Computational Geometry (236719)
Prof. Gill Barequet (
Amir Vaxman (
Fall 2008-09


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

(31/Jan/09) Ex3 was posted in the web page. Enjoy!

(26/Jan/09) There is a minor bug-fix in ex4. You may reload the entire graphics package.

(19/Jan/09) Ex4 was posted in the web page. Enjoy!

(14/Jan/09) The recitation planned for Thursday 15/1/09 is canceled; it will be given in Thursday 22/1/09.

(05/Jan/09) Ex2 was posted in the web page. Enjoy!

(24/Dec/08) Amir will give the recitation of 08/01/09.

(17/Dec/08) Ex1 was posted in the web page. Enjoy!

(15/Dec/08) The lecture of Monday 22/12 will take place in T-201.

(17/Nov/08) The next recitation will take place in Thursday 4/12.

(10/Nov/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!)

(10/Nov/08) The classes of both today (10/11) and next week (17/11) are canceled. Compensation lectures will be announced. We meet next time in 24/11.

(10/Nov/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, and I will announce its posting when this happens.


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: Monday 16/Feb/08, 9am, T-4; Moed B: Wednesday 25/mar/08, 5:30pm, T-7.)

Assignment 1 (dry): given 17/12/08, due 01/01/09.

Assignment 2 (dry): given 05/01/09, due 19/01/09.

Assignment 3 (dry): given 31/01/09, due 10/02/09.

Assignment 4 (wet): given 19/01/09, due 19/02/09 (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 definitions. 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.

(c) Space Partitioning
BSP tress, quadtress.

(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.