 ## 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 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

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

Revised : November 7, 1996