Introduction to Combinatorial and Computational Geometry

Introduction to Combinatorial and Computational Geometry 2024/2025

Zaklady kombinatoricke a vypocetni geometrie 2024/2025

Lecture: Tuesday at 14:00 in S3. The lectures are in English this year, but you can also ask questions in Czech or Slovak.
Prednaska: V utery od 14:00 v S3. Prednaska je letos v anglictine, muzete se ale ptat i cesky nebo slovensky.

Exercise sessions: after the lecture at 15:40 in S8 according to the schedule on the exercise webpage.
Cviceni: po prednasce 15:40 v S8 podle rozvrhu na strance cviceni.

Extent of teaching: WINTER semester 2/2 (Exam and Exercise credit). Entry in SIS, syllabus
Rozsah vyuky: ZIMNI semestr 2/2 Z, Zk. Stranka v SISu, sylabus

Annotation
Discrete geometry investigates combinatorial properties of geometric objects such as finite point sets or convex sets in Euclidean spaces. Computational geometry considers the design of efficient algorithms for computing with geometric configurations, and discrete geometry serves as its mathematical foundation. Part I of the course is a concise introduction. The lecture will be very similar to the last year. The contents of Part II varies among the years, each year covering a few selected topics in more depth.

Anotace
Vypocetni geometrie se zabyva navrhem efektivnich algoritmu pro geometricke problemy v rovine i ve vicedimenzionalnim prostoru (napr. je-li dano n bodu v rovine, jak co nejefektivneji najit dvojici bodu s nejmensi vzdalenosti). Takove problemy jsou motivovany aplikacemi v pocitacove grafice, prostorovem modelovani (napr. molekul, budov, soucastek), geografickych informacnich systemech a pod. Pri analyze takovych algoritmu se potrebuje kombinatoricka geometrie, studujici kombinatoricke vlastnosti geometrickych konfiguraci, konvexnich mnozin a pod. Vysledky jsou dulezite i z ciste matematickeho hlediska, napr. v teorii cisel. V teto uvodni prednasce se probiraji zakladni pojmy a metody, s durazem na matematicky zaklad (jine mozne podani by bylo z vice "informatickeho" hlediska, s durazem na datove struktury, implementaci algoritmu apod.). O naplni prednasky si muzete udelat lepsi predstavu podle latky probirane v minulych letech.

Literature

[M1] Jiri Matousek's lecture notes Introduction to Discrete Geometry, ITI Series 2003-150, available in the Computer Science library of MFF UK. Here is a postscript file with two pages of text on one A4 page, intended for printing. Here is a PDF file for readers and tablets.
The lecture notes are, in fact, a selection of topics from the book J. Matousek: Lectures on Discrete Geometry.
A text for the part about incremental geometric algorithms is available here (12 pages).
Chan's algorithm for the construction of the convex hull: doi:10.1007/BF02712873
A standard introductory text about computational geometry: M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf: Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 1997. The computer science library of MFF UK has a few copies.
J. Pach, P. Agarwal: Combinatorial Geometry, Cambridge University Press 1995

Literatura

Skripta Jiriho Matouska Introduction to Discrete Geometry vysla v ITI Seriich (preprintova rada Institutu teoreticke informatiky MFF UK) pod cislem 2003-150 a je k dispozici v informaticke knihovne MFF UK. Tady jsou take jako postscriptovy soubor, v nemz jsou pro usporu mista 2 male stranky na jedne strance A4. Je urcen hlavne pro dvoustranny tisk a ma vlevo okraj na svazani. A zde PDF soubor mysleny pro ctecky a tablety.
Predchozi text je ve skutecnosti vyber casti obvykle probiranych v prednasce z knihy J. Matousek: Lectures on Discrete Geometry.
Text k casti o inkrementalnich geometrickych algoritmech je k dispozici zde (12 str).
Chanuv algoritmus pro konstrukci konvexniho obalu: doi:10.1007/BF02712873
Kapitola "Geometricke algoritmy" ve skriptech Martina Marese
Standardni uvodni text o vypocetni geometrii je M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf: Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin, 1997. Informaticke oddeleni knihovny ma nekolik exemplaru.
J. Pach, P. Agarwal: Combinatorial Geometry, Cambridge University Press 1995

Topics covered:

1.10. (JK)

Introductory information
Basic notions: d-dimensional Euclidean space R^d (the set of d-tuples of real numbers), hyperplane, closed halfspace
Afinne subspace, afinne hull, afinne combination, afinne (in)dependence
Convex set, convex hull, it is the set of all convex combinations (sketch of proof)
Caratheodory's theorem (only statement, the proof is an exercise)
Hyperplane separation theorem (statement)

8.10. (JK)

Hyperplane separation theorem: sketch of proof for the case of two compact sets, idea for the case of bounded sets
Radon's theorem
Helly's theorem
Infinite version of Helly's theorem
Existence of a centerpoint
The ham sandwich theorem (only statement)

15.10. (JK)

Center transversal theorem (only statement)
Drawing of a graph, crossing number of a graph
Crossing lemma
Incidences between points and lines, examples

22.10. (JK)

Lower bound on the number of incidences between n points and n lines
Szemeredi–Trotter theorem about maximum number of incidences between points and lines
Upper bound O(n^{4/3}) on the number of unit distances (only statement)
Lower bound n^{1+c/(log log n)} on the number of unit distances (description of the construction, no proof)
Minkowski's theorem for the integer lattice
Application of Minkowski's theorem: looking out of a regular forest

...

3.12. (JK)

Voronoi diagram, regions (cells), faces; regions are convex polyhedra
Applications of Voronoi diagrams
Delaunay triangulation (two definitions), properties without proofs
Voronoi diagrams in R^d are projections of polyhedra in R^{d+1} (proof using hyperplanes tangent to the unit paraboloid)
An asymptotic upper bound on the number of faces of all regions of the Voronoi diagram
Power diagrams, their properties without proofs

10.12. (JK)

Arrangement of hyperplanes in R^d, cells, faces
Simple arrangement of hyperplanes, the exact number of cells of a simple arrangement of n hyperplanes (proof by double induction on n, d)
Second proof for the exact number of cells of a simple arrangement of n hyperplanes (by induction on d)
Arrangement of arbitrary subsets of R^d
Arrangement of zero sets of polynomials in d variables, sign patterns of a collection of polynomials, Milnor–Thom theorem, asymptotic and more precise version (no proof)

Temata prednasek: