(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, and Minkowski sums. Geometric data structures: Doubly-connected edge lists, quad-edges, face lattices, trapezoidal maps, conflict graphs, history DAGs, spatial search trees (a.k.a. range search), binary space partitions, quadtrees and octrees, 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 | Assignment Due | |
1: January 23 | Two-dimensional convex hulls | Chapter 1, Erickson notes | . |
2: January 28 | Line segment intersection | Sections 2, 2.1 | . |
3: January 30 | Overlay of planar subdivisions | Sections 2.2, 2.3, 2.5 | . |
4: February 4 | Polygon triangulation | Sections 3.2–3.4 | . |
5: February 6 | Delaunay triangulations | Sections 9–9.2 | . |
6: February 11 | Delaunay triangulations | Sections 9.3, 9.4, 9.6 | . |
7: February 13 | Voronoi diagrams | Sections 7, 7.1, 7.5 | . |
February 18 | President's Day | . | . |
8: February 20 | Planar point location | Chapter 6 | Homework 1 |
9: February 25 | Duality; line arrangements | Sections 8.2, 8.3 | . |
10: February 27 | Zone theorem; discrepancy | Sections 8.1, 8.4 | . |
11: March 4 | Polytopes | Matoušek Chapter 5 | . |
12: March 6 | Polytopes and triangulations | Seidel Upper Bound Theorem | Homework 2 |
13: March 11 | Small-dimensional linear programming | Seidel T.R.; Sections 4.3, 4.6 | . |
14: March 13 | Small-dimensional linear programming | Section 4.4; Seidel appendix | . |
15: March 18 | Higher-dimensional convex hulls | Seidel T.R.; Secs. 11.2 and 11.3 | . |
16: March 20 | Higher-dimensional Voronoi; point in polygon | Secs. 11.4, 11.5 | . |
March 25–29 | Spring Recess | ||
17: April 1 | k-d trees | Sections 5–5.2 | . |
18: April 3 | Range trees | Sections 5.3–5.6 | Homework 3 |
19: April 8 | Interval trees; closest pair in point set | Sections 10–10.1; Smid Sec. 2.4.3 | . |
20: April 10 | Segment trees | Section 10.3 | . |
21: April 15 | Geometric robustness | Lecture notes | . |
22: April 17 | Binary space partitions | Sections 12–12.3 | Homework 4 |
23: April 22 | Binary space partitions | Sections 12.5, 2.4, BSP FAQ | . |
24: April 24 | Robot motion planning | Sections 13–13.2 | . |
25: April 29 | Minkowski sums | Sections 13.3–13.5 | Project |
26: May 1 | Visibility graphs | Chapter 15; Khuller notes | . |
27: May 6 | Nearest neighbor search; order k Voronoi | . | Homework 5 |
For January 23, here are Jeff Erickson's lecture notes on two-dimensional convex hulls.
For March 4 and 6, 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 6, 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 11 & 13), the Clarkson–Shor convex hull construction algorithm (October 19), 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 March 13, 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 April 8, 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 15, here are my Lecture Notes on Geometric Robustness.
For April 22, here is the BSP FAQ.
For May 1, 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.)