# Generative Modeling for Computer Graphics and CAD: Symbolic Shape
Design Using Interval Analysis

John M. Snyder

California Institute of Technology

San Diego: Academic Press, 1992.

ISBN 0-12-654040-3

## Based on:

Snyder, J. * Generative Modeling: An Approach to High
Level Shape Design for Computer Graphics*, Ph.D. Thesis, Computer Science
Department, California Institute of Technology, 1992.

## Abstract

Generative modeling is an approach to computer-assisted geometric modeling.
The goal of the approach is to
allow convenient and high-level specification of shapes, and provide
tools for rendering and analysis of the specified shapes. Shapes include curves,
surfaces, and solids in 3D space, as well as higher-dimensional entities such
as surfaces deforming in time, and solids with a spatially varying mass density.
Shape specification in the approach involves combining low-dimensional entities,
especially 2D curves, into higher-dimensional shapes.
This combination is specified through
a powerful shape description language which builds multidimensional parametric functions.
The language is based on a set of primitive operators on parametric functions which include
arithmetic operators, vector and matrix operators,
integration and differentiation, constraint solution and global optimization.
Although each primitive operator is fairly simple, high-level
shapes and shape building operators can be defined using recursive combination
of the primitive operators.

The approach encourages the modeler to build
parameterized families of shapes rather than single instances. Shapes
can be parameterized by scalar parameters (e.g., time or joint angle)
or higher-dimensional parameters (e.g., a curve controlling
how the scale of a cross section varies as it is translated).
Such parameterized shapes allow easy modification of the design,
since the modeler can interact with parameters that
relate to high-level properties of the shape. In contrast,
many geometric modeling systems use a much lower-level specification,
such as through sets of many 3D control points.

Tools for rendering and analysis of generative models are developed
using the concept of interval analysis. Each primitive operator on parametric
functions has an inclusion function method, which produces an interval bound
on the range of the function, given an interval bound on its domain.
With these inclusion functions, robust algorithms exist for computing
solutions to nonlinear systems of constraints and global minimization
problems, when these problems are expressed in the modeling language.
These algorithms, in turn, are developed into robust approximation techniques
to compute intersections, CSG operations, and offset operations.