Computational Geometry (236719)
Prof. Gill Barequet (barequet@cs.technion.ac.il)
Amir Vaxman (avaxman@cs.technion.ac.il)
Spring 2006-07

Syllabus

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/07) The lecture of Monday, July 30, is canceled.

(16/JUL/07) Ex3 is availbale. Enjoy!

(25/JUN/07) Both Ex2 and Ex4 are availbale. Enjoy!

(18/JUN/07) The due date of Ex1 was set to next Monday 25/6.

(14/May/07) Ex1 is found in the web page of the course. It will formally be distributed when the strike is over.

(11/Apr/07) I will not be able to deliver the lecture of 16/4 (Yom HaZikaron LaSho'a VelaGvura). A compensation lecture will be given in 18/4, 12:30, hall T-8. (modulo strikes 8-).

(10/Apr/07) Those who did not register yet to the mailing list of the course are kindly requetsed to do this ASAP!

(16/Mar/07) CG students are required to join the mailing-list of the course! In order to do this, please send me by e-mail the following details: full name, id, faculty, and degree. (This is in addition to the formal registration to the course!)

Bibliography

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

Library links

Grading Policy

3 Home assignments: 12% (Takef, submission in singletons!!);
Running project: 13% (same);
Final exam: 75% (Moed A: Sunday 09/09/07, 9am, hall T-6; Moed B: Tuesday 23/10/07, ???, ???)

Assignment 1 (dry): given ??/05/07, due ??/??/07.

Assignment 2 (dry): given 26/06/07, due 15/07/07.

Assignment 3 (dry): given 16/07/07, due 09/08/07.

Assignment 4 (wet): given 24/06/07, due 24/07/07 (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.

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

(b) Ex4
Striaght skeleton, ex4's description.

(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.

(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.