Generative modeling

This paperdiscusaes a new, symbolic approach to geometric modeling called generative modeling. Tbc approach allows specification, rendering, and maiysis of a wide variety of ahapes including 3D curves, strrfaees, and solids, as well as higfrer-dimenaionsd shapes such as surfaces deforming in time, and volumes with a spatially varying mass &nsity. The system also supports powerful operations on shapes such as '"repammeterize this curve by arclengtb", "compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces", or "solve this constraint or ODE system". The system haa been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds mtrkidimerrsiorralparametric functions. Tbc Imgustgc is baaed on a set of symbolic operators on continuous, pieccwise differentiable parametric functions. We present several shape examples to show bow conveniently shapes can be specified in the system. We also discuss tbe kinds of operators useful in a geometric modeling system, including arithmetic operators, vector and matrix operators, imegration, differentiation, constraint solution, and constrained minimisation. Associated with each operator are several methmta, which compute proprties about the parametric functions represented with the operxom. We show how marrypowerful rendering and snafyricsf operations can be supported with only three methods: evahsation of the parametric function at a point, symbolic dlffererrtiation of the parametric function, md evacuation of artinclusion function for the parametric function.