(Untitled, Till Rickert, Shift 2005 Calendar.)

CS 274
Computational Geometry

Jonathan Shewchuk

Spring 2005
Mondays and Wednesdays, 1:00-2:30 pm
320 Soda Hall

Combinatorial geometry: Polygons, polytopes, triangulations, planar and spatial subdivisions. Constructions: triangulations of polygons, convex hulls, intersections of halfspaces, Voronoi diagrams, Delaunay triangulations, arrangements of lines and hyperplanes, Minkowski sums; relationships among them. Geometric duality and polarity. Numerical predicates and constructors. Upper Bound Theorem, Zone Theorem.

Algorithms and analyses: Sweep algorithms, incremental construction, divide-and-conquer algorithms, randomized algorithms, backward analysis, geometric robustness. Construction of triangulations, convex hulls, halfspace intersections, Voronoi diagrams, Delaunay triangulations, arrangements, Minkowski sums.

Geometric data structures: Doubly-connected edge lists, quad-edges, face lattices, trapezoidal maps, history DAGs, spatial search trees (a.k.a. range search), binary space partitions, visibility graphs.

Applications: Line segment intersection and overlay of subdivisions for cartography and solid modeling. Triangulation for graphics, interpolation, and terrain modeling. Nearest neighbor search, small-dimensional linear programming, database queries, point location queries, windowing queries, discrepancy and sampling in ray tracing, robot motion planning.

Here are Homework 1, Homework 2, Homework 3, Homework 4, and Homework 5.

The best related sites:

Resources for dealing with robustness problems (in increasing order of difficulty):


Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (presently known as Otfried Cheong), Computational Geometry: Algorithms and Applications, second revised edition, Springer-Verlag, 2000. ISBN # 3-540-65620-0.
Known throughout the community as the Dutch Book.


Homeworks will be irregularly assigned, and are due at the start of class on a Wednesday. Homeworks are to be done alone, without help from or discussion with other humans.

Topic Readings Due Wednesday
1: January 19 Two-dimensional convex hulls Chapter 1, Erickson notes .
2: January 24 Line segment intersection Sections 2, 2.1 .
3: January 26 Overlay of planar subdivisions Sections 2.2, 2.3, 2.5 .
4: January 31 Polygon triangulation Sections 3.2-3.4 .
5: February 2 Delaunay triangulations Sections 9-9.2 .
6: February 7 Delaunay triangulations Sections 9.3, 9.4, 9.6 .
7: February 9 Voronoi diagrams Sections 7, 7.1, 7.3 .
8: February 14 Planar point location Chapter 6 .
9: February 16 Duality; line arrangements Sections 8.2, 8.3 Homework 1
February 21 Presidents' Day . .
10: February 23 Zone theorem; discrepancy Sections 8.1, 8.4 .
11: February 28 Polytopes Matoušek Chapter 5 .
12: March 2 Polytopes and triangulations Seidel Upper Bound Theorem Homework 2
13: March 7 Small-dimensional linear programming Sections 4.3, 4.6; Seidel T.R. .
14: March 9 Small-dimensional linear programming Section 4.4; Seidel appendix .
15: March 14, 12:40 Carlo Séquin on splines, 203 McLaughlin Carlo's lecture notes 1 .
16: March 16, 12:40 Carlo Séquin on subdivision, 203 McLaughlin Carlo's lecture notes 2 .
March 21-25 Spring Recess
17: March 28 Higher-dimensional convex hulls Seidel T.R.; Secs. 11.2 and 11.3 .
18: March 30 Higher-dimensional Voronoi; point in polygon Secs. 11.4, 11.5; Smid Sec. 2.4.3 Homework 3
19: April 4 k-d trees Sections 5-5.2 .
20: April 6 Range trees Sections 5.3-5.6 .
21: April 11 Interval trees Sections 10-10.1 .
22: April 13 Segment trees Section 10.3 .
23: April 18 Binary space partitions Sections 12-12.3 .
24: April 20 Binary space partitions Sections 12.4, 2.4, BSP FAQ .
25: April 25 Robot motion planning Sections 13-13.2 .
26: April 27 Minkowski sums Sections 13.3-13.5 .
27: May 2 Visibility graphs Chapter 15; Khuller notes Homework 4
28: May 4 Geometric robustness Lecture notes Project
29: May 9 Constrained triangulations . .
May 13 . . Homework 5

For January 19, here are Jeff Erickson's lecture notes on two-dimensional convex hulls.

For February 28 and March 2, if you want to supplement my lectures, most of the material comes from Chapter 5 of Jirí Matoušek, Lectures on Discrete Geometry, Springer (New York), 2002, ISBN # 0387953744. He has several chapters online; unfortunately Chapter 5 isn't one of them.

For March 2, I will hand out Raimund Seidel, The Upper Bound Theorem for Polytopes: An Easy Proof of Its Asymptotic Version, Computational Geometry: Theory and Applications 5:115-116, 1985. Don't skip the abstract: the main theorem and its proof are both given in their entirety in the abstract, and are not reprised in the body at all.

Seidel's linear programming algorithm (March 7 & 9), the Clarkson-Shor convex hull construction algorithm (March 28), and Chew's linear-time algorithm for Delaunay triangulation of convex polygons are reported in Raimund Seidel, Backwards Analysis of Randomized Geometric Algorithms, Technical Report TR-92-014, International Computer Science Institute, University of California at Berkeley, February 1992. Warning: online paper is missing the figures. I'll hand out a version with figures in class.

For March 9, I will hand out the appendix from Raimund Seidel, Small-Dimensional Linear Programming and Convex Hulls Made Easy, Discrete & Computational Geometry 6(5):423-434, 1991. For anyone who wants to implement the linear programming algorithm, I think this appendix is a better guide than the Dutch Book.

On March 30, I will teach a randomized closest pair algorithm from Section 2.4.3 of Michiel Smid, Closest-Point Problems in Computational Geometry, Chapter 20, Handbook on Computational Geometry, J. R. Sack and J. Urrutia (editors), Elsevier, pp. 877-935, 2000. Note that this is a long paper, and you only need pages 12-13.

For April 18, here is the BSP FAQ.

For April 27, here are Samir Khuller's notes on visibility graphs.

For May 2, here are my Lecture Notes on Geometric Robustness.

For the Project, read Leonidas J. Guibas and Jorge Stolfi, Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams, ACM Transactions on Graphics 4(2):74-123, April 1985. Feel free to skip Section 3, but read the rest of the paper. See also this list of errors in the Guibas and Stolfi paper, and Paul Heckbert, Very Brief Note on Point Location in Triangulations, December 1994. (The problem Paul points out can't happen in a Delaunay triangulation, but it's a warning in case you're ever tempted to use the Guibas and Stolfi walking-search subroutine in a non-Delaunay triangulation.)

Geometry Applets

These applets can be quite helpful in establishing your geometric intuition for several basic geometric structures and concepts.



Supported in part by the National Science Foundation under Awards ACI-9875170, CMS-9980063, CCR-0204377, CCF-0430065, and EIA-9802069, and in part by a gift from the Okawa Foundation.
(Radiolarian Color Painting. Ernst Haeckel, zoologist, 1834-1919.)