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.

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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

Original: November 7, 1996

Revised : October 3, 2020