# CS3621 Introduction to Computing with Geometry Notes

### Professor Department of Computer Science Michigan Technological University

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Last update: May 4, 2011

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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 Winged-Edge Data Structure
The Euler-Poincaré 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: B-spline Curves
Motivation
B-spline Basis Functions
Definition
Important Properties
Computation Examples
B-spline Curves
Definition
Open Curves
Closed Curves
Important Properties
Computing the Coefficients
A Special Case
Moving Control Points
Modifying Knots
Derivatives of a B-spline Curve
Important Algorithms for B-spline Curves
Knot Insertion
Single Insertion
Inserting a Knot Multiple Times
De Boor's Algorithm
De Casteljau's and de Boor's Algorithms
Subdividing a B-spline Curve
Problems
References
Unit 7: NURBS Curves
Motivation
Definition
Important Properties
Modifying Weights
Important Algorithms for B-spline 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
B-spline 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