Computational Geometry (236719)
Prof. Gill Barequet (
Amir Vaxman (
Spring 2007-08


Fundamental techniques, data structures, and algorithms for solving geometric problems such as computing convex hulls, intersection of line segments, the Voronoi diagram and Delaunay triangulation of a point set, polygon triangulation, range search, linear programming, and point location. Some topics of discrete geometry, e.g., the crossing number of a graph and its applications, are also covered.

News and Messages

(28/JUL/08) Ex3 was posted. Enjoy!

(13/JUL/08) Ex4 was posted. Enjoy!

(03/JUL/08) Ex2 was posted. Enjoy!

(03/JUL/08) In Monday 7/7, there will be a 3-hour lecture, starting at 9:30.
In Monday 14/7, the 2-hour lecture will be given from 9:30 to 11:30, followed by a 1-hour tutorial (11:30-12:30).

(10/Jun/08) A compensation lecture will be given in Wednesday 18/6 at 12:30, T-5.

(09/Jun/08) The classes of 16/6 and 30/6 are canceled. Compensation lecture(s) will be announced later.

(02/Jun/08) The next tutorial will take place in 23/6. It will cover straight skeletons and medial axes of polygons.

(01/Jun/08) Ex1 was posted. Enjoy!

(23/May/08) I put in this web page links to slightly old (last year's) versions of the course presentations. The "new edition" will only include a few minor typo fixes.

(23/May/08) Those who did not register yet to the mailing list of the course are kindly requetsed to do this ASAP! Please e-mail me your full name, id #, faculty, and degree toward which you study. (This is in addition to the formal registration to the course!)


Main text book: Computational Geometry: Algorithms and Applications (3rd ed.), M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Springer-Verlag, 2008.
For background: Computational Geometry in C (2nd ed.), J. O'Rourke, Cambridge University Press, 2000.

Library links

Grading Policy

3-4 Home assignments: ~12.5% (Takef, submission in singletons!!);
Running project: ~12.5% (same);
Final exam: 75% (Moed A: Tuesday 09/Sep/08, time TBA, hall TBA; Moed B: Tuesday 11/Nov/08, hopefully no need to).

Assignment 1 (dry): given 02/06/08, due 19/06/08.

Assignment 2 (dry): given 07/07/08, due 28/07/08.

Assignment 3 (dry): given 28/07/08, due 18/08/08.

Assignment 4 (wet): given 14/07/08, due 14/08/08 (Graphics files, FAQ file)

Course summary and slides

(1) Introduction
What is Computational Geometry? Example problems and motivations. Naive, incremental, and divide-and-conquer convex-hull algorithms.

(2) Vectors, Plane sweep
Vector cross product and orientation test. Segment-intersection test. Convex-polygon queries. Plane-sweep paradigm. Segment-intersection algorithm.

(a) Planar graphs
Graph definition, planar graphs, Euler's formula, the DCEL structure.

(3) Polygon triangulation
The art-gallery theorem. Partitioning a simple polygon into monotone pieces. Triangulating a monotone polygon.

(4) Linear programming
What is linear programming. A D&C algorithm for half-planes intersection. An incremental algorithm for half-planes intersection. Randomized linear programming. Unbounded linear programming. Smallest enclosing disk of a 2D point set.

(b) Polygonal skeletons
Straight skeleton of a polygon and a polyhedron. Their complexities, and algorithms to compute them.

(5) Orthogonal range searching
1D range searching. 2D kd-trees. 2D Range trees.

(6) Point location
Slabs structure. Trapezoidal map. A randomized incremental algorithm for computing a trapezoidal map. Worst- and average-case Time/Space analysis of the algorithm. Handling degeneracies.

(7) Voronoi diagram
Definition and variants. A plane-sweep algorithm for computing the Voronoi diagram of a point set.

(c) Voronoi diagram
More and alternative efinitions. Lloyd's algorithm.

(8) Duality
A point-line duality in the plane and its properties.

(9) Line Arrangements
Line arrangements and their properties. The zone theorem. Computing an arrangement of lines. Levels in line arrangements. Halfspace discrepancy and its dual problem.

(10) Delaunay triangulation
Triangulation of a point set. Angle vector and the triangulation that maximizes it. Delaunay triangulation and its relation to the angle vector. A randomized incremental algorithm for computing the Delaunay triangulation.

(11) The crossing-number lemma
The crossing-number lemma and a few of its applications.

(12) 2-point site Voronoi diagrams
Some 2-point site distance functions and their respective Voronoi diagrams.

(13) A few theorems
The upper-bound theorem. Interpretations of Voronoi diagrams. Zone theorems. Envelopes of lines and planes.