Introduction to Geometry 15-499C / 15-881A
Fall 1997

Instructors: Michael Erdmann, with guest lectures by Yanxi Liu
TA: German
Location: Scaife 324
Time: TR 10:30-11:50

Click here (23K postscript) for the course schedule.

This course will cover elementary differential and computational geometry. The purpose of the course is to prepare a student for advanced geometrical work in robotics and computer science. Increasingly, cutting-edge results in these areas require a working knowledge of differential geometry, algebraic geometry, algebraic topology, and computational geometry. Much of this work is inaccessible to a student just entering the field. In this course we will convey the basic tools, definitions, and results of differential geometry and the basic algorithms of computational geometry, so that a student can, either by self-study or through further courses, understand and implement the advanced results in computer science and robotics discovered in the past decade. We will touch upon some of these applications in the course. In particular, we will consider the robot motion planning problem as a core application.

The topics are:

• Motion Planning: Configuration Space, Visibility Graph, Non-Holonomic Motion Planning, Forces in Cspace.
  
  
• Frame Fields: Curves, Frenet Formulas, Covariant Derivatives, Differential Forms, Connection Forms, Structural Equations.
  
  
• Calculus on Surfaces: Surfaces, Patches, Tangent Vectors, Mappings, Differential Forms, Integration on Surfaces, Manifolds.
  
  
• Shape Operators: Surface Shape Operator, Normal Curvature, Gaussian Curvature.
  
  
• Point and Range Queries: One shot, repeated query, slab method, multidimensional binary tree.
  
  
• Convex Hull Algorithms: Lower bounds, sorting reduction, 2D algorithms (Graham's Scan, Jarvis's March, Quickhull, Mergehull), dynamic convex hull, higher-dimensional algorithms (gift-wrapping, beneath-beyond), 3D convex hull.
  
  
• Proximity Algorithms: Voronoi diagram, triangulations, Euclidean minimum spanning tree.
  
  
• Plane Sweep: Polygon union and intersection.
  
  

Prerequisites for this course are minor, but important. Calculus and linear algebra should be sufficient from the mathematics side, while fundamental data structures and algorithms should be sufficient from the computer science side. The course will consist of a series of intensive lectures. It is expected that a student will spend a few hours each day digesting the material of the lectures and solving some related homework problems. Grades will be based primarily on the homework, and possibly an exam. The course may be taken at either the undergraduate or the graduate levels. Students taking the class for graduate credit will be asked to prepare a project, either a paper or an implementation of an advanced topic.

Bibliography

• W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
  
  
• H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin, 1987.
  
  
• J.-C. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Boston, 1991.
  
  
• B. O'Neill, Elementary Differential Geometry, Academic Press, New York, 1966. 2nd Edition: 1997.
  
  
• F. P. Preparata and M. I. Shamos, Computational Geometry, Springer-Verlag, New York, 1985. (Corrected and expanded printing: 1988.)
  
  
• M. Spivak, Differential Geometry, Publish or Perish, Berkeley, 1979.
  
  

Aug 26 Overview, Robot Motion Planning

Aug 31 Calculus on Euclidean Space: space, tangent vectors, curves

Sep 02 Dual Spaces, 1-Forms

Sep 07 Differential forms

Sep 09 Mappings

Sep 14 Curves, speed, unit speed, [T, N, B]

Sep 16 Frenet Formulas

Sep 21 Covariant Derivatives, Lie Brackets, Frame Fields

Sep 23 Non-holonomic Motions, Connection Forms, Structural Equations

Sep 28 Surfaces: definitions, examples, patches, manifolds

Sep 30 Manifolds: configuration space, embeddings, differential forms

Oct 05 open

Oct 07 Mappings, pullbacks, line integrals, 2D integration, Stokes' Theorem

Oct 12 Shape Operator, examples, normal acceleration

Oct 14 Curvature: normal, Gaussian, mean, methods of computation

Oct 19 Gauss Theorem, surface area

Oct 21 Datastructure preliminaries (e.g., segment tree, DCEL)

Oct 26 Geometric preliminaries (e.g., homog. coords., duality, edge-edge inter.)

Oct 28 Range searching, point location

Nov 02 Range query and point location (cont)

Nov 04 Convex Hull (definitions, reductions, extreme points, etc.)

Nov 09 Convex Hull in 2D (definitions, theorems, Graham's scan)

Nov 11 Convex Hull in 2D (Jarvis' March, Quickhull, Mergehull, Dynamic)

Nov 16 Higher Dimensional Convex Hull

Nov 18 Proximity problems in the plane, Voronoi Diagram definition

Nov 23 open

Nov 25 Thanksgiving

Nov 30 Voronoi Diagram algorithm, Euclidean Minimum Spanning Tree, Triang.

Dec 02 Polygon union and intersection, plane sweep