Deformation of n-dimensional objects

Abstract

This paper presents a new technique for computing space deformations that interpolate a set of user-defined constraints. Deformations are represented by a polynomial mapping from K" to K". Constraints are specified by indicating the images of selected points. The deformation is formulated as the product of a polynomial function/of 05" into a higher-dimensional space, R™, with a linear projection from OS"1 back to K". The projection matrix is computed using a pseudo-inverse technique so as to satisfy all constraints when m is sufficiently large and to provide a least-square optimal solution when m is too small given the number of specified constraints. For sufficient m, the additional degrees of freedom may be used to optimize potential functions controlled by attracting and repulsing points of K". A prototype implementation is presented, which demonstrates the application of this technique to the interactive design of free-form shapes (when n = 3) and of deformation processes in space-time domain (when n = 4). For graphic purposes, the deformation is simply applied to the vertices of a triangulation of the object's faces. Greater control of the deformation is achieved by using a B-spline basis for the imbedding, rather than a power basis.