9 Vorlesungen uber Differentialgeometrie. II. Affine Differentialgeometrie, Springer-Verlag, Berlin. [] Pogorelov, A.V. (969 Exterior Geometry of Convex Surfaces [in Russian]. Nauka, Moscow. [4] Alexandrov, A.D. (956 Uniqueness Theorems for Surfaces in the Large [in Russian]. Vesti Leningrad State University, 9, 5-4. [5] Bakelman, I.Ya., Verner, A.L. and Kantor, B.E. (97 Differential Geometry in the Large [in Russian]. Nauka, Moskow. [6] Firey, W.J. (97 Intermediate Christoffel-Minkowski Problems for Figures of Revolution. Israel Journal of Mathematics, 8, [7] Berg, C. (969 Corps convexes et potentiels spheriques. Matematisk-fysiske Meddelelser Udgivet af. Det Kongelige Danske Videnskabernes Selska, 7, 64. [8] Wiel, W. and Schneider, R. (98 Zonoids and Related Topics. In: Gruber, P. and Wills, J., Eds., Convexity and Its Applications, Birkhauser, Basel, [9] Gardner R.J. and Milanfar, P. ( Reconstruction of Convex Bodies from Brightness Functions. Discrete & Computational Geometry, 9, [] Ryabogin, D. and Zvavich, A. (4 Reconstruction of Convex Bodies of Revolution from the Areas of Their Shadows. Archiv der Mathematik, 5, [] Leichtweiz, K. (98 Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin. [] Ambartzumian, R.V. (99 Factorization Calculus and Geometrical Probability. Cambridge University Press, Cambridge. [] Aramyan, R.H. ( An Approach to Generalized Funk Equations I [in Russian]. Izvestiya Akademii Nauk Armenii. Matematika [English Translation: Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences], 6, [4] Aramyan, R.H. ( Generalized Radon Transform on the Sphere. Analysis International Mathematical Journal of Analysis and Its Applications,, [5] Aramyan, R.H. ( Solution of an Integral Equation by Consistency Method. Lithuanian Mathematical Journal, 5, -9. [6] Blaschke, W. (956 Kreis und Kugel, (Veit, Leipzig. nd Edition, De Gruyter, Berlin. 95