CIS6930: Computational Geometry

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

HW4 is out. It is due by Thu Dec 4 (beginning of the class).
HW3 is out. It is due by Tue Nov 11 (beginning of the class).
HW2 is out. It is due by Thu Oct 23 (beginning of the class).
The class on Oct 2 is cancelled.
The deadline for HW1 is extended to Sep 30.
HW1 is out. It is due by Thu Sep 25 (beginning of the class).
For the implementation problems use the input output file format of the triangulation software Triangle.
The test files for the first problem are r1000.node, r5000.node, Kuzmin10K.node.
The test files for the second problem are puertorico.poly, fishnoeye.poly, florida4.poly.
First class meeting is on Aug 26, Tuesday 10:30am at MCB1108. Here is the course syllabus.

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

Voronoi Game and the Applet
Sweepline Algorithm Animation for Voronoi Diagrams
Convex Hulls by Alejo Hausner [Graham's scan, Jarvis march, and quickhull]
Convex Hull Algorithms by Tim Lambert [incremental, gift-wrapping, divide and conquer, and quickhull]
Sweepline algorithm for Line Segment Intersection by Wei Qian
An Applet on the Art gallery problem
Voronoi Diagram Illustrations
Duality Demo
PTAS_TSP slides
Visibility Applet

syllabus announcements schedule projects references

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


[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.

[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.

[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

COT5520: COMPUTATIONAL GEOMETRY

Announcements

Schedule

References