Homeomorphisms and Metamorphosis of Polyhedral Models Using Fields of Directions Defined on Triangulations

Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In this article we present a new approach which uses fields of directions defined on triangulations(FDTs) to treat both the problem of getting an homeomorphism between the models and that of morphing them. Consider that an scaled version(P1) of one of the objects, has already been adequately placed in the interior of the other(P2). The objective of the first part of the approach, is to obtain a field of unit vectors defined on a triangulation of the space between P1 and P2. This field must have no singularities and the trajectories determined by it will be later used to get warping and morphing transformations between P1 and P2. The morphing transformations obtained have the good property of being topology preserving ones but it can be hard to get an FDT defined on a triangulation of P1 P2 and the intermediate models can have a very large number of faces. To illustrate those aspects, transformations between simple models are presented.