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Computational Geometry, Fall 2022–2023


Lecture: Monday, 16:10-19:00  @ Britannia 005

Recitation: Monday, 19:10-20:00 @ Dach Auditorium

InstructorDan Halperin , [email protected]
Office hours by appointment; we can have zoom sessions, meet in my office (CheckPoint 434), or correspond by email

Teaching Assistant: Michal Kleinbort, [email protected]
Office hours by appointment

Grader: Michael Moshe Bilevich, [email protected]

The course covers fundamental algorithms for solving geometric problems such as computing convex hulls, intersection of line segments, Voronoi diagrams, polygon triangulation, and linear programming in low dimensional space. We will also discuss several applications of geometric algorithms to solving problems in robotics, GIS (geographic information systems), computer graphics, and more.


The main textbook of the course is:

A bibliographic list for the course


I will often use slides that accompany the main textbook of the course. The slides are by Marc van Kreveld and they can be found at the following site: Geometric Algorithms

Assignments, Examination and Grade

The standard assignments will account for 10% of the final grade, in case they improve the final grade.

The grade in the final exam will account for the remaining 90% of the grade in the course.

Exam(ple), [pdf] traditional style

Although the assignment grades will be used only for improving the final grade, submission of the assignments is mandatory and a prerequisite for taking the exam. About halfway through the course (probably earlier), we will present a programming assignment.

This year’s programming assignment is Minimum Coverage by Convex Polygons as described in the Computational Geometry Challenge 2023. We will use the same input and output format. More details on the assignment will be provided soon.

We encourage you to submit the programming assignment as well. This is not mandatory.

If however you do submit it, then your grade breakdown will be 10% the standard assignments,
15% the programming assignment, and 75% the final exam. Here as well,
the assignments grades will be counted toward the final grade only if they improve it.

The assignments will appear in the Moodle page of the course; the solutions should be submitted as a pdf file there.

Course Summary

Below you’ll find a very brief summary of what was covered in class during the semester.
This should not be taken as a complete description of the course’s contents.
For an outline of the course, see for example, the course summary in the Spring 2022 computational geometry course.

What is Computational Geometry?
Motivation and techniques. A slow convex hull algorithm. Graham’s O(n log n) algorithm (Chapter 1 in CGAA). (slides1.) Lower bound. Gift-wrapping algorithm for computing the convex hull, Jarvis’s March (Preparata-Shamos, Section 3.3.3).Orientation (Side-of-line) test, course mechanics and overview (slides).

Output sensitive algorithm to compute the intersections of line segments: Bentley Ottmann’s plane sweep (CGAA, Chapter 2), (slides2a).The Doubly Connected Edge List, DCEL (CGAA, Chapter 2). (slides2b).

The Doubly Connected Edge List, DCEL, cont’d, and map overlay  (CGAA, Chapter 2). (slides2b).Euler’s formula for plane graphs, proof without induction (Proofs from THE BOOK, Aigner-Ziegler, 5th Ed., Chapter 13)The Douglas-Peucker algorithm (see, e.g., wikipedia).Polygon triangulation and the art gallery problem (slides), up till Slide #13

Polygon triangulation and the art gallery problem, cont’d (slides)Introduction to CGAL, guest talk by Efi Fogel (slides)21.11.22Casting, half-plane intersection and Linear Programming in low dimensions (CGAA Chapter 4, slides4a).Supplementary material (Slides) , up till Unbounded LP.

Unbounded LP2D, LP3D (Slides, revised version).Smallest enclosing circle (CGAA Section 4.7, slides4b). GeoGebra files: SEC is unique, SEC new point on boundary

Orthogonal range searching I: kd-trees (CGAA, Chapter 5, slides5a)Orthogonal range searching II: range trees (CGAA, Chapter 5, slides5b)

Orthogonal range searching, continued: fractional cascading (CGAA, Chapter 5, slides5b)Trapezoidal maps and planar point location (CGAA, Chapter 6, slides6)

The plan: Trapezoidal maps and planar point location, cont’d (CGAA, Chapter 6, slides6)Guaranteed logarithmic-time point location (slides, paper)Point-Line duality, arrangements of lines in the plane, the zone theorem and incremental construction of the arrangement (CGAA, Chapter 8, slides8).

Point-Line duality, arrangements of lines in the plane, the zone theorem and incremental construction of the arrangement, cont’d (CGAA, Chapter 8, slides8).More on arrangements and duality (slides).The minimum area triangle problem (CGAL Arrangements and Their Applications, Section 4.2.1)The casting problem revisited (slides) (paper); Voronoi diagrams (CGAA, Chapter 7, slides7_combined)

Voronoi diagrams (CGAA, Chapter 7, slides7_combined) Arcs on the beach line (slides)

Triangulating planar point sets and Delaunay triangulations (CGAA, Chapter 9) (slides)Tiling with squares and packing dominos in polynomial time, guest talk by Mikkel Abrahamsen (slides, new version 09/01/23)Computing convex hulls in 3D (CGAA, Chapter 11) (slides)

The plan:Connections between major geometric structures (slides)

The main textbook used in the course:

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf,
Computational Geometry: Algorithms and Applications , 3rd Edition
Springer, 2008.

J. O’Rourke,
Computational Geometry in C 2nd Edition
Cambridge University Press, 1998 

J.-D. Boissonnat and M. Yvinec
Algorithmic Geometry
Cambridge University Press, 1995, 1998 (English version)
J.E. Goodman , J. O’Rourke, and Csaba Toth (editors)
Handbook of Discrete and Computational Geometry , 3rd Edition
CRC Press LLC, Boca Raton, FL, 2017.
J.-R. Sack and J. Urrutia (editors)
Handbook of Computational Geometry
North Holland, 2000.
M. Sharir and P.K. Agarwal
Davenport-Schinzel Sequences and Their Geometric Applications
Cambridge University Press, New York, 1995
K. Mulmuley
Computational Geometry: An Introduction Through Randomized Algorithms
Prentice Hall, Englewood Cliffs, NJ, 1994
F.P. Preparata and M.I. Shamos
Computational Geometry: An Introduction
Springer-Verlag, New York, NY, 1985.
H. Edelsbrunner
Algorithms in Combinatorial Geometry Springer Verlag, Heidelberg, 1987.

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