CIS 700, Spring 2002
More Advanced Geometric Methods in Computer Science
Course Information
January 5, 2002
Coordinates:
Moore 222, Mon. 11:001:00, Wed. 12:001:00.
Instructor:
Jean H.
Gallier, MRE 176, 84405, jean@saul
Office Hours: TuTh 4:305:30, or TBA
TBA
Prerequesites:
Basic Knowledge in linear algebra and geometry
(talk to me).
Textbook:
Home page for book
Springer NY
and
Amazon.com
Review in MathSciNet (item 2)
MathSciNet
SIAM Geometric Design Conference (GD01)
Mathematical Viruses, by David Hestenes
Geometric Calculus, R&D (ASU)
Grades:
Project, presentation of paper(s), problem Sets.
A Word of Advice :
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 followup 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:

Spectral Theorems
Normal Linear Maps,
SelfAdjoint and Other Special Linear Maps,
Normal and Other Special Matrices

Singular Value Decomposition (SVD) and Polar Form
Polar Form,
Singular Value Decomposition (SVD)

Applications of Euclidean Geometry
Applications to Least Squares Problems,
Lagrange Multipliers

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

Differential geometry of curves
Curvature, torsion,
osculating planes, the Frenet frame, osculating circles,
osculating spheres.

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 CodazziMainardi,
Bonnet's theorem, lines of curvatures, geodesic torsion,
asymptotic lines, geodesic lines, local GaussBonnet theorem.
Additional References:
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 CohnVossen, 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, McGrawHill, 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

Hermitian Spaces (slides)

Spectral Theorems (Symmetric, SkewSymmetric, Normal matrices) (slides)

Polar Form and SVD (slides)

Least squares, Pseudoinverses, Minimization of
quadratic functions using Lagrange multipliers

Lie Groups and Lie Algebras, the exponential map, part I

Lie Groups and Lie Algebras, the exponential map, part II

Differential geometry of curves, osculating circles, curvature, osculating plane, etc., part I

Differential geometry of curves, normal plane, rectifying plane, torsion, Frenet frame, etc., part II
 Differential geometry of curves, more on Frenet frames of nD curves, part III
(ps)
(pdf)
 Differential geometry of surfaces, tangent plane, normal, first fundamental form, part I
(ps)
(pdf)
 Differential geometry of surfaces, normal curvature, second fundamental form, geodesic curvature,
Christoffel symbols, part II
(ps)
(pdf)
 Differential geometry of surfaces, principal curvatures, Gaussian curvature,
mean curvature, part III
(ps)
(pdf)
 Differential geometry of surfaces, The Gauss map, the Dupin indicatrix, the Theorema Egregium
of Gauss, part IV
(ps)
(pdf)
 Differential geometry of surfaces, lines of curvature, geodesic torsion,
asymptotic lines, part V
(ps)
(pdf)
 Differential geometry of surfaces, geodesic lines,
local GaussBonnet theorem, covariant derivatives, part VI, VII
(ps, part VI)
(pdf)
 Covariant derivatives, parallel vector fields, parallel
transports, geodesics revisited, part VII
(ps)

Isometries of Hermitian Spaces and Hilbert Spaces (notes)
 Clifford algebras, Clifford groups, and the groups
Pin and Spin (notes)
(ps)
(pdf)

Bibliography (from book))

Basic Linear Algebra (Appendix 1)

Determinants (Appendix 2)
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,
crossratios, 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.
The table of contents of my book can be found
by clicking there:
Table of contents
For more information, visit
Geometric Methods and Applications
For Computer Science and Engineering
published by: