Prerequisite:
Algorithms
Announcements:
Instructor:
Matya Katz ( matya@cs.bgu.ac.il )
Office hours: Monday 12:15-13:00 (and by
appointment) Office hours
Teaching Assistant:
Kerem
Geva
Class Time:
Monday 10-12
Wednesday 12-14
Course Description:
This
is an introductory course to computational geometry and its applications. We
will present data structures, algorithms and general techniques for solving
fundamental geometric problems, such as convex hull computation, line segment
intersection, orthogonal range searching, construction of Voronoi diagram and Delaunay
triangulation, polygon triangulation, and linear programming. We will also
present several geometric (optimization) algorithms (both exact and
approximate) for problems in robotics, computer graphics, GIS (geographic
information systems), communication networks, facility location, and VLSI
systems design.
The main textbook of the course is
[dBCvKO] Computational Geometry: Algorithms and Applications (3rd edition),
M. de Berg, O. Cheong, M. van Kreveld and M. Overmars, Springer-Verlag, 2008.
Additional textbooks
[DO] Discrete and Computational
Geometry, S. Devadoss
and J. O’Rourke, Princeton University Press, 2011.
[E]
Algorithms in Combinatorial Geometry, H. Edelsbrunner,
Springer-Verlag, 1987.
[M]
Computational Geometry: An Introduction Through
Randomized Algorithms, K. Mulmuley, Prentice
Hall, 1994.
[O]
Computational Geometry in C (2nd
edition), J. O'Rourke, Cambridge University Press, 1998.
[PS]
Computational Geometry: An Introduction (2nd edition), F. Preparata and M. Shamos,
Springer-Verlag, 1988.
The final grade will be determined by 4-5 homework
assignments (5% each) and a final exam.
You
may hand in the assignments either by yourself or with one other student.
Some old exams: exam 2005 A;
exam 2005 B;
exam 2007 A;
exam 2007 B;
exam 2009 A;
exam 2009 B;
exam 2016 A
The following list of topics is tentative.
The convex hull of
a set of points in the plane (applications: computing the diameter and width of
a point set).
An output sensitive
algorithm for computing the intersection points of a set of line segments; the plane
sweep technique.
A representation for planar
maps (based on doubly-connected edge lists).
Computing the overlay
of two planar maps; Boolean operations on two polygons (union, intersection,
and difference).
The art gallery theorem;
introduction to polygon triangulation.
An O(n
log n) polygon triangulation algorithm (partitioning a polygon into y-monotone
pieces; triangulating a y-monotone polygon).
Orthogonal range
searching.
Computing the intersection
of n half planes in O(n log n) time.
Linear programming -
introduction; A randomized incremental algorithm for linear programming in the
plane.
Planar point location,
vertical decomposition / trapezoidal map, a randomized incremental algorithm.
Nearest site queries,
nearest site Voronoi diagram.
Triangulation of a set of
points in the plane; the Delaunay triangulation.
Arrangement of
lines; duality; computing the discrepancy of a set of points in
the unit square.
Segment trees;
computing the area of a set of n axis-parallel rectangles in O(n
log n) time.
Simplex range
searching.
Hidden surface removal:
problem definition, image space / object space, the z-buffer algorithm, depth
order, the painter's algorithm. Output sensitive hidden surface removal
algorithm for horizontal fat triangles.
Introduction to geometric
optimization through facility location optimization and wireless
networks.
Measures of similarity
between curves.
Below you will
find, after each class, a brief summary of the topics covered in class.
This should not be taken as a complete description of the course's content.
19.10.20
Introduction
21.10.20
The convex hull of a set of points in the plane (gift wrapping, quickhull, O(n log n)-time incremental algorithm and divide and conquer algorithm
26.10.20
The diameter and width of a set of
points; rotating calipers
28.10.20
Sweeping; an output sensitive
algorithm for line segment intersection
2.11.20
Sweeping with a ray
Implementing the decision problem “do two line segments
intersect?” and a more general discussion on implementation issues and general
position.
4.11.20
The DCEL representation for planar maps; Map overlay (sketch); Boolean operations on polygons
9.11.20
The art gallery theorem, guarding –
a brief survey, introduction to polygon triangulation
11.11.20
Partitioning a polygon into
y-monotone pieces
16.11.20
Triangulating a y-monotone polygon
18.11.20
Orthogonal range searching – Kd-trees
23.11.20
Orthogonal range trees
25.11.20
Segment trees
Last
update: November 29, 2020