Instructor: Prof. Xiangmin Jiao
Time: Mon. 12:50pm--2:10pm, Fri. 2:20pm--3:40pm, Spring 2008
Location: Old Chem. 135
Office hours: Mon. 2:15pm--3:30pm, Fri. 1:00pm--2:00pm, 1-115 Math Tower
Note: This webpage is for public access only. Registered students, please access course materials by logging onto Blackboard. Students who are interested in auditing the course, please send me email so that I can grant you access to the materials.
Geometry is the foundation of many advanced numerical computations for physics, engineering, and computer-aided design, and it is remarkably rich and intriguing. Unfortunately, some classical theories in geometry (in particular, differential geometry) do not lend themselves to accurate and efficient numerical computations over discretized surfaces, and some classical formulas were not scrutinized carefully in terms of numerical stability. Robust solution of dynamic surfaces is an area that is critically important for many scientific and engineering applications and requires numerical solutions to geometric problems. Combined expertise in geometry and numerical analysis is required for the development of numerical solutions for such problems that involve geometric or physical differential equations and complex geometry or dynamic surfaces. This course aims to introduce students to geometry and numerical analysis through a unified approach.
The course has two complementary goals: 1) It introduces students to the fundamentals of differential geometry and differential and integral calculus on surfaces, by taking a fresh look from a numerical analysis point of view. We refer to this new approach as numerical differential geometry (analogous to numerical linear algebra). 2) The course exposes students to some outstanding research issues for both static and dynamic surfaces that are important to scientific computations.
The course is organized as four parts: 1) fundamentals of numerical geometry (geometry of curves and surfaces, discretizations of curves and surfaces, singularities), 2) numerical calculus on surfaces (generalization of gradient, divergence, curl, and Laplacian to surfaces, numerical surface integrations, differential forms, discrete and finite-element exterior calculus), 3) dynamic surfaces (fundamentals of dynamic surfaces, front tracking, face offsetting, level-set methods, geometric differential equations, collision detection and topological changes), and 4) surface parameterization and optimization (homeomorphism and isometric, surface parameterization, cross parameterizations, surface mesh optimization, anisotropic mesh adaptation). The first two parts will take the first half of the semester, and the remainder will take the second half.
Textbook: None. The material will be extracted from various textbooks (see below) and research papers.
Prerequisite: Basic knowledge of numerical linear
algebra, multivariate calculus, programming in C/C++ and MAT LAB.
Course format: Lectures, homework assignments, term project (with
proposal, mid-term report, and final report and presentation).
Grading policy: Participation (20%), homework assignments (40%), project (40%)
Some Reference Books (Incomplete)
G. B. Arfken and H. J. Weber. Mathematical Methods for Physicists. Academic Press, 5th edition, 2001.
M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
G. Farin. Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, 5th edition, 2002.
J. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms, Morgan Kaufmann, 1999.
J. H. Hubbard and B. B. Hubbard. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Prentice Hall, 2nd edition, 2001.
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, 2003.
H. M. Schey. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. W.W. Norton, New York, 4th edition, 2005.