## Prof. Christoph M. Hoffmann

### Course Information

Computer-Aided Geometric Design (CAGD) is a technology that is underlying virtually all shapes designed and manufactured. Design of cars, planes and ships are commonly-known exaples. Less widely known fields where CAGD plays a role are architecture, consumer goods such as bottles, movies and games.

We will study the basic techniques including shape design and manipulation of Bézier and B-spline curves and surfaces, as well as subdivision curves and surfaces.

The course will consider theory, applications and examples. Homework includes both written questions and programs reinforcing concepts. There will be a midterm, a final, and either a term paper or a larger project implementation, plus a final. Class participation is valued.

Expected weights are as follows:
Written homework, demo programs, 30%
Midterm, 20%
Final, 25%
Project/term paper, 25%

### Agenda

There will be three parts, each on the order of 5 weeks.

1. Basics everyone should know:

Projective geometry and linear algebra; implicit algebraic and parametric curves and surfaces; Bezout's theorem; Liming's construction; conversion betwen implicit and parametric curves; differential properties; geodesics; medial axis and transforms; scalar fields and level sets; orientation and quaternions; Chasles' theorem and compliant motion; linear and nonlinear shape models; direct manipulation interfaces; geometric constraints.

2. Bezier and B-spline curves and surfaces:

Splines in general; partition of unity; integral Bezier curves, basic properties; DeCasteljau and domain extension and reduction; degree raising; Hermite problem for cubic Bezier and B-splines; rational curves; Haar basis, knots and deBoor points; Cox-deBoor algorithm; NURBS; surface continuity; triangle patches.

3. Subdivision curves and surfaces (to be expanded):

Concepts, examples and applications; open vs. closed meshes; limit theorems; and more.