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Planar Convex Hull 2013 / 5 / 9 Group 4 Sungheon Park Jeongho Son CS504 Presentation [CS504 Presentation]

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Contents CS504 Presentation Definition of convex hull Bruteforce algorithm Graham’s scan Divide and conquer Quickhull Jarvis’ method

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What is convex hull? CS504 Presentation Let S be a set of points in the plane. Intuition: Imagine the points of S as being pegs; the convex hull of S is the shape of a rubber-band stretched around the pegs.

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What is convex hull? CS504 Presentation

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Applications of convex hull CS504 Presentation computer visualization, ray tracing path finding Geographical Information Systems (GIS) Visual pattern matching

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Orientation test CS504 Presentation

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Graham’s Scan CS504 Presentation

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Graham’s Scan CS504 Presentation

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Graham’s Scan CS504 Presentation

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Original Graham’s scan CS504 Presentation Initially, points are sorted in increasing angular value If the point is not convex (concave), it removes the current point from the perimeter list

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Divide-and-Conquer CS504 Presentation

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Divide-and-Conquer CS504 Presentation

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Quickhull CS504 Presentation

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Quickhull CS504 Presentation

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Quickhull CS504 Presentation

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Jarvis’s March CS504 Presentation Build the hull using “gift wrapping” process

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Jarvis’s March CS504 Presentation

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Jarvis’s March CS504 Presentation

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Applet CS504 Presentation Java applet –http://www.cse.unsw.edu.au/~lambert/java/3d/ hull.htmlhttp://www.cse.unsw.edu.au/~lambert/java/3d/ hull.html –http://www.cs.princeton.edu/courses/archive/s pr09/cos226/demo/ah/JarvisMarch.htmlhttp://www.cs.princeton.edu/courses/archive/s pr09/cos226/demo/ah/JarvisMarch.html

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CS504 Presentation Chan’s algorithm Jeongho Son

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Planar Convex Hull CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation Stage 1 n = 32 Set m = 8

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Chan’s Algorithm CS504 Presentation Stage 1 n = 32 Set m = 8 r = 4

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation Stage 2 (After Stage 1) m = 8 r = 4

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Chan’s Algorithm CS504 Presentation Stage 2 Using Graham’s Scan

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Chan’s Algorithm CS504 Presentation Stage3 : Jarvis’s March How to merge these r hulls into a single hull? IDEA : treat each hull as a “fat point” and run Jarvis’s March! # of iteration is at most m –to guarantee the time complexity O(nlogh)

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Chan’s Algorithm CS504 Presentation (-inf,0) -> lowest pt lowest pt

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Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1

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Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1 2

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Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1 2 3

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation FAIL EXAMPLE – too small value m m = 4 4 iteration

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Chan’s Algorithm CS504 Presentation In 4(a), how to find such points?

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Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1

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Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in a hull

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Chan’s Algorithm CS504 Presentation 1 2 3 4 5

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Chan’s Algorithm CS504 Presentation

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Lower bound for convex hull CS504 Presentation n points in the x-axis

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Lower bound for convex hull CS504 Presentation lifting up to 2D plane

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Lower bound for convex hull CS504 Presentation lower convex hull

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Quiz CS504 Presentation

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Quiz CS504 Presentation

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Summary CS504 Presentation Finding the convex hull of a set of points is an important problem that is often part of a larger problem Many different algorithms –Graham’s Scan –Quickhull –Divide-and-Conquer –Jarvis’s March –Chan’s algorithm

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Q&A CS504 Presentation Any question?

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