( Lime-basil Triangulation, Dinara Kasko, 2016.) CS 274 Computational Geometry Jonathan Shewchuk Spring 2026 Mondays and Wednesdays, 2:30-4:00 pm 320 Soda Hall My office hours: Mondays, 5:10–6 pm, 529 Soda Hall, Wednesdays, 5:10–6 pm, 529 Soda Hall, and by appointment. (I'm often free after the lectures too.)

Update: the Project is due 2:40 pm, April 29, 2026.

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

Algorithms and analyses: Sweep algorithms, incremental construction, divide-and-conquer algorithms, randomized algorithms, backward analysis. Numerical predicates and constructors, geometric robustness. Construction of triangulations, convex hulls, halfspace intersections, Voronoi diagrams, Delaunay triangulations, arrangements, and Minkowski sums. Small-dimensional linear programming and quadratic programming.

Geometric data structures: Doubly-connected edge lists, quad-edges, face lattices, trapezoidal maps, conflict graphs, history DAGs, 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. Mesh generation for graphics and scientific computing. Nearest neighbor search. Point location queries. Discrepancy and sampling in ray tracing. Curve reconstruction and surface reconstruction. Robot motion planning.

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

The best related sites:

• David Eppstein's Geometry in Action and Geometry Junkyard.
• Jeff Erickson's Computational Geometry Pages.
• Lists of open problems in computational geometry from Erik Demaine et al., Jeff Erickson, and David Eppstein.

• My robust predicates page (floating-point inputs, C).
• Chee Yap's CORE Library (C/C++).
• David Bailey's extensive MPFUN arbitrary precision arithmetic package (floating-point, C++ or Fortran).
• Olivier Devillers' predicates (integer inputs).
• Stefan Näher et al.'s LEDA contains several arbitrary precision numerical types, including integers and floating-point (C++). Commercial; you have to pay for it.

Textbook

Lectures

For January 21, here are Jeff Erickson's lecture notes on two-dimensional convex hulls.

For February 4, you might (optionally) also be interested in Chapter 2 from my book: Siu-Wing Cheng, Tamal Krishna Dey, and Jonathan Richard Shewchuk, Delaunay Mesh Generation, CRC Press (Boca Raton, Florida), December 2012.

For March 2 and 4, if you want to supplement my lectures, most of the material comes from Chapter 5 of Jiří 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 4, 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 9 & 11), the Clarkson–Shor convex hull construction algorithm (March 16), 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.

For March 11, 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.

For March 30, here are my Lecture Notes on Geometric Robustness.

For April 6 and 8, here is the compilation of BSP Tree Frequently Asked Questions.

For April 8, the best paper I know about how to implement a k-d tree is Sunil Arya and David M. Mount, Algorithms for Fast Vector Quantization, Data Compression Conference, pages 381–390, March 1993. See also Sunil Arya, David M. Mount, Nathan S. Netanyahu, Ruth Silverman, and Angela Y. Wu, An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions, Journal of the ACM 45(6):891–923, November 1998.

For April 15, here are Samir Khuller's notes on visibility graphs.

On April 20, we study the Delaunay refinement algorithm for triangular mesh generation by Jim Ruppert, A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation, Journal of Algorithms 18(3):548–585, May 1995.

On April 22, we study the octree-based algorithm for tetrahedral mesh generation by François Labelle and Jonathan Richard Shewchuk, Isosurface Stuffing: Fast Tetrahedral Meshes with Good Dihedral Angles, ACM Transactions on Graphics 26(3), August 2007. (Special issue on Proceedings of SIGGRAPH 2007.)

On April 27, we study the NN-Crust algorithm by Tamal K. Dey and Piyush Kumar, A Simple Provable Algorithm for Curve Reconstruction, Proceedings of the Tenth Annual Symposium on Discrete Algorithms (Baltimore, Maryland), pages 893–894, January 1999. My lecture includes Lemma 1 from this pioneering paper, which Dey and Kumar use in their correctness proof: Nina Amenta, Marshall Bern, and David Eppstein, The Crust and the Beta-Skeleton: Combinatorial Curve Reconstruction, Graphical Models and Image Processing 60/2(2):125–135, 1998.

For April 29, I suggest reading this paper about the Cocone algorithm. Feel free to skip the proofs, but read the theorems. Nina Amenta, Sunghee Choi, Tamal K. Dey, and N. Leekha, A Simple Algorithm for Homeomorphic Surface Reconstruction, International Journal of Computational Geometry and Applications 12(1–2):125–141, 2002.

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.)

Prerequisites

• CS 170 (Advanced Algorithms) or the equivalent. In particular, you should know and understand amortized analysis; how to solve recurrences; sorting algorithms; graph algorithms like Dijkstra's shortest path algorithm, connected components, and topological sorting; and basic data structures like binary heaps, hash tables, and balanced binary search trees (splay trees or AVL trees or red-black trees or 2-3-4 trees or B-trees). Every one of these will make an appearance at least once.
• A basic course in probability.
• Experience doing mathematical proofs. If you've never taken a class where you did lots of proofs, consider working your way through Bruce Ikenaga's notes and Larry Cusick's notes and exercises.

Grading

• 80% for the homeworks.
• 20% for the project: a Delaunay triangulation implementation, or an alternative by arrangement with the instructor.