Solid-Interpolating Deformations: Construction and animation of PIPs

Computer programs that simulate the deformations of geometric shapes have played a key role in the increasing popularity of software tools for artistic animation. Previously published techniques for specifying and animating deformations are either limited in their domain or ill suited for interactive editing and visualization. This is because the effects of alterations performed by the animator on the model's parameters may not always be anticipated, and because realtime animation may only be produced by visualizing pre-computed sequences of 3D frames, which are obtained by a slow process and require vast amounts of storage. To support an interactive environment for animation design, we have developed a new, simple, and efficient animation primitive: a Parameterized Interpolating Polyhedron, or PIP for short. PIPs are easily specified and edited by providing their initial and final shapes, which may be any polyhedra, and need not have corresponding boundary elements. PIPs may be efficiently animated on standard graphic hardware because a PIP is a smoothly varying family of polyhedra bounded by faces that evolve with time. The faces have constant orientations and vertices that each move on a straight line between a vertex of the initial shape and a vertex of the final one. The cost of recalculating the time dependant information of a PIP is small in comparison to the display cost. We provide simple and efficient algorithms, based on Minkowski sum operations, for computing PIPs. When both the initial and final shapes are convex, the resulting faces are the true boundary of the deforming object, otherwise subsets of the resulting faces may lie inside the object. In both cases, correct images are automatically generated using standard depth-buffer hardware. The tools we have developed are convenient for interactively designing animation sequences that show the metamorphosis of 3D shapes. They may also be used to simulate the geometric effect of a variety of manufacturing operations, and for interactively selecting the optimal compromise between two or more shapes. They are being integrated in the LAMBADA design and inspection environment for animated assemblies, where deformations and rigid-body motions may be easily combined and synchronized using a hierarchical representation.