CIS 700, Spring 2002 More Advanced Geometric Methods in Computer Science Course Information January 5, 2002

Coordinates:

Moore 222, Mon. 11:00-1:00, Wed. 12:00-1:00.

Instructor:

Jean H. Gallier, MRE 176, 8-4405, jean@saul

TBA

Prerequesites:

Basic Knowledge in linear algebra and geometry (talk to me).

Textbook:

Review in MathSciNet (item 2) MathSciNet

Project, presentation of paper(s), problem Sets.

Expect to be held to high standards, and conversely! In addition to transparencies, I will distribute lecture notes. Please, read the course notes regularly, and start working early on the problems sets. They will be hard! Take pride in your work. Be clear, rigorous, neat, and concise. Preferably, use a good text processor, such as LATEX, to write up your solutions. You are allowed to work in small teams of at most three. We will have special problems sessions, roughly every two weeks, during which we will solve the problems together. Be prepared to present your solutions at the blackboard. I am hard to convince, especially if your use blatantly ``handwaving'' arguments.

Brief description:

The course should be of interest to anyone who likes geometry.

This course is basically a follow-up to CIS610. The goal is to present more of the advanced geometric methods used in geometric modeling, computer graphics, computer vision, robotics, etc.

The focus will be on SVD and polar forms, least squares, basics of Lie groups, and elementary differential geometry, with applications to geometric modeling using curves and surfaces (manufacturing, medical, ...)

plan to cover the following topics:

1. Spectral Theorems

Normal Linear Maps, Self-Adjoint and Other Special Linear Maps, Normal and Other Special Matrices

2. Singular Value Decomposition (SVD) and Polar Form

Polar Form, Singular Value Decomposition (SVD)

3. Applications of Euclidean Geometry

Applications to Least Squares Problems, Lagrange Multipliers

4. Basics of Classical Lie Groups

The Exponential Map, Some Classical Lie Groups, Symmetric and Other Special Matrices, Exponential of Some Complex Matrices, Hermitian and Other Special Matrices, The Lie Group SE(n) and the Lie Algebra se(n), Finale: Lie Groups and Lie Algebras, Applications of Lie Groups and Lie Algebras

5. Differential geometry of curves

Curvature, torsion, osculating planes, the Frenet frame, osculating circles, osculating spheres.

6. Differential geometry of surfaces

First fundamental form, normal curvature, second fundamental form, geodesic curvature, Christoffel symbols, principal curvatures, Gaussian curvature, mean curvature, the Gauss map and its derivative dN, the Dupin indicatrix, the Theorema Egregium, equations of Codazzi-Mainardi, Bonnet's theorem, lines of curvatures, geodesic torsion, asymptotic lines, geodesic lines, local Gauss-Bonnet theorem.

Ge'ome'trie 1, English edition: Geometry 1, Berger, Marcel, Universitext, Springer Verlag, 1990

Ge'ome'trie 2, English edition: Geometry 2, Berger, Marcel, Universitext, Springer Verlag, 1990

Metric Affine Geometry, Snapper, Ernst and Troyer Robert J., Dover, 1989, First Edition

A vector space approach to geometry, Hausner, Melvin, Dover, 1998

Geometry and Topology for Mesh Generation, Edelsbrunner, Herbert, Cambdridge U. Press, 2001

Computational Line Geometry, Pottman H. and Wallner J., Springer, 2001

Topological Geometry, I.R. Porteous, Cambridge University Press, 1981

Geometry, A comprehensive course, Pedoe, Dan, Dover, 1988, First Edition

Introduction to Geometry, H.S.M. Coxeter, Wiley, 1989, Second edition

Geometry And The Immagination, Hilbert, D. and Cohn-Vossen, S., AMS Chelsea, 1932

Polyhedra, Peter Cromwell, Cambridge University Press, 1999

Methodes Modernes en Geometrie, Jean Fresnel, Hermann, 1996

Computational Geometry in C, O'Rourke, Joseph, Cambridge University Press, 1998, Second Edition

Convex Analysis, Rockafellar, Tyrrell, Princeton University Press, 1970

Convex Sets, Valentine, Frederick, McGraw-Hill, 1964

Introduction to Applied Mathematics, Strang, Gilbert, Wellesley Cambridge Press, 1986, First Edition

Linear Algebra and its Applications, Strang, Gilbert, Saunders HBJ, 1988, Third Edition

Applied Numerical Linear Algebra, Demmel, James, SIAM, 1997

Numerical Linear Algebra, L. Trefethen and D. Bau, SIAM, 1997

Matrix Analysis, R. Horn and C. Johnson, Cambridge University Press, 1985

Matrix Computations, G. Golub and C. Van Loan, Johns Hopkins U. Press, 1996, Third Edition

A Course in Differential Geometry, W. Klingenberg, Springer, 1978, GTM No. 51

Differential Geometry of Curves and Surfaces, do Carmo, Manfredo P., Prentice Hall, 1976

Differential Geometry, Kreyszig, E., Dover, 1961

Differential Geometry, Guggenheimer, H. W., Dover, 1977

Lectures on Classical Differential Geometry, Struik, D., Dover, 1961

Modern Differential Geometry of Curves and Surfaces, Gray, A., CRC Press, 1997, Second Edition

Differential geometry, manifolds, curves, and surfaces, Berger, M. and Gostiaux, B., Springer, 1992, GTM No. 115

Riemannian Geometry, A Bginner's guide<\I>, Morgan, F., A.K. Peters, 1998

Geometry And Music

In mathematics, and especially in geometry, beautiful proofs have a certain ``music.'' I will play short (less than 2mn) pieces of classical music, or Jazz, whenever deemed appropriate by you and me!

Some Slides and Notes

Other possible topics are (nonexhaustive list):

• Introduction to projective geometry.
This includes projective spaces and subspaces, frames, projective maps, multiprojectve maps, the projective completion of an affine space, cross-ratios, duality, and the complexification of a real projective space.
• Applications to Rational curves and surfaces. Control points for Rational curves. Rectangular and Triangular rational patches. Drawing closed rational curves and surfaces.
• Calculus of variations (applied to robotics, vision, computer graphics)
• Fourier analysis and Wavelets and their applications to image analysis and computer graphics
Optimization methods
• Matrix analysis methods, Numerical methods for solving ODE's, PDE's,
• Finite elements method.