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Graduate Student Courses

 

l   Fundamentals of Applied Geometric Modeling

 

Time

2003-2015  Winter Quarter (32 Teaching Hours)

 

Description

Geometric modeling includes shape representation and processing. It is fundamentals of computer graphics, CAD, digital media and entertainment, scientific visualization, GIS, bio-informatics. The course includes the following contents such as mathematical principles and fundamentals of geometric modeling, parametric representation, implicit representation, solid modeling etc. This course is designed for both Ph.D. students and master students.

 

Preliminary

Calculus, Linear algebra, Numerical analysis, Data structure, C/C++

 

Examination

Project and Report

 

Contents

 

l   Vector and Affine Spaces

1.          Coordinate Systems

2.          Vector and Affine Spaces

3.          Frames

l   Differential Geometry of Curves

1.          Representation of Curves and Surfaces

2.          Differential Geometry of Curves

l   Differential Geometry of Surfaces (1)

1.          Tangent plane and surface normal

2.          First fundamental form I (metric)

3.          Second fundamental form II (curvature)

l   Differential Geometry of Surfaces(2)

1.          Principal curvatures

2.          Gaussian and mean curvatures

3.          Euler's theorem and Dupin's indicatrix

l   Bézier Curves and Surfaces (1)

1.          Bernstein Polynomials

2.          A Divide-and-Conquer Method for Drawing a Bézier Curve

3.          Quadratic Bézier Curves

4.          Cubic Bézier Curve

5.          A Matrix Representation for Cubic Bézier Curves

l   Bézier Curves and Surfaces (2)

1.          Reparameterizing Bézier Curves

2.          Bézier Control Polygons for a Cubic Curve

3.          The Equations for a Bézier Curve of Arbitrary Degree

4.          Bézier Patches

5.          Bézier Curves on Bézier Patches

6.          Subdivision of Bézier Patches

7.          A Matrix Representation of the Cubic Bézier Patch

8.          Advanced Topics on Bézier Curves/Patches

l   B-Spline Curves and Surfaces (1)

1.          The Analytic and Geometric Definition of a B-Spline Curve

2.          The Uniform B-Spline Blending Functions

3.          The DeBoor-Cox Calculation

l   B-Spline Curves and Surfaces (2)

1.          Uniform B-Spline Definition: Convolution Form

2.          The Two-Scale Relation for Uniform B-Splines

3.          A Proof of the Two-Scale Relation for Uniform Splines

4.          B-Spline Curves

l   B-Spline Curves and Surfaces (3)

1.          Rational B-Spline Curves

2.          Non-Rational B-Spline Surfaces

3.          Rational B-Spline Surfaces

4.          Catmull-Rom Spline

l   B-Spline Interpolation and Approximation

1.          Parameter Selection and Knot Vector Generation

2.          Global Curve Interpolation

3.          Global Curve Approximation

4.          Global Surface Interpolation

5.          Global Surface Approximation

l   Subdivision Curves and Surfaces (1)

1.          Subdivision/Refinement Process

2.          Chaikin’s Curves

3.          Quadratic Uniform B-Spline Curve Refinement

4.          Cubic Uniform B-Spline Curve Refinement

5.          Vertex and Edge Points

6.          Developing a Matrix Equation for Refinement

7.          Eigen-Analysis for Refinement Matrices

8.          Direct Calculation of Points on Cubic Subdivision Curves

9.          Calculating the Tangent Vectors at a Point

l   Subdivision Curves and Surfaces (2)

1.          Subdivision Methods for Quadratic B-Spline Surfaces

2.          Doo-Sabin Surfaces

3.          Subdivision Methods for Cubic B-Spline Surfaces

4.          Catmull-Clark Surfaces

5.          Loop Surfaces

l   Implicit Surface: Metaball and Quadratic Algebraic Surface

1.          Oveview of implicit surface modeling

2.          Metaball and blobby model

3.          Quadratic algebraic surfaces

l   Implicit Surface: A-patch

1.          Introduction

2.          Mathematical preliminaries

3.          Curve linear mesh schemes

4.          Simplex- and box-based schemes

5.          Subdivision based schemes

l   Implicit Surface: Displaying implicit surfaces

1.          Surface Tilling

2.          Raytracing

3.          Using Particles to Sample and Control More Complex Implicit Surfaces

l   Solid Modeling

1.          Solid Representations: An Introduction

2.          Wireframe Models

3.          Boundary Representations

4.          Constructive Solid Geometry

5.          References

 

 

l   Computer Graphics

 

Time

2007-2015  Fall and Winter Quarters (64 Teaching Hours)

 

Description

The course will focus on introductions of fundamental concepts, classical algorithms, research progress in computer graphics. The topics to be covered are: overview of the graphics process, rasterization process, transformation and clipping, geometric modeling, visible surface determination, realistic images synthesis, fundamental of scientific visualization. This course is designed for both Ph.D. students and master students.

 

Preliminary

Calculus, Linear algebra, Data structure, C/C++

 

Examination

Project and Report