CIS6930: Computational Geometry

CIS6930: COMPUTATIONAL GEOMETRY
Term: Fall 2004
Time: Tuesday 11:45am-1:40pm, Thursday 12:50pm-1:40pm
Location: Turlington 2336
Office hours: Tuesday 1:45pm-2:45pm, Thursday 1:45pm-2:45pm
Professor: Alper Üngör

syllabus announcements schedule projects references

Announcements

Final Exam Grades are available now.
HW#4[ps, pdf] is out. It is due by Dec 2, Thursday.
Homework1-3 Grades are available now.
HW#3[ps, pdf] is out. It is due by Nov 4, Thursday.
See the new links for more project ideas. Proposals are due next Tuesday Oct 19.
HW#2[ps, pdf] is out. It is due by Oct 14, Thursday.
There is no class and office hour on Sep 21, Tuesday due to conference travel.
Office hour is cancelled on Sep 16, Thursday due to ADG2004 invited talk by Ileana Streinu.
HW#1[ps, pdf] is out. It is due by Sep 28, Tuesday.
All UF classes cancelled on Sep 7, Tuesday (thanks to Hurricane Frances). HW#0 is due by Sep 9, Thursday.
HW#0[ps, pdf] is out. It is due by Sep 7, Tuesday.
First class meeting is on Aug 24, Tuesday 11:45am at Turlington 2336.

Schedule

Date	Lecture Topic	Assignments	Speaker
Aug 24 Tu	Introduction, syllabus, course structure, etc.	HW#0 out [ps, pdf]	A
	CONVEX HULLS ALGORITHMS [BKOS00, Chapter 1]
Aug 26 Th	Convex hulls Orientation test; Degenaracy; Jarvis' march [J73]		A
Aug 31 Tu	Convex hulls Divide & conquer; Graham's scan [G72, A79]; Chan's alg.[C96]		A
	PLANE-SWEEP ALGORITHMS [BKOS00, Chapters 2 and 3 ]
Sep 2 Th	Line segment intersections Plane-sweep [BO79];		A
Sep 7 Tu	Hurricane Frances (all UF classes cancelled)
Sep 9 Th	Doubly-connected Edge list [BKOS00] Half-edge data structure [CGAL]	HW#0 due	A
Sep 14 Tu	Subdivision Overlays Map overlays, intersection, union, difference [BKOS00]	HW#1[ps, pdf] out	A
Sep 16 Th	Art Gallery Problems [BKOS00] Art gallery theorem [C83]		A
Sep 21 Tu	INTERNATIONAL MESHING ROUNTABLE (no class)
Sep 23 Th	Polygon Triangulation Partitioning simple polygons [LP77]; triangulating monotone polygons [GJPT78]		A
	ORTHOGONAL SEARCH [BKOS00, Chapters 5 and 10]
Sep 28 Tu	Geometric data structures; Range search Quad-tree; kd-tree [B75];	HW#1 due	A
Sep 30 Th	Improvements on Range Searching Range tree; fractional cascading [CG86]		A
Oct 5 Tu	Inverse Range Search Segment tree [B77]; interval tree [E83]; priority search tree [M85]		A
Oct 7 Th	Open problems and Project Ideas	HW#2 out [ps, pdf] HW#1 graded	A
	VORONOI DIAGRAMS & DELAUNAY TRIANGULATIONS [BKOS00, Chapters 7, 9, and 14]
Oct 12 Tu	Voronoi diagrams Voronoi diagrams [AK00]; largest empty disk; Fortune's algorithm		A
Oct 14 Th	Delaunay triangulation	HW#2 due	A
Oct 19 Tu	Project proposal presentations	Proposals due
Oct 21 Th	Project proposal presentations
Oct 26 Tu	Optimal triangulations Empty circles, local Delaunayhood [D34], edge-flip [L77], lifting, analysis, maxmin angles	HW#2 graded	A
Oct 28 Th	Randomized incremental algorithm Incremental construction [GKS92]; backward analysis [S93]	HW#3 out [ps, pdf]	A
Nov 2 Tu	Steiner triangulations Steiner triangulation [BE95]; quality measure; quad-trees [BE95]; local feature size		A
Nov 4 Th	Delaunay refinement Circumcenter insertion [R95];	HW#3 due	A
Nov 9 Tu	Surface reconstruction and simplification Crust algorithm; Edge contraction [GH97]; topology preservation [DEGN99];	HW#3 graded	A
Nov 11 Th	VETERANS DAY
	ARRANGEMENTS [BKOS00, Chapter 8]
Nov 16 Tu	Zones Duality; line arrangements; complexity; incremental algorithm; zone theorem [ESS93]		A
Nov 18 Th	Levels and discrepancy Super-sampling for rendering; Half-plane discrepancy[EG89]		A
	OTHER GEOMETRY APPLICATIONS [BKOS00, Chapters 13 and 15]
Nov 23 Tu	Motion Planning Robotics; Configuration Space; Connectedness; Visibility Graphs; Protein Docking	HW#4 out [ps, pdf]	A
Nov 25 Th	THANSGIVING BREAK
Nov 30 Tu	Project Presentations
Dec 2 Th	Project Presentations	HW#4 due Reports due
Dec 7 Tu	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, 2nd edition, 2000. [course textbook]

Surveys

[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.

[B75] 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.

[CG86] B. Chazelle and L. J. Guibas. Fractional cascading. Algorithmica 1:133-162 and 163-191, 1986.

[C83] V. Chvatal. A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B 18:39-41, 1975.

[D34] B. N. Delaunay. Sur la Sphere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793-800, 1934.

[DEGN99] T. K. Dey, H. Edelsbrunner, S. Guha and D. V. Nekhayev. Topology preserving edge contraction. Publications de l'Institut Mathematique (Beograd) (N. S.) 66:23-45, 1999.

[E83] H. Edelsbrunner. A new approach to rectangle intersections. International Journal Computational Mathematics 13:209-219 and 221-229, 1983.

[EG89] H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. Journal Computer and System Sciences 38:165-194, 1989. Corrigendum. Journal Computer and System Sciences 42:249-251, 1991.

[ESS93] H. Edelsbrunner, R. Seidel and M. Sharir. On the zone theorem for hyperplane arrangements. SIAM Journal on Computing 22:418-429, 1993.

[ESU04] D. Eppstein, J. Sullivan and A. Üngör. Computational Geometry: Theory and Applications, 27:237-255, 2004.

[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.

[GH97] M. Garland and P. S. Heckbert. Surface simplification using quadratic error metrics. Computer Graphics, Proc. SIGGRAPH, 209-216, 1997.

[G72] R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132-133, 1972.

[G72] B. Grunbaum. Arrangements and Spreads. American Mathematical Society, Providence, Rhode Island, 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.

[KM58] L. M. Kelly and W. O. J. Moser. On the number of ordinary lines deternined by n points. Canadian Journal Mathematics, 10:210-219, 1958.

[Kir83] D. G. Kirkpatrick. Optimal search in planar subdivisions. SIAM J. Computing, 12:28-35, 1983.

[L77] C. L. Lawson. Software for C1 surface interpolation. Mathematical Software III, J. Rice ed., Academic Press, New York, 1977, 161-194.

[LP77] D. T. Lee and F. P. Preparata. Location of a point in a planar subdivision and its applications. SIAM J. Computing, 6:594-606, 1977.

[M85] E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.

[NP82] J. Nievergelt and F. P. Preparata. Plane-sweep algorithms for intersecting geometric figures. Communications of the ACM, 25:739-747, 1982.

[R95] J. Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms, 18:548-585, 1995.

[S93] R. Seidel. Backward analysis of randomized geometric algorithms. New Trends in Computational Geometry, J. Pach ed., Springer-Verlag, Berlin, 37-67, 1993.

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

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]

syllabus announcements schedule projects references

Alper Üngör (ungor@cise.ufl.edu) August 2004


[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.
[B75]	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.
[CG86]	B. Chazelle and L. J. Guibas. Fractional cascading. Algorithmica 1:133-162 and 163-191, 1986.
[C83]	V. Chvatal. A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B 18:39-41, 1975.
[D34]	B. N. Delaunay. Sur la Sphere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793-800, 1934.
[DEGN99]	T. K. Dey, H. Edelsbrunner, S. Guha and D. V. Nekhayev. Topology preserving edge contraction. Publications de l'Institut Mathematique (Beograd) (N. S.) 66:23-45, 1999.
[E83]	H. Edelsbrunner. A new approach to rectangle intersections. International Journal Computational Mathematics 13:209-219 and 221-229, 1983.
[EG89]	H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. Journal Computer and System Sciences 38:165-194, 1989. Corrigendum. Journal Computer and System Sciences 42:249-251, 1991.
[ESS93]	H. Edelsbrunner, R. Seidel and M. Sharir. On the zone theorem for hyperplane arrangements. SIAM Journal on Computing 22:418-429, 1993.
[ESU04]	D. Eppstein, J. Sullivan and A. Üngör. Computational Geometry: Theory and Applications, 27:237-255, 2004.
[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.
[GH97]	M. Garland and P. S. Heckbert. Surface simplification using quadratic error metrics. Computer Graphics, Proc. SIGGRAPH, 209-216, 1997.
[G72]	R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132-133, 1972.
[G72]	B. Grunbaum. Arrangements and Spreads. American Mathematical Society, Providence, Rhode Island, 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.
[KM58]	L. M. Kelly and W. O. J. Moser. On the number of ordinary lines deternined by n points. Canadian Journal Mathematics, 10:210-219, 1958.
[Kir83]	D. G. Kirkpatrick. Optimal search in planar subdivisions. SIAM J. Computing, 12:28-35, 1983.
[L77]	C. L. Lawson. Software for C1 surface interpolation. Mathematical Software III, J. Rice ed., Academic Press, New York, 1977, 161-194.
[LP77]	D. T. Lee and F. P. Preparata. Location of a point in a planar subdivision and its applications. SIAM J. Computing, 6:594-606, 1977.
[M85]	E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.
[NP82]	J. Nievergelt and F. P. Preparata. Plane-sweep algorithms for intersecting geometric figures. Communications of the ACM, 25:739-747, 1982.
[R95]	J. Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms, 18:548-585, 1995.
[S93]	R. Seidel. Backward analysis of randomized geometric algorithms. New Trends in Computational Geometry, J. Pach ed., Springer-Verlag, Berlin, 37-67, 1993.

[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

CIS6930: COMPUTATIONAL GEOMETRY

Announcements

Schedule

References