MCS 481: Computational Geometry

David Dumas

University of Illinois at Chicago
Spring 2014

General information

Instructor	David Dumas (ddumas@math.uic.edu)
Lectures	MWF 10am in Taft Hall 207
CRN	35483 (undergraduate), 35484 (graduate)
Text	Computational Geometry: Algorithms and Applications, 3ed., de Berg, Cheong, van Kreveld, and Overmars (Springer-Verlag, 2008) ISBN-13: 978-3540779735
Office hours	Mon & Fri, 2-3pm in SEO 503

Course Materials

• Course syllabus

Assignments

Reading

In the main textbook unless otherwise noted.

Date (announced)	Reading
Mon, Jan 13	Chapters 1 and 2
Mon, Feb 3	Chapter 3
Wed, Feb 12	Chapter 6
Mon, Feb 24	Chapter 7
Wed, Mar 5	Chapter 9
Monday, Mar 17	Chapter 5
Monday, Mar 31	Chapter 8
Monday, Apr 21	Chapter 11
Friday, Apr 25	Paper folding

Homework

Problem numbers refer to exercises in the textbook. Notation such as "9.14ac" means parts a and c of problem 9.14, while "9.14" means the entire problem 9.14.

Hwk #	Due	Problems
0	Fri, Jan 24	Install CGAL and confirm that it is working by compiling and running one of the included example programs. Take a screenshot and email it to ddumas@math.uic.edu. (See sample screenshot and explanation.)
1	Mon, Jan 27	1.4ab, 1.6ab, 1.7ad, 1.9, 1.10bcd
2	Wed, Feb 12	2.2, 2.5, 2.8, 2.11, 2.14 + additional problem
3	Mon, Feb 24	3.3, 3.4, 3.14, 6.1, 6.2
4	Mon, March 10	6.5, 6.6, 7.5, 7.10, 7.11
5	Fri, Mar 21	9.2, 9.5a, 9.9, 9.11, 9.13
6	Mon, Apr 7	5.1(lower bound only), 5.3, 5.4, 5.5ab
7	Mon, Apr 21	8.2, 8.4, 8.11 + additional problems

Projects

Project #	Due
1	Mon, Feb 10	Convex hull and line segment intersection Project description: projdesc1.pdf Source files: projdesc1-files-1.0.tar.gz
2	Fri, Mar 7	Polygons Project description: projdesc2.pdf Source files: projdesc2-files-1.0.tar.gz
3	Fri, Apr 4	Voronoi and Delaunay Project description: projdesc3.pdf Source files: projdesc3-files-1.0.tar.gz
Final	Fri, Apr 11 (proposals) Fri, May 2 (reports)	Final project details

Software

• CGAL - Computational Geometry Algorithms Library
• Dependencies of CGAL (full details)
  • A supported C++ compiler, such as gcc/g++
  • cmake build system
  • boost general-purpose C++ library

CGAL and its dependencies are available as binary packages for some operating systems, including several versions of GNU/Linux, and this is often the easiest way to install these programs.

For example, on recent versions of Ubuntu GNU/Linux (e.g. 12.04, where the instructions below were tested), all of the necessary packages can be installed from a regular user account by entering the following commands in a terminal:

sudo apt-get update
sudo apt-get install g++ cmake libcgal-demo

After these commands complete, a compressed archive of the source code of the CGAL example programs is located in /usr/share/doc/libcgal-demo/examples.tar.gz. Thus the following command would decompress the example programs in the current directory (e.g. your home directory):

tar --gzip -xvf /usr/share/doc/libcgal-demo/examples.tar.gz

The exact installation procedure and file locations for other Linux distributions may vary. Check the documentation provided with the CGAL package for your distribution for details.

Grading

Your course grade will be determined on the following basis:

Homework	30%
Projects	30%
Final project	40%

Computational Geometry & Algorithms Resources

• Online resources for background in analysis of algorithms
  • MIT OpenCourseWare for SMA 5503 - Introduction to Algorithms - C. Leiserson and E. Demaine, including:
    • Lecture 2 covers: O, Ω,Θ notation and the Master Theorem on recurrences
  • Analysis of Algorithms notes by Steven Skiena --- lecture notes and audio
• Computational Geometry in general
  • Preparata, F. and Shamos, I.  Computational Geometry: An Introduction, Springer-Verlag, 1990.
    ISBN-13: 978-3540961314
  • O'Rourke, J.  Computational Geometry in C, Cambridge University Press, 2001.
    ISBN-13: 978-0521649766
    • This is a good reference for details of implementation in a low-level language like C
  • Cormen, T., Leiserson, C., Rivest, R., and Stein, C.  Introduction to Algorithms, 2nd ed., MIT Press, 2001.
    ISBN-13: 978-0262032933
    • This general reference on data structures and algorithms covers some computational geometry in chapter 33
• Convex hulls
  • Notes on 2D convex hulls by Jeff Erickson.
  • Demonstration of the quick-hull algorithm
  • qhull, a C implementation of quick-hull in any number of dimensions
  • Classroom examples of robustness problems in geometric computations (Kettner, Mehlhorn, Pion, Schirra, and Yap).
    • See Figure 1 on page 4 for a vivid demonstration of the failure of the naive floating-point orientation predicate.
  • Fast robust predicates for computational geometry (Shewchuk)
    • Paper and C implementation of adaptive precision to detect and correct inaccuracies of the naive floating-point orientation predicate.
• Segment Intersection and Map Overlay
  • Demo applet: Plane sweep for segment intersection by Wei Qian
    • Select "Data Set 1" then use the horizontal scrollbar to visualize the sweep.
  • The DCEL representation of a planar subdivision is discussed in Section 1.2.3 of Preparata and Shamos (see reference above)
  • An important special case of the overlay of planar subdivision is the polygon boolean operation problem, i.e. computing the intersection, difference, or union of two polygons. The intersection case is also called polygon clipping.
    • Polygon clipping applet by Christopher Andrews
    • Comparison of various implementations of polygon boolean operations
• Polygon Triangulation and the Art Gallery Problem
  • Art Gallery Theorems and Algorithms by Joseph O'Rourke
    • Includes discussion of NP-hardness of the minimal art gallery problem (chapter 9)
  • Art gallery applet and discussion by Thierry Dagnino
  • References for the triangulation of monotone polygons using mountains:
    • Lecture notes by Sariel Har-Peled. Discussed: polygon -> monotone -> mountains -> triangles
    • Lecture notes by Jeff Erickson. Discussed: mountains -> triangles, monotone -> triangles
• Voronoi diagrams
  • Real-time javascript Voronoi with d3.js
  • Video demo of Fortune's algorithm (video by Kevin Schaal)
    • There are many other online videos illustrating Fortune's algorithm, but I think Schaal's might be the most clear. A few others for more complicated point collections:
      • Fortune's Algorithm - Part 2 by Eddie Lee
      • A nice 3D description of the beach line
  • Voronoi Diagrams and a Day at the Beach by David Austin (AMS feature column, Aug 2006)
  • Java applets demonstrating Fortune's algorithm (by David Austin):
    • Interactive Voronoi diagram
    • The beach line
    • Break points sweep out Voronoi edges
  • A visual implementation of Fortune's Voronoi algorithm by Allan Odgaard & Benny Kjaer Nielsen
• Orthogonal Range Searching
  • KD-tree demo applet by H. Akcan
• Arrangements of lines and discrepancy
  • Proof of the zone theorem using Davenport-Schinzel sequences, notes by Samir Khuller.
  • Book: Davenport-Schinzel sequences and their geometric applications by Sharir and Agarwal. (Link goes to book information with links to several online retailers.)
  • Survey: Davenport-Schinzel sequences and their geometric applications by Sharir and Agarwal. (gzipped postscript format)
• Binary Space Partition
  • Toth's work on the complexity of BSP for segments in the plane:
    • A note on binary plane partitions - Worst case BSP size Ω(n log(n) / log(log(n))), 2001
    • Binary plane partitions for disjoint line segments - BSP of size O(n log(n) / log(log(n))) exists (2009)
  • Wikipedia article IdTech1 with detailed discussion of the use of BSP in the computer game Doom.
• Paper folding
  • Erik Demaine's Fold and Cut page, including links to:
    • Geometric Folding Algorithms (book)
    • Paper on the disk-packing method by Bern, Demaine, and Eppstein
  • Jeff Erickson's straight skeleton page

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