Support Function Representation of Convex Bodies, Its Application in Geometric Computing, and Some Related Representations

Abstract

The importance of thesupport functionin representation, manipulation, and analysis of convex bodies can indeed be compared with that of the Fourier transform in signal processing. The support function, in intuitive terms, is the signed distance of a supporting plane of a convex body from the origin point. In this paper we show that, just as simple multiplication in the Fourier transform domain turns out to be the convolution of two signals, similarly simple algebraic operations on support functions result in a variety of geometric operations on the corresponding geometric objects. In fact, since the support function is areal-valuedfunction, these simple algebraic operations are nothing but arithmetic operations such as addition, subtraction, reciprocal, and max–min, which give rise to geometric operations such as Minkowski addition (dilation), Minkowski decomposition (erosion), polar duality, and union–intersection. Furthermore, it has been shown in this paper that a number of representation schemes (such as the Legendre transformation, the extended Gaussian image, slope diagram representation, the normal transform, and slope transforms), which appear to be very disparate at first sight, belong to the same class of the support function representation. Finally, we indicate some algebraic manipulations of support functions that lead to new and unsuspected geometric operations. Support function like representations for nonconvex objects are also indicated.