Computational Geometry (236719)
Lecturer: Prof. Gill Barequet (barequet@cs.technion.ac.il)
TA: Maor Grinberg (maorg@cs.technion.ac.il)
Fall 2012-13

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

(26/Dec/12) Ex3 was posted in the web page of the course.

(18/Dec/12) Ex4 was posted in the web page of the course.

(02/Dec/12) Ex2 was posted (well ahead of time) in the web page of the course.

(27/Nov/12) Recitation slides about fractional cascading were posted in the web page.

(20/Nov/12) The class of today is canceled.

(04/Nov/12) Ex1 was posted in the web page of the course.

(21/Sep/12) All students (including free listeners) are kindly requested to join the mailing list of the course. (This is in addition to the formal registration to the course! One can leave this mailing list at any time.) For joining the mailing list, please e-mail me (barequet@cs) your full name, id #, faculty, and degree toward which you study.

(20/Sep/12) Since I will be abroad at that time, the second term (Moed B) of the course will NOT be held in March 13, 2013! Hopefully there will not be a need for such an exam. If yes, an alternative date will be coordinated with the relevant student(s).

Bibliography

	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 Home assignments (dry): ~12.5% (Takef, submission in singletons!!);
Running project (wet): ~12.5% (same);
Final exam: 75% (Moed A: Tuesday 05/Feb/13, time TBA, location TBA; no Moed B!)

Assignment 1 (dry): given 06/Nov/12, due 20/Nov/12.

Assignment 2 (dry): given 11/Dec/12, due 28/Dec/12.

Assignment 3 (dry): given 01/Jan/13, due 22/Jan/13.

Assignment 4 (wet): given 18/Dec/12, due 29/Jan/13 (Graphics files, sample input 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.

(c) Fractional cascading

Fractional cascading for range searching.

(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.

(d) 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.

(e) Space Partitioning

BSP trees, 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 applications of it.

(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.