Generative modeling: a symbolic system for geometric modeling

This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higher-dimensional shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as 'reparameterize this curve by arclength', '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 has 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 multidimensional parametric functions. The language is based on a set of symbolic operators on continuous, piecewise differentiable parametric functions. We present several shape examples to show how conveniently shapes can be specified in the system. We also discuss the kinds of operators useful in a geometric modeling system, including arithmetic operators, vector and matrix operators, integration, differentiation, constraint solution, and constrained minimization. Associated with each operator are several methods, which compute properties about the parametric functions represented with the operators. We show how many powerful rendering and analytical operations can be supported with only three methods: evaluation of the parametric function at a point, symbolic differentiation of the parametric function, and evaluation of an inclusion function for the parametric function. Like CSG, and unlike most other geometric modeling approaches, this modeling approach is closed, meaning that further modeling operations can be applied to any results of modeling operations, yielding valid models. Because of this closure property, the symbolic operators can be composed very flexibly, allowing the construction of higher-level operations without changing the underlying implementation of the system. Because the modeling operations are described symbolically, specified models can capture the designer's intent without approximation error.