On-Line Geometric Modeling Notes Computer Science Department, University of California, Davis

Overview

These are topic papers on geometric modeling set up and maintained by the faculty and students of the UC Davis Computer Graphics Group. The notes cover a wide range of basic topics in the geometric modeling area and are continually expanding. They were initially started by Professor Ken Joy as a service to the Computer Science Department's geometric modeling courses. In most cases, we have provided both hypertext and postscript versions of the notes.

These notes are similar in content to some of those contained in the on-line computer graphics notes.

Enjoy.......

Coordinate Systems

Vector and Affine Spaces

Vector Spaces

Points and Vectors

Affine-Spaces, Barycentric Coordinates and Convex Combinations

Frames

Bernstein Polynomials

Bezier Curves and Patches

A Divide-and-Conquer Method for Drawing a Bezier Curve

Quadratic Bezier Curves

A Matrix Representation for Cubic Bezier Curves

Reparameterizing Bezier Curves

Bezier Control Polygons for a Cubic Curve

The Equations for a Bezier Curve of Arbitrary Degree

Bezier Patches

A Matrix Representation of the Cubic Bezier Patch

Bezier Curves on Bezier Patches

Subdivision of Bezier Patches

B-Spline Curves and Patches

The Analytic and Geometric Definition of a B-Spline Curve

The Uniform B-Spline Blending Functions (with examples)

The DeBoor-Cox Calculation

The Support of a Blending Function

Writing Uniform B-Spline Blending Functions as Convolutions

The Two-Scale Relation for Uniform Splines

A Proof of the Two-Scale Relation for Uniform Splines

Other Curve/Patch Representations

The Catmull/Rom Spline

Subdivision/Refinement

What are Subdivision/Refinement Methods?

Introduction to Subdivision Curves

Subdivision Methods for Quadratic B-Spline Curves

Subdivision Methods for Cubic B-Spline Curves

Defining Cubic Refinement by Vertex and Edge Points

Directly Calculating Points on the Cubic Curve

Directly Calculating Tangents of the Cubic Curve

Eigenvalue Calculation for Refinement Matrices

Introduction to Subdivision Surfaces

Subdivision Methods for Quadratic B-Spline Surfaces

Doo-Sabin Surfaces

Subdivision Methods for Cubic B-Spline Surfaces

Catmull-Clark Surfaces

Loop Surfaces

Please Write us and tell us how you are using these notes.

This page maintained by Ken Joy
Comments to the Author: joy@cs.ucdavis.edu