Function-Based Haptic Interaction in Cyberworlds

Polygon and point based models dominate in virtual reality. These models also affect haptic rendering algorithms which are often based on collision with polygons. We use mathematical functions to define and implement geometry (curves, surfaces and solid objects), visual appearance (3D colors and geometric textures) and various tangible physical properties (elasticity, friction, viscosity, and force fields). The function definitions are given as analytical formulas (explicit, implicit and parametric), function scripts and procedures. Since the defining functions are very small we can efficiently use them in collaborative virtual environments to exchange between the participating clients. We proposed an algorithm for haptic rendering of virtual scenes including mutually penetrating objects with different sizes and arbitrary location of the observer without a prior knowledge of the scene to be rendered. The algorithm is based on casting multiple haptic rendering rays from the Haptic Interaction Point (HIP), and it builds a stack to keep track on all colliding objects with the HIP. The algorithm uses collision detection based on implicit function representation of the object surfaces. The proposed approach allows us to be flexible when choosing the actual rendering platform. The function-defined objects and parts constituting them can be used together with other common definitions of virtual objects such as polygon meshes, point sets, voxel volumes, etc. We implemented an extension of X3D and VRML which allows for defining complex geometry, appearance and haptic effects in virtual scenes by functions and common polygon-based models, with various object sizes, mutual penetrations, arbitrary location of the observer and variable precision.