Yazd Univ. Computational Geometry (Master Course) Computational Geometry Course Outline Mohammad Farshi Textbook Grading Prerequisties Department of Computer Science, Introduction What is CG? Yazd University Research on CG Journals 1392-1 Conferences 1 / 17
Textbook: Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars, Computational Geometry Algorithms and Yazd Univ. Applications , 3rd Edition, Springer-Verlag Berlin Heidelberg, 2008. Computational Geometry Course Outline Textbook Grading Prerequisties Introduction What is CG? Research on CG Journals Conferences 2 / 17
Grading: Yazd Univ. Computational Grading: Geometry Midterm exam: 6 Course Outline Final exam: 7 Textbook Grading Presentation: 3 Prerequisties Introduction Homework: 4 What is CG? Important: For passing the course, one should get Research on CG Journals at least 8 from midterm + final exams. Conferences 3 / 17
Online Resources: Yazd Univ. CG on Web: Course Webpage: Computational Geometry cs.yazd.ac.ir/farshi/Teaching/CG3921/CG.html (Jeff Erickson) Course Outline compgeom.cs.uiuc.edu/ ∼ jeffe/compgeom/courses.html Textbook Grading Prerequisties (David Eppstein) www.ics.uci.edu/ ∼ eppstein/geom.html Introduction (Godfried Toussaint) What is CG? Research on CG www-cgrl.cs.mcgill.ca/ ∼ godfried/teaching/cg-web.html Journals Conferences Computational Geometry Pages www.computational-geometry.org and much more ... 4 / 17
Prerequisites: Yazd Univ. What you need to know: Computational Geometry Basic Algorithms and Algorithm Analysis: O , Θ notations, sorting, searching. Course Outline Textbook Basic Data Structures: Priority Queue (Heap), Binary Grading Prerequisties Search Tree, ... and their analysis. Introduction Basic Probability theory: Expected value, ... What is CG? Research on CG Not needed: Application of CG, programming, Journals Conferences knowledge in Geometry. 5 / 17
Prerequisites: Yazd Univ. Computational Geometry Course Outline Textbook Grading Prerequisties Introduction What is CG? Research on CG Journals Conferences 6 / 17
Yazd Univ. Computational Geometry Introduction Course Outline Textbook Grading Prerequisties Introduction What is CG? Research on CG Journals Conferences 7 / 17
What is CG? Yazd Univ. Computational Geometry is a branch of computer science devoted to the study Computational Geometry of algorithms which can be stated in terms of geometry. Course Outline is a subfield of the Design and Analysis of Algorithms Textbook Grading Prerequisties deals with efficient data structures and algorithms for Introduction geometric problems What is CG? Research on CG is only about 30 years old Journals Conferences started out by developing solid theoretical foundations, but became more and more applied over the last years 8 / 17
What is CG? Yazd Univ. Applications Computational Computer graphics, Geometry Computer-aided design and manufacturing (CAD/CAM), Course Outline Textbook Robotics (motion planning and visibility problems), Grading Prerequisties Geographic Information Systems (GIS) (geometrical Introduction What is CG? location and search, route planning), Research on CG Journals Integrated Circuit design (IC geometry design and Conferences verification), and so on. 9 / 17
What is CG? The main branches of computational geometry are: Combinatorial computational geometry , also Yazd Univ. called algorithmic geometry , which deals with Computational geometric objects as discrete entities. A Geometry groundlaying book in the subject by Preparata and Shamos dates the first use of the term Course Outline "computational geometry" in this sense by 1975. Textbook Grading Prerequisties Numerical computational geometry , also called Introduction machine geometry , computer-aided geometric What is CG? design (CAGD), or geometric modeling, which deals Research on CG Journals primarily with representing real-world objects in Conferences forms suitable for computer computations in CAD/CAM systems. This branch may be seen as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD. 10 / 17
What is CG? Combinatorial computational geometry The primary goal is to develop efficient algorithms Yazd Univ. and data structures for solving problems stated in Computational terms of basic geometrical objects: points, line Geometry segments, polygons, polyhedra, etc. Example: The closest pair problem: Given n points in Course Outline Textbook the plane, find the two with the smallest distance Grading Prerequisties from each other. The brute-force algorithm takes Introduction O ( n 2 ) time. A classic result: an algorithm that takes What is CG? O ( n log n ) time. Also randomized algorithms that Research on CG Journals take O ( n ) expected time, as well as a deterministic Conferences algorithm that takes O ( n log log n ) time. Computational geometry focuses heavily on computational complexity since the algorithms are meant to be used on very large data sets containing tens or hundreds of millions of points. 11 / 17
What is CG? Combinatorial computational geometry The primary goal is to develop efficient algorithms Yazd Univ. and data structures for solving problems stated in Computational terms of basic geometrical objects: points, line Geometry segments, polygons, polyhedra, etc. Example: The closest pair problem: Given n points in Course Outline Textbook the plane, find the two with the smallest distance Grading Prerequisties from each other. The brute-force algorithm takes Introduction O ( n 2 ) time. A classic result: an algorithm that takes What is CG? O ( n log n ) time. Also randomized algorithms that Research on CG Journals take O ( n ) expected time, as well as a deterministic Conferences algorithm that takes O ( n log log n ) time. Computational geometry focuses heavily on computational complexity since the algorithms are meant to be used on very large data sets containing tens or hundreds of millions of points. 11 / 17
What is CG? Combinatorial computational geometry The primary goal is to develop efficient algorithms Yazd Univ. and data structures for solving problems stated in Computational terms of basic geometrical objects: points, line Geometry segments, polygons, polyhedra, etc. Example: The closest pair problem: Given n points in Course Outline Textbook the plane, find the two with the smallest distance Grading Prerequisties from each other. The brute-force algorithm takes Introduction O ( n 2 ) time. A classic result: an algorithm that takes What is CG? O ( n log n ) time. Also randomized algorithms that Research on CG Journals take O ( n ) expected time, as well as a deterministic Conferences algorithm that takes O ( n log log n ) time. Computational geometry focuses heavily on computational complexity since the algorithms are meant to be used on very large data sets containing tens or hundreds of millions of points. 11 / 17
What is CG? Problem classes Yazd Univ. Static problems Computational Convex hull: Given a set of points, find the smallest 1 Geometry convex polyhedron/polygon containing all the points. Line segment intersection: Find the intersections 2 Course Outline between a given set of line segments. Textbook Grading Voronoi diagram: Given a set of points, partition the 3 Prerequisties space according to which point is closest. Introduction Closest pair of points: Given a set of points, find the 4 What is CG? two with the smallest distance from each other. Research on CG Journals Euclidean shortest path: Connect two points in a 5 Conferences Euclidean space (with polyhedral obstacles) by a shortest path. Polygon triangulation: Given a polygon, partition its 6 interior into triangles 12 / 17
What is CG? Problem classes Yazd Univ. Geometric query problems Computational Geometry Range searching: Preprocess a set of points, in 1 order to efficiently count the number of points inside a query region. Course Outline Textbook Point location: Given a partitioning of the space into 2 Grading cells, produce a data structure that efficiently tells in Prerequisties Introduction which cell a query point is located. What is CG? Nearest neighbor: Preprocess a set of points, in 3 Research on CG order to efficiently find which point is closest to a Journals Conferences query point. Ray tracing: Given a set of objects in space, produce 4 a data structure that efficiently tells which object a query ray intersects first. 13 / 17
What is CG? Yazd Univ. Computational Geometry Problem classes Dynamic problems Course Outline Textbook Variations Grading Prerequisties Point in polygon: Decide whether a point is inside or Introduction outside a given polygon. What is CG? Research on CG Journals Conferences 14 / 17
What is CG? Yazd Univ. Computational Geometry Problem classes Dynamic problems Course Outline Textbook Variations Grading Prerequisties Point in polygon: Decide whether a point is inside or Introduction outside a given polygon. What is CG? Research on CG Journals Conferences 14 / 17
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