On-Line Geometric Modeling Notes

Computer Science Department,
University of California, Davis


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.


 Coordinate Systems

Vector and Affine Spaces

 Vector Spaces
 Points and Vectors
 Affine-Spaces, Barycentric Coordinates and Convex Combinations


 Bernstein Polynomials

Bezier Curves and Patches

 A Divide-and-Conquer Method for Drawing a Bezier Curve
 Quadratic Bezier Curves
 Cubic 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

 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

Revised : November 7, 1996