Deformation of n-dimensional objects

This paper presents deformations that Paul Borrel and Dominique Bechmann Interactive Geometric Modeling IBM Research Division Thomas J. Watson Research Center P.O. Box 704 Yorktown Heights, NY 10598 a new technique for computing space interpolate a set of user-defined constraints. Deformations” are represented by a polynomial mapping from N’ to R“. Constraints are specified by indicating the images of selected points. The deformation is formulated as the product of a polynomial function ~ of 0?” into a higher-dimensional space, Rm, with a linear projection from R“ back to U?”. The projection matrix is computed using a pseudo-inverse technique so as to satisfy all constraints when m is suflicient!y large and to provide a least-square optimal solution when m is too small given the number of specified constraints. For sutlicient m, the additional degrees of freedom may be used to optimize potential functions controlled by attracting and repulsing points of R“. A prototype implementation is presented, whtch 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. Permission to copy without fee atl or part of this material is granted provided that the copies arc not made or distributed for direct cormnercird advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machine~. To copy otherwise, or to republish, requires a fee and/or specific permission.