Computational Geometry (236719)
Lecturer: Prof. Gill Barequet (barequet@cs.technion.ac.il)
TA: Mr. Gil Ben-Shachar (gilbe@cs.technion.ac.il)
Spring 2019

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

(09/Jun/19) There will be no class tomorrow, Monday, June 10, 2019. Instead, there will be a guided-reading of the paper whose link was distributed by email. Enjoy!

(03/Jun/19) EX3 was published in the web page. Enjoy!

(03/Jun/19) The recitations of Wednesdays 5/6 and 19/6 will not be held.

(14/May/19) EX2 was published in the web page. Enjoy!

(07/Apr/19) The recitations were moved to Ullmann 708.

(03/Apr/19) EX1 and EX4 were published in the web page. Enjoy!

(15/Mar/19) The first lecture of the semester, planned for Monday, March 18, 2019, will be given by the TA of the course, Mr. Gil Ben-Shachar.

(14/Mar/19) Students who intend to take the course are kindly requested to follow my request from March 3, 2019.

(03/Mar/19) All students (including free listeners) are kindly requested to join the mailing list of the course. (This is in addition, and irrelated, 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.

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, 1998.

Grading Policy

3 Home assignments (dry): ~12.5% (compulsory, submission in singletons!!);
Running project (wet): ~12.5% (same);
Final exam: 75% (1st term: Sunday 07/Jul/19, 09:00, at T-301. 2nd term: hopefully no need to!)

Assignment 1 (dry): published 03/Apr/19, due 17/Mar/19

Assignment 2 (dry): published 13/May/19, due 27/May/19

Assignment 3 (dry): published 03/Jun/19, due 17/Jun/19

Assignment 4 (wet): published 03/Apr/19, due 15/May/19 (before ex3!)

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.