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

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Presentation transcript:

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

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

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.

What is convex hull? CS504 Presentation

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

Orientation test CS504 Presentation

Graham’s Scan CS504 Presentation

Graham’s Scan CS504 Presentation

Graham’s Scan CS504 Presentation

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

Divide-and-Conquer CS504 Presentation

Divide-and-Conquer CS504 Presentation

Quickhull CS504 Presentation

Quickhull CS504 Presentation

Quickhull CS504 Presentation

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

Jarvis’s March CS504 Presentation

Jarvis’s March CS504 Presentation

Applet CS504 Presentation Java applet – hull.htmlhttp:// hull.html – pr09/cos226/demo/ah/JarvisMarch.htmlhttp:// pr09/cos226/demo/ah/JarvisMarch.html

CS504 Presentation Chan’s algorithm Jeongho Son

Planar Convex Hull CS504 Presentation

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

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

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

Chan’s Algorithm CS504 Presentation

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

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

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)

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

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

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

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

Chan’s Algorithm CS504 Presentation

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

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

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

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

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

Chan’s Algorithm CS504 Presentation

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

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

Lower bound for convex hull CS504 Presentation lower convex hull

Quiz CS504 Presentation

Quiz CS504 Presentation

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

Q&A CS504 Presentation Any question?