VOLUME-PRESERVING FREE-FORM SOLIDS

Some important trends in geometric modeling are the reliance on solid models rather than surface-based models and the enhancement of the exp ressive power of models, by using free-form objects in addition to the usual geometric primitives and by incorporating physical principles. An additional trend is the emphasis on interactive performance. In thi s paper we integrate all of these requirements in a single geometric p rimitive by endowing the tri-variate tensor product free-form solid wi th several important physical properties, including volume and interna l deformation energy. Volume preservation is of benefit in several app lication areas of geometric modeling, including computer animation, in dustrial design and mechanical engineering. However, previous physics- based methods, which usually have used some forms of ''energy,'' have neglected the issue of volume (or area) preservation. We present a nov el method for modeling an object composed of several tenser-product so lids while preserving the desired volume of each primitive and ensurin g high-order continuity constraints between the primitives. The method utilizes the Uzawa algorithm for non-linear optimization, with object ive functions based on deformation energy or least squares. We show ho w the algorithm can be used in an interactive environment by relaxing exactness requirements while the user interactively manipulates free-f orm solid primitives. On current workstations, the algorithm runs in r eal-time for tri-quadratic volumes and close to real-time for tn-cubic volumes.