Dynamic Implicit Surfaces for Fast Proximity Queries in Physically Based Modeling

In this paper we deal with the problem of distance computation between static or dynamic objects, which is an important problem when dealing with the effects of collisions in the context of physically based modeling to obtain a fast numerical integration of the underlying ordinary differential equation. We use an implicit formulation of the (potential) collisions already in the physical laws of the simulated system. The new idea of our approach for distance computation is to use a special implicit representation recently used by Raviv and Elber (in the context of free-form sculpting). In the context of proximity queries this implicit representation has two main advantages: First, any given boundary representation can be approximated quite easily, no high-degree polynomials and complicated approximation algorithms are needed. Second, the evaluation of the corresponding implicit trivariate tensor product B-spline function is very fast and independent of the size of the object. In the paper we first describe the approach in detail and discuss its space and time complexity. Examples from different areas show the advantages of the approach even for dynamic scenes and objects in practice. Our method is fast enough to maintain a 1000 Hz refresh rate, as is commonly required by haptic rendering, for small objects in very complex environments on standard PC hardware.