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

Rafik Aramyan

doi:10.4236/ajcm.2015.52007

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.

Highlights

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 [1]
The formula is written in terms of spherical harmonics
The purpose of the present paper is to find a necessary and sufficient condition that ensures a positive answer to both Problems 1,2 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

Summary

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 [1]. If Equation (1) has a solution H there exists a convex body B with projection curvature radius function F, whose support function is H. The purpose of the present paper is to find a necessary and sufficient condition that ensures a positive answer to both Problems 1,2 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 (1). Theorem 3 A positive 2 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), (10) and the extension (to R3 ) of the function F defined by (11) is convex

The Consistency Condition

Averaging

A Representation for Support Functions of Convex Bodies

Proof of Theorem 3