COT5520: COMPUTATIONAL GEOMETRY

Term: Fall 2008
Time: Tuesday 10:40am-11:30am, Thursday 10:40am-12:35pm
Location: MCCB1108
Professor: Alper Üngör
Office hours: Tue 11:30am-1:30pm

syllabus announcements schedule projects references

Announcements

Schedule

Date Lecture Topic Assignments Speaker
Aug 26 Tu Introduction, syllabus, course structure, etc.    
  CONVEX HULLS ALGORITHMS [BKOS00, Chapter 1]    
Aug 28 Th Convex hulls
Orientation test; Degenaracy; Jarvis' march [J73]

A
Sep 2 Tu Convex hulls
Divide & conquer; Graham's scan [G72, A79] Chan's alg.[C96]
  A
  PLANE-SWEEP ALGORITHMS [BKOS00, Chapters 2 and 3 ]    
Sep 4 Th Line segment intersections
Plane-sweep [BO79];
  A
Sep 9 Tu Doubly linked edge list, Overlay subdivisions
  A
Sep 11 Th Polygon Triangulation
Triangulating monotone polygons [GJPT78]
  A
Sep 16 Tu Polygon Triangulation
Partitioning simple polygons
  A
Sep 18 Th Convex Partitioning
Lower and upper bounds, A factor 4 approximation algorithm
  A
  LINEAR PROGRAMMING [BKOS00, Chapter 4 ]    
Sep 23 Tu Manufacturing with Molds
Necessary and Sufficient condition, Half-Plane Intersections
  A
Sep 25 Th Linear Programming
Fesaible Region, Optimal solution;
Incremental and randomized algorithms
  A
  ORTHOGONAL SEARCH [BKOS00, Chapters 5 and 10]    
Sep 30 Tu Geometric data structures; Range search
Quad-tree; kd-tree [B75];
  A
Oct 7 Tu Improvements on range searching
Range tree; fractional cascading [CG86]
Inverse Range Search
Segment tree [B77]; interval tree [E83]; priority search tree [M85]
  A
  VORONOI DIAGRAMS & DELAUNAY TRIANGULATIONS
[BKOS00, Chapters 7, 9, and 14]
   
Oct 9 Th Voronoi diagrams
Voronoi diagrams [AK00]; furthest point Voronoi diagram, Other distance metrics
  A
Oct 16 Th Voronoi diagrams
Fortune's plane sweep algorithm
  A
Oct 21 Tu Delaunay triangulation
Empty circles, local Delaunayhood [D34],
edge-flip [L77], lifting, analysis, maxmin angles
  A
Oct 23 Th Randomized incremental algorithm
Incremental construction [GKS92]; backward analysis [S93]
  A
Oct 28 Tu Project proposal presentations and discussions HW#3 is out
 
Oct 30 Th Project proposal presentations and discussions  
Nov 4 Tu Point Location
DAG structure for point location in triangulations
[BKOS00, Chapters 6 and 9]
Steiner triangulations
Steiner triangulation [BE95]; quality measure; quad-trees [BE95];
  A
Nov 6 Th Delaunay refinement
Circumcenter insertion [R95]; Sphere packing argument
  A
  ARRANGEMENTS [BKOS00, Chapter 8]    
Nov 11 Tu Zones
Duality; line arrangements; complexity;
incremental algorithm; zone theorem [ESS93]
HW#3 due A
Nov 13 Th Levels and discrepancy
Super-sampling for rendering; Half-plane discrepancy[EG89]
  A
  OTHER GEOMETRY APPLICATIONS [BKOS00, Chapters 13 and 15]    
Nov 18 Tu Geometric Approximation Algorithms
TSP, Metric TSP, Euclidean TSP
Polynomial Time Approximation Scheme (PTAS)
HW#4 is out
A
Nov 20 Th Motion Planning
Trapezoidel Maps; Robotics; Configuration Space; Connectedness;
Visibility Graphs
  A
Nov 25 Tu Project Presentations   A
Nov 27 Th THANSGIVING BREAK
Dec 2 Tu Project Presentations
Reports due  
Dec 4 Th Final Exam (in-class, closed book)    

References

Books
[BKOS00] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008. [course textbook]
Surveys
[E02] Lecture Notes on ConvexHulls by J. Erickson.
[E07] Lecture Notes on Polygon Triangulations by J. Erickson.
[BE95] M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry (2nd ed.), D.-Z. Du and F. Hwang (eds.), World Scientific, 1995, 47-123.
[E00] H. Edelsbrunner. Triangulations and meshes in computational geometry. Acta Numerica (2000), 133-213.

Other Papers
[A79] A. M. Andrew. Another efficient algorithm for convex hulls in two dimensions. Information Processing Letters, 9:216-219, 1979. [A left-to-right variant of Graham's scan]
[AK00] F. Aurenhammer and R. Klein. Voronoi Diagrams. Handbook of Computational Geometry, Ed. J. Sack, J. Urrutia (eds.), 2000, 201-290.
[BDH96] B. Barber, D. Dobkin, and H. Huhdanpaa. The Quickhull Algorithm for Convex Hulls. ACM Transactions on Mathematical Software Vol. 22, No. 4, December 1996, Pages 469¿483.
[B75] J. L. Bentley. Multidimensional binary search trees used for associative searching. Commun. ACM, 18:509-517, 1975.
[B77] J. L. Bentley. Solution to Klee's rectangle problems. Tech. Rep., Carnegie-Mellon Univ., Pittsburgh, 1975.
[BO79] J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers, C-28:643-647, 1979.
[C96] T. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete and Computational Geometry, 16:361-368, 1996.
[CE92] B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. Journal of the ACM 39:1-54, 1992.
[CG86] B. Chazelle and L. J. Guibas. Fractional cascading. Algorithmica 1:133-162 and 163-191, 1986.
[D34] B. N. Delaunay. Sur la Sphere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793-800, 1934.
[E83] H. Edelsbrunner. A new approach to rectangle intersections. International Journal Computational Mathematics 13:209-219 and 221-229, 1983.
[GJPT78] M. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan. Triangulating a simple polygon. Information Processing Letters, 7:175-179, 1978.
[G72] R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132-133, 1972.
[GKS92] L. J. Guibas, D. E. Knuth, M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7:381-413, 1992.
[J73] R. A. Jarvis. On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters, 2:18-21, 1973.
[KS86] D. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM J. Computing, 12:1:287-299, 1986.
[L77] C. L. Lawson. Software for C1 surface interpolation. Mathematical Software III, J. Rice ed., Academic Press, New York, 1977, 161-194.
[M85] E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.

Links
[TOPP] The Open Problems Project by Erik D. Demaine - Joseph S. B. Mitchell - Joseph O'Rourke
[OP] Open Problems by Jeff Erickson
[OPDCG] Open Problems on Discrete and Computational Geometry by Jorge Urrutia
[SP] Sample Computational Geometry Projects from McGill University

Animations

syllabus announcements schedule projects references


Alper Üngör (ungor@cise.ufl.edu) Sep 2006