Combinatorial 3-Manifolds from Sets of Tetrahedra

We seek to further expand the shared collaborative potential of cyberworlds by using haptic force- feedback in shared virtual scenes. We propose how to define density of the objects, together with their geometry and appearance, by using mathematical functions. We illustrate this concept by developing software which allows us to touch and feel surfaces of VRML and X3D objects, convert them to solid objects as well as create any other solid objects using the function-based extension of VRML and X3D. We define geometry, appearance and density of the solid objects by implicit, explicit and parametric functions straight in the VRML/X3D code or in dynamic-link libraries. Since the function-based models are small in size, it is possible to perform their collaborative interactive modifications with concurrent synchronous visualization at each client computer with any required level of detail. We illustrate the proposed with several application examples.