Generative Modeling for Computer Graphics and Cad - 1st Edition - ISBN: 9780126540406, 9781483260358

Generative Modeling for Computer Graphics and Cad

1st Edition

Symbolic Shape Design Using Interval Analysis

Authors: John M. Snyder
eBook ISBN: 9781483260358
Imprint: Academic Press
Published Date: 4th August 1992
Page Count: 334
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Generative Modeling for Computer Graphics and Cad: Symbolic Shape Design Using Interval Analysis presents a symbolic approach to shape representation that is useful to the CAD/CAM and computer graphics communities.

This book discusses 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 that compute properties about the parametric functions represented with the operators.

This text also elaborates how numerous 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.

This publication is intended for people working in the area of computational geometry who are interested in a robust class of algorithms for manipulating shapes and those who want to know how human beings can specify and manipulate shape.

Table of Contents

Foreword by James Τ. Kajiya



Index of Figures

1 Introduction

1.1 User vs. Macliine Shape Representations

1.2 Criteria for Evaluation of a Shape Representation

1.3 Previous Work in Shape Representation

1.3.1 Polyhedra

1.3.2 Piecewise Parametric Polynomial Shapes

1.3.3 Algebraic Shapes

1.3.4 Sweeps

1.3.5 Deformations

1.3.6 Non-Polynomial Parametric Shapes

1.3.7 Non-Polynomial Implicit Shapes

1.4 Areas for Improvement in Shape Representation

1.5 The Generative Modeling Approach

1.6 Generative Modeling: The State of the Art

1.6.1 Previous Work Related to Generative Modeling

1.6.2 New Work Described in this Book

1.6.3 Future Work

2 Shape Representation

2.1 Generative Models: A Domain of Shapes

2.1.1 Why Generative Models?

2.2 Specifying Generative Models

2.2.1 Parametric Functions and the Closure Property

2.2.2 Using Symbolic Operators to Specify Parametric Functions

2.2.3 Specific Operators

2.2.4 Operator Metliods

2.3 Development of tlie Generative Modeling Representation

2.3.1 System 1-Nonrecursive Transformations and Generators

2.3.2 System 2-Limited Recursive Transformations

2.3.3 System 3-Fully Recursive Transformations and Generators

2.3.4 System 4-Using a General Purpose Language

3 Shape Specification Examples

3.1 GENMOD Preliminaries

3.2 Generative Surfaces

3.2.1 Affine Cross Section Transformations

3.2.2 Non-Affine Cross Section Transformations

3.2.3 Boolean Operations on Planar Cross Sections

3.2.4 Parameterizing Cross Sections

3.3 Other Generative Shapes

3.3.1 Solids

3.3.2 Time-Dependent Shapes

3.3.3 Vector Fields on Surfaces

4 Shape Rendering

4.1 Categorizing Generative Models for Rendering

4.1.1 Curves

4.1.2 Surfaces

4.1.3 Solids

4.1.4 Shapes of High Input Dimension

4.1.5 Shapes of High Output Dimension

4.2 Sampling Generative Models

4.2.1 Uniform Sampling

4.2.2 Adaptive Sampling

4.3 Controlling Visualization of Generative Models

4.3.1 Visualization Methods

4.3.2 Fast Visualization

4.3.3 Non-Precomputed Visualization

4.3.4 Non-Uniformly Sampled Visualization

4.3.5 Interactive Rendering Implementation

5 Interval Methods for Shape Synthesis and Analysis

5.1 Why Interval Analysis?

5.2 Inclusion Functions

5.2.1 Terminology and Definitions

5.2.2 Inclusion Functions for Arithmetic Operations

5.2.3 Natural Interval Extensions

5.2.4 Inclusion Functions for Relational and Logical Operators

5.2.5 Mean Value and Taylor Forms

5.2.6 An Inclusion Function for the Integration Operator

5.2.7 Inclusion Functions Based on Monotonicity

5.3 Constraint Solution Algorithm

5.3.1 The Problem of Indeterminacy

5.3.2 Subdivision Methods

5.3.3 Solution Aggregation

5.3.4 Termination and Acceptance Criteria for Constraint Solution

5.3.5 Interval Newton Methods

5.3.6 Existence of Solutions

5.3.7 A Constraint Evaluation Enhancement

5.4 Constrained Minimization Algorithm

5.4.1 Termination and Acceptance Criteria for Constrained Minimization

5.4.2 Monotonicity Test

6 Applying Interval Methods to Geometric Modeling

6.1 Offset Operations

6.1.1 The B-Offset: A Tighter Representation for the S-Offset Boundary

6.1.2 A Constraint-Based Approach for Computing B-Offsets

6.2 Approximating Implicit Curves

6.2.1 An Implicit Curve Approximation Algorithm

6.2.2 A Robust Test for Global Parameterizability

6.2.3 A Heuristic Test for Global Parameterizability

6.2.4 Relaxing the Approximation Algorithm's Restrictions

6.3 Approximating Parametric Shapes Using Adaptive Criteria

6.3.1 Kd-Trees

6.3.2 Generating a Triangular Mesh

6.4 CSG-Like Operations with Trinuned Surfaces

6.4.1 An Algorithm for Approximating CSG-like Operations

6.4.2 Example Application of the CSG-Like Operation Algorithm

6.4.3 Kd-Tree Algorithms for CSG-like Operations

6.5 Constructive Solid Geometry with Trinuned Surfaces

6.6 Approximating Implicit Surfaces

6.6.1 An Implicit Surface Approximation Algorithm

7 Conclusion

Appendices: A The GENMOD Language

A.1 Language Extensions

A. 1.1 New Operators

A. 1.2 Overloaded Operators

A.2 Language Types

A.3 Language Primitive Operators

A.3.1 Constants and Parametric Coordinates

A.3.2 Arithmetic Operators

A.3.3 Elementary Operators

A.3.4 Vector Operators

A.3.5 Matrix Operators

A.3.6 Integral and Derivative Operators

A.3.7 Curves and Tables

A.3.8 Relational Operators

A.3.9 Logical Operators

A.3.10 Conditional and Branching Operators

A.3.11 Substitution Operators

A.3.12 Inverse Operator

A.3.13 Constraint Solution and Constrained Minimization Operators

A.3.14 ODE Solution Operator

A.4 language Extensibility: Building Higher-Level Operators

A.4.1 Interpolation Operators

A.4.2 Concatenation Operators

A.4.3 Reparameterization Operator

A.4.4 Closed Offset Operator

A.5 Operator Libraries

A.5.1 Affine Transformation Library

A.5.2 Line/Are Library

A.5.3 Physical Properties of Rigid Bodies Library

Β GENMOD Code Examples

B.1 Sphere/Cylinder Fillet Example

B.2 Screwdriver Tip Examples

B.3 Bottle Example

B.4 Teddy Bear Example

C Theorems in Interval Analysis

C.1 Second Order Convergence of the Mean Value Form

C.2 Convergence of the Constraint Solution Algorithm

C.3 Convergence of the Minimization Algorithm

C.4 Existence of Zeroes

C.5 Interval Implicit Function Theorem




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© Academic Press 1992
Academic Press
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About the Author

John M. Snyder

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