COMPUTATIONAL GEOMETRY INTRODUCTION AND CONVEX HULLS ZAHRA NILFOROUSHAN ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT, FACULTY OF ENGINEERING, KHARAZMI UNIVERSITY, FALL 2016

COMPUTATIONAL GEOMETRY 8th Winter School On CG.,Feb , Study of algorithms, data structures and computational complexity of computational problems in geometry. 2

LANDSCAPE OF CG Applications: Computer Graphics Robotics Vision GIS CAD / manufacturing VLSI (chip layout) Pattern Recognition Optimization Transportation Architecture Statistics Molecular modelling... Algorithmic Tools: general incremental divide-&-conquer space sweep topological sweep prune and search random sampling locus approach multidimensional search dynamization... Data Structures: general interval trees range trees segment trees priority search trees K-d trees fractional cascading persistent D.S.... 8th Winter School On CG.,Feb , Implementation Issues : Degeneracy (symbolic perturbation) Robustness (inexact arithmetic)...

Polygon Triangulation (Guarding an Art Gallery) Linear Programming (Manufacturing with Molds) CG OVERVIEW 8th Winter School On CG.,Feb ,

Orthogonal Range Searching (Querying a Database) Point Location (Knowing Where You Are) CG OVERVIEW 8th Winter School On CG.,Feb ,

Voronoi Diagrams (The Post Office Problem) Arrangements and Duality (Super sampling in Ray Tracing) CG OVERVIEW 8th Winter School On CG.,Feb ,

Delaunay Triangulations (Height Interpolation) More Geometric Data Structures (Windowing) CG OVERVIEW 8th Winter School On CG.,Feb ,

Binary Space Partitions (The Painter’s Algorithm) Robot Motion Planning (Getting Where You Want to Be) CG OVERVIEW 8th Winter School On CG.,Feb ,

Quad trees (Non-Uniform Mesh Generation) Visibility Graphs (Finding the Shortest Route) CG OVERVIEW 8th Winter School On CG.,Feb ,

Hidden Surface Removal (Multi-window user systems) VLSI Design (Checking no bugs in design) D A B C E D A B C E F F CG OVERVIEW 8th Winter School On CG.,Feb ,

Digitizing 3-Dimensional Objects (Scan the objects) CG OVERVIEW 8th Winter School On CG.,Feb ,

Bio Geometry (Protein folding) CG OVERVIEW 8th Winter School On CG.,Feb ,

Medial Axis & Skeleton CG OVERVIEW 8th Winter School On CG.,Feb ,

Computational Geometry Algorithms Library C++ library of efficient data structures and algorithms for many computational geometry problems supports round off-free exact computations 8th Winter School On CG.,Feb ,

CG HISTORY 8th Winter School On CG.,Feb ,

(The Canadian Conference on Computational Geometry ) (The European Workshop on Computational Geometry) (Laboratory of Algorithms and Computational Geometry (Room 213), Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran.) USEFUL LINKS 8th Winter School On CG.,Feb ,

CONVEX HULLS BASIC NOTIONS CLASSIC ALGORITHMS APPLICATIONS 8th Winter School On CG.,Feb ,

CONVEXITY 8th Winter School On CG.,Feb ,

CONVEX HULL 8th Winter School On CG.,Feb ,

CONVEX HULL Imagine that the points are nails sticking out of the plane, take an elastic rubber band, hold it around the nails, and let it go. It will snap around the nails, minimizing its length. The area enclosed by the rubber band is the convex hull of P. 8th Winter School On CG.,Feb ,

CONVEX HULL PROBLEM 8th Winter School On CG.,Feb , The output has at least 4 and at most 2n coordinates, so it has size between O(1) and O(n) 21

8th Winter School On CG.,Feb , BRUTE FORCE ALGORITHM 22

8th Winter School On CG.,Feb , BRUTE FORCE ALGORITHM 23

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

INCREMENTAL ALGORITHM 8th Winter School On CG.,Feb ,

GIFT-WRAPPING ALGORITHM 1.Start at lowest point 2.Rotate the line until we hit another point All other points will lie on one side of this line Look for the point that gives you the largest angle with the current line 3.Repeat 4.You’re done when you get back to the starting point 8th Winter School On CG.,Feb ,

Start with the lowest point (anchor point) 8th Winter School On CG.,Feb , GRAHAM SCAN 41

Now, form a closed simple path traversing the points by increasing angle with respect to the anchor point 8th Winter School On CG.,Feb , GRAHAM SCAN 42

GRAHAM SCAN  Remove points at concave angles (one at a time, backtracking one step when any point is removed).  Continue until you get all the way around. 8th Winter School On CG.,Feb ,

Stack S p7p6p5p4p3p2p1p7p6p5p4p3p2p1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 pipi pipi GRAHAM SCAN: A SNAPSHOT 8th Winter School On CG.,Feb ,

DIVIDE & CONQUER Preprocessing: sort the points by x- coordinate Divide the set of points into two sets A and B: A contains the left  n/2  points, B contains the right  n/2  points Recursively compute the convex hull of A Recursively compute the convex hull of B Merge the two convex hulls A B 8th Winter School On CG.,Feb ,

QUICK HULL ALGORITHM Form initial quadrilateral Q, with left, right, top, bottom. Discard points inside Q. 8th Winter School On CG.,Feb ,

QUICK HULL ALGORITHM For an edge ab, find the farthest outside point c. Discard points inside triangle abc. Split remaining points into “outside” points for ac and bc. 8th Winter School On CG.,Feb ,

QUICK HULL ALGORITHM 8th Winter School On CG.,Feb ,

SUMMARY 8th Winter School On CG.,Feb ,

Motion planning Find an optimal route that avoids obstacles for a robot obstacle start end 8th Winter School On CG.,Feb , SOME APPLICATIONS 50

8th Winter School On CG.,Feb , Bounding Box Obtain a closer bounding box in computer graphics Bounding boxes are used to an approximate location of an object in question and as a very simple descriptor of its shape. Convex hull can be a candidate for replacement of bounding boxes in these situations. SOME APPLICATIONS 51

8th Winter School On CG.,Feb , SOME APPLICATIONS Visual Pattern Matching pattern matching is the act of checking a given sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact. Compare two objects using their convex hulls 52

8th Winter School On CG.,Feb , SOME APPLICATIONS Diameter Computation Find the diameter of a set of points, which is the pair of points a maximum distance apart The diameter will always be the distance between two points on the convex hull pjpj pipi CH(S) 53

8th Winter School On CG.,Feb , SOME APPLICATIONS A B A B Linear Separability Separate two set of points by a line A & B are linearly separable  CH(A) & CH(B) have disjoint interiors 54

8th Winter School On CG.,Feb , SOME APPLICATIONS Smallest enclosing circle Minimax Facility Location Problem Determine location X where a facility should be located so as to minimize the distance from X to its furthest customer 55

8th Winter School On CG.,Feb , SOME APPLICATIONS Convex Layers Onion Layers of a set of points Registration and verification stages of two onions layers 56

8th Winter School On CG.,Feb , SOME APPLICATIONS Shape Morph Find “an average” between two objects Feature matching! Smoothly turn a concave shape into a convex one before blending, and then the other way around when the blend is complete. 57

8th Winter School On CG.,Feb , SOME APPLICATIONS 58

8th Winter School On CG.,Feb , SOME APPLICATIONS 59

THANKS FOR YOUR ATTENTION 8th Winter School On CG.,Feb ,