Dynamic Implicit Surfaces for Fast Proximity Queries in Physically Based Modeling Bernd Eberhardt WSI-GRIS Jens Hahn WSI-GRIS Reinhard Klein IGD Wolfgang Straßer WSI-GRIS Andreas Weber IGD WSI-GRIS: Arbeitsbereich Graphisch-Interaktive Systeme Wilhelm-Schickard-Institut f¨ ur Informatik Auf der Morgenstelle 10 Universit¨ at T ¨ ubingen, T ¨ ubingen, Germany IGD: Fraunhofer-Institut f ¨ ur Graphische Datenverarbeitung Rundeturmstr. 6, D-64283 Darmstadt, Germany Abstract 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 phys- ically based modeling to obtain a fast numerical integration of the underlying ordinary differential equation. We use an implicit for- mulation 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 com- plicated approximation algorithms are needed. Second, the evalua- tion 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 ob- jects 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. Keywords: implicit surfaces, collision detection, physically based modeling, proximity queries, cloth-modeling, haptic rendering 1 Introduction The problem of collision detection and other proximity queries has attracted substantial research during the last years; see e. g. [16] beber@gris.uni-tuebingen.de jhahn@gris.uni-tuebingen.de rklein@igd.fhg.de strasser@gris.uni-tuebingen.de aweber@igd.fhg.de for a survey. For convex polyhedra algorithms that work in ex- pected linear time have been developed [11, 20, 7]. Other meth- ods that overcome the restriction of convexity have been developed on the basis of various hierarchical bounding volume approaches: oriented bounding boxes used in OBBtrees [10], swept sphere vol- umes [14], discretely oriented polytopes (k-DOPs) [13]. In [2] also lower bounds for the complexity of a bounding volume approach are proved. Using these approaches interactive response times are possible on present hardware for models consisting of several thousand tri- angles. To achieve interactive response times in simulations that are governed by ODEs more sophisticated techniques are needed that allow for fast distance computations in even shorter times. The benefit of implicit surfaces for collision detection, distance computation and physically based modeling of deformable objects has been known for a long time [24, 21]. In particular an easy inside-outside test involving just a few computations makes the use of implicit surfaces desirable—provided the object of interest can be represented by an implicit function of the desired class. For physically based modeling, the collision model, in its implicit rep- resentation, can be incorporated into the formulation of the dif- ferential equation (potential field around the object, force field). Boundary conditions of the system (which are given by the collision object) can again be implicitly formulated in the ODE. The numer- ical integration of the ODE is then fast with large time-steps. The problem however is to find a cheap evaluation of a potential func- tion which represents the collision model as an iso-surface. With the approach described here we can find such a function for arbi- trary collision models given as a boundary representation, even for dynamic models. The use of implicit surfaces for animation purposes involves usu- ally the computation of an animated skeleton. In several settings simulating just the very limited number of particles of the skele- ton reduces computation time compared to the calculation of the many vertices of an explicitly given surface. However, examples of such animations are usually “blobby” and of an artificial nature.