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Generative Modeling: A Symbolic System for Geometric Modeling

John M. Snyder, James T. Kajiya

California Institute of Technology
From * Proceedings of SIGGRAPH 1992*, Association for Computing
Machinery Special Interest Group on Computer Graphics (ACM SIGGRAPH),
1992, p. 369-378.
## Abstract

This paper discusses a new, symbolic approach to geometric modeling
called generative modeling. The approach allows specifications,
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
multi-dimensional 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
operators 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.