(Untitled, Till Rickert,
Shift 2005
Calendar.)
CS 274
Jonathan Shewchuk 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:
Topic | Readings | Due Wednesday | |
1: August 28 | Two-dimensional convex hulls | Chapter 1, Erickson notes | . |
2: August 30 | Line segment intersection | Sections 2, 2.1 | . |
September 4 | Labor Day | . | . |
3: September 6 | Overlay of planar subdivisions | Sections 2.2, 2.3, 2.5 | . |
4: September 11 | Polygon triangulation | Sections 3.2-3.4 | . |
5: September 13 | Delaunay triangulations | Sections 9-9.2 | . |
6: September 18 | Delaunay triangulations | Sections 9.3, 9.4, 9.6 | . |
7: September 20 | Voronoi diagrams | Sections 7, 7.1, 7.3 | . |
8: September 25 | Planar point location | Chapter 6 | . |
9: September 27 | Geometric robustness | Lecture notes | Homework 1 |
10: October 2 | Duality; line arrangements | Sections 8.2, 8.3 | . |
11: October 4 | Zone theorem; discrepancy | Sections 8.1, 8.4 | . |
12: October 9 | Polytopes | Matoušek Chapter 5 | . |
13: October 11 | Polytopes and triangulations | Seidel Upper Bound Theorem | Homework 2 |
14: October 16 | Small-dimensional linear programming | Sections 4.3, 4.6; Seidel T.R. | . |
15: October 18 | Small-dimensional linear programming | Section 4.4; Seidel appendix | . |
16: October 23 | Higher-dimensional convex hulls | Seidel T.R.; Secs. 11.2 and 11.3 | . |
17: October 25 | Higher-dimensional Voronoi; point in polygon | Secs. 11.4, 11.5 | . |
18: October 30 | k-d trees | Sections 5-5.2 | . |
19: November 1 | Range trees | Sections 5.3-5.6 | Homework 3 |
20: November 6 | Interval trees; closest pair in point set | Sections 10-10.1; Smid Sec. 2.4.3 | . |
21: November 8 | Segment trees | Section 10.3 | . |
22: November 13 | Binary space partitions | Sections 12-12.3 | . |
23: November 15 | Binary space partitions | Sections 12.4, 2.4, BSP FAQ | Homework 4 |
24: November 20 | Robot motion planning | Sections 13-13.2 | . |
25: November 22 | Minkowski sums | Sections 13.3-13.5 | . |
26: November 27 | Visibility graphs | Chapter 15; Khuller notes | . |
27: November 29 | Constrained triangulations | . | Project |
28: December 4 | Dobkin-Kirkpatrick hierarchies | . | . |
29: December 6 | Homework review | . | Homework 5 |
For August 28, here are Jeff Erickson's lecture notes on two-dimensional convex hulls.
For September 27, here are my Lecture Notes on Geometric Robustness.
For October 9 and 11, 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 October 11, 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 (October 16 & 18), the Clarkson-Shor convex hull construction algorithm (October 23), and Chew's linear-time algorithm for Delaunay triangulation of convex polygons are surveyed 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 October 18, 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 November 6, 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 November 15, here is the BSP FAQ.
For November 27, here are Samir Khuller's notes on visibility graphs.
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.)