3. How does the rubber band look when it snaps tight?
4. We represent the convex hull as the sequence of points on the convex hull polygon, in counter-clockwise order.By Ravikirankalal
5. Defination Informal definition: Convex hull of a set of points in plane is the shape taken by a rubber band stretched around the nails pounded into the plane at each point Convex hull of a set of points S is the set of all convex combinations of points of S Convex hull of S is denoted by convS, sometimes the notation (S) is also used By Ravikirankalal
6. Extreme Points The extreme points of a set S of points in the plane are the vertices of the convex hull at which the interior angle is less than π Also a point is extreme iff there exists a line through that point that other wise does not touch the convex hull By Ravikirankalal
7. Extreme Edges for each i do for each j ≠ i do for each k ≠ i ≠ j do if pkis not left or on (pi ,pj) then (pi ,pj)isnot extreme There are three nested loops in this algorithm Hence the order is O(n3) For each of the n2 pair of points, the test for extremeness costs n The vertices that are extreme can now be found By Ravikirankalal
8. Applications Computer Visualization, Ray Tracing, Video Games. Geographical Information Systems (GIS) - Computing Accessibility Maps Visual Pattern Matching - Detecting Car License Plates Path Finding - Embedded AI of Mars mission Rovers Replacement of Bounding Boxes By Ravikirankalal
24. Finding the lower tangent in O(n) time 9 3 a = rightmost point of A b = leftmost point of B while T=ab not lower tangent to both convex hulls of A and B do{ while T not lower tangent to convex hull of A do{ a=a-1 } while T not lower tangent to convex hull of B do{ b=b+1 } } 4=b 4 2 3 5 5 a=2 6 1 7 1 0 0 A B By Ravikirankalal
25.
26. Divide the set of points into two sets A and B:
31. Merge the two convex hullsO(1) T(n/2) T(n/2) O(n) By Ravikirankalal
32.
33. Quickhull QuickHull uses a divide and conquer approach similar to the QuickSort algorithm. Benchmarks showed it is quite fast in most average cases. Recursive nature allows a fast and yet clean implementation. By Ravikirankalal
34. Initial input The initial input to the algorithm is an arbitrary set of points. By Ravikirankalal
35. First two points on the convex hull Starting with the given set of points the first operation done is the calculation of the two maximal points on the horizontal axis. By Ravikirankalal
36. Recursively divide Next the line formed by these two points is used to divide the set into two different parts. Everything left from this line is considered one part, everything right of it is considered another one. Both of these parts are processed recursively. By Ravikirankalal
37. Max distance search To determine the next point on the convex hull a search for the point with the greatest distance from the dividing line is done. This point, together with the line start and end point forms a triangle. By Ravikirankalal
38. Point exclusion All points inside this triangle can not be part of the convex hull polygon, as they are obviously lying in the convex hull of the three selected points. Therefore these points can be ignored for every further processing step. By Ravikirankalal
39. Recursively divide Having this in mind the recursive processing can take place again. Everything right of the triangle is used as one subset, everything left of it as another one. By Ravikirankalal
40. Abort condition At some point the recursively processed point subset does only contain the start and end point of the dividing line. If this is case this line has to be a segment of the searched hull polygon and the recursion can come to an end. By Ravikirankalal
41. Running time The running time of Quickhull, as with QuickSort, depends on how evenly the points are split at each stage. T(n) = 1 if n = 1 T(n1) + T (n2) otherwise where n1+n2<=n If we assume that the points are ``evenly'' distributed, the running time will solve to O(n log n). if the splits are not balanced, then the running time can easily increase to O(n^2). By Ravikirankalal
42. Think of wrapping a gift. Put the paper in contact with the gift and continue to wrap around from one surface to the next until you get all the way around. By Ravikirankalal
46. The point set is wrapped by a string that bends the that bends with minimum angle from previous to next hull edge By Ravikirankalal
47. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
48. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
49. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
50. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
51. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
52. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
53. Jarvis March - Example p10 p6 p9 p5 p7 p12 p3 p4 p11 p1 p8 p2 p0 By Ravikirankalal
54. Running time we can find the point q in O(n) time. After repeating this h times, we will return back to the starting point and we are done. Thus, the overall running time is O(nh). Worst case efficiency will be n2. By Ravikirankalal
![Convex Hull 1 4 5 3 6 2 1 0 ,[object Object]](https://image.slidesharecdn.com/adaconvex-110501124117-phpapp02/85/convex-hull-1-320.jpg)
