CS3621 Introduction to Computing with Geometry Notes
Dr. C.K. Shene
Professor
Department of Computer Science
Michigan Technological University
© 19972011 C.K. Shene
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since July 1, 1998
Last update: May 4, 2011
Select the topics you wish to review:
 Unit 1: Course Overview

Why Is Computing with Geometry Important?

The Theme of this Course

The Complexity of Geometric Problems

Computing with Floating Point Numbers

Problems

References
 Unit 2: Geometric Concepts

Coordinate Systems, Points, Lines and Planes

Simple Curves and Surfaces

Homogeneous Coordinates

Geometric Transformations

Problems

References
 Unit 3: Solid Models

Solid Representations: An Introduction

Wireframe Models
Boundary Representations


Manifolds

The WingedEdge Data Structure

The EulerPoincaré Formula

Euler Operators
Constructive Solid Geometry


Interior, Exterior and Closure

Regularized Boolean Operators

A CSG Design Example

Problems

References
 Unit 4: Parametric Curves

Parametric Curves: A Review

Tangent Vector and Tangent Line

Normal Vector and Curvature

Continuity Issues

Rational Curves

Problems

References
 Unit 5:
Bézier Curves

An Introduction

Construction

Moving Control Points

De Casteljau's Algorithm

Why Is de Casteljau's Algorithm Correct?

Derivatives of a Bézier Curve

Subdividing a Bézier Curve

Why Is the Subdivision Algorithm Correct?

Degree Elevation of a Bézier Curve

Why Is the Degree Elevation Algorithm Correct?

Problems

References
 Unit 6:
Bspline Curves

Motivation
Bspline Basis Functions

Definition

Important Properties

Computation Examples
Bspline Curves

Definition

Open Curves

Closed Curves

Important Properties

Computing the Coefficients

A Special Case

Moving Control Points

Modifying Knots

Derivatives of a Bspline Curve
Important Algorithms for Bspline Curves
Knot Insertion

Single Insertion

Inserting a Knot Multiple Times

De Boor's Algorithm

De Casteljau's and de Boor's Algorithms

Subdividing a Bspline Curve

Problems

References
 Unit 7:
NURBS Curves

Motivation

Definition

Important Properties

Modifying Weights
Important Algorithms for Bspline and NURBS Curves

Knot Insertion: Single Insertion

De Boor's Algorithm

Rational Bézier Curves

Rational Bézier Curves: Conic Sections

Circular Arcs and Circles

Problems

References
 Unit 8: Surfaces

Basic Concepts
Bézier Surfaces


Construction

Important Properties

De Casteljau's Algorithm
Bspline Surfaces


Construction

Important Properties

De Boor's Algorithm
 Unit 9: Interpolation
and Approximation
Parameter Selection and Knot Vector Generation


Overview

The Uniformly Spaced Method

The Chord Length Method

The Centripetal Method

Knot Vector Generation

The Universal Method

Parameters and Knot Vectors for Surfaces

Solving Systems of Linear Equations
Curve Interpolation


Global Interpolation
Curve Approximation


Global Approximation
Surface Interpolation


Global Interpolation
Surface Approximation


Global Approximation
 Mesh Related Information in Slides (PDF):
These slides will be converted to HTML pages in the future

Mesh Basics (March 28, 2010, 1.24MB, 45 pages)

Subdivision Surfaces (April 6, 2010, 1.6MB, 49 pages)

Mesh Simplification (April 8, 2010, 3.77MB, 61 pages)

Multiresolution Modeling (very) Basics (April 14, 2010, 82K, 12 pages)

Mesh Compression under construction
Please send comments and suggestions to
shene@mtu.edu