Announcements Computational Geometry for Thursday, March 2 the - PDF document

Announcements Computational Geometry for � Thursday, March 2 the Tablet PC � ABET Program review � Give important feedback on the CSE 481b program � Free Donuts! Lecture 17 Overview Tablet Geometry � Computational � Basic structure – Geometry on the Stroke: sequence of Tablet PC points � Geometric primitives � Himetric coordinates � Intersections � Sampled 150 times per second � Polygons � Convexity � Coordinates stored in � Voronoi Diagram an array Points Computational Geometry Basic geometry p 2 � Algorithms for geometric computation � Point p � p � Numerical issues with coordinates p 1 � Line segment � Importance of degenerate cases � (p 1 , p 2 ) � Distance � Examples of degenerate cases p 3 � Dist(p 1 , p 2 ) p 2 � Basic Test � Three lines intersecting at a point � LeftOf(p 1 , p 2 , p 3 ) � Segments overlapping � CCW(p 1 , p 2 , p 3 ) p 1 � Three points co-linear 1

Counter Clockwise Test Segment intersection � CCW(p 1 , p 2 , p 3 ) � Find intersection of (p 1 ,p 2 ) and (p 3 ,p 4 ) � Q = α p 1 + (1- α )p 2 public static bool CcwTest(Point p1, Point p2, Point p3){ � Q = β p 3 + (1- β )p 4 int q1 = (p1.Y - p2.Y)*(p3.X - p1.X); � Solve for α , β int q2 = (p2.X - p1.X)*(p3.Y - p1.Y); return q1 + q2 < 0; � Two equations, two unknowns } p 2 � Intersect if 0 < α < 1 and 0 < β < 1 � Derived points � In general, try to avoid computing derived points in geometric algorithms p 1 p 3 Making intersection test more Problem efficient � Determine if two line segments (p 1 , p 2 ) and (p 3 ,p 4 ) � Take care of easy cases intersect just using CCW Tests using coordinate comparisons � Only use CCW tests if bounding boxes intersect Student Submission Segment intersection Computing intersections algorithm � How many self intersections can a � Run time O(nlog n + Klog n) for finding K intersections � Sweepline Algorithm single stroke with n points have? 3 6 5 1 2 7 4 Student Submission 2

Sweepline Algorithm Sweepline example 3 � Event queue � Start Segment 6 � Start Segment (S 2 ) � Insert in list � End Segment (E 2 ) � Check above and below 5 1 for intersection � Intersection (I 2,4 ) � End Segment � Move sweepline to next 2 7 event � Remove from list 4 � Check newly adjacent � Maintain vertical order segments for intersection of segments as line � Intersection sweeps across � Reorder segments � Check above and below for intersection Activity: Identify when each of the intersections is detected Polygons � Sequence of points representing a D closed path A F � Simple polygon – closed path with no C self intersections E B Student Submission Describe an algorithm to test if a point q is in a polygon P Polygon inclusion test � Is the point q inside the Polygon P? 3

Convexity Convex polygons � Defn: Set S is convex if whenever p 1 , � P = {p 0 , p 1 , . . . p n-1 } and p 2 are in S, the segment (p 1 , p 2 ) is � P is convex if contained in S � CCW(p i , p i+1 , p i+2 ) for all i � Interpret subscripts mod n � Also holds for CW (depending on how points are ordered) Problem: Test if a point q is inside a Convex hull convex polygon P using CCW Tests � Smallest enclosing convex figure � Rubber band “algorithm” Student Submission Compute the Convex Hull Algorithms � Convex hull algorithms: O(nlog n) � Related to sorting � Insertion algorithm � Gift Wrapping (Jarvis’s march) � Divide and Conquer � Graham Scan Student Submission 4

Convex Hull Algorithms Gift wrapping Divide and Conquer Polar sort the red points around q (Start with p, number the points in CCW Graham Scan order) � Polar sort the points around a point inside the hull � Scan points in CCW order � Discard any point that causes a CW turn p q � If CCW advance � If !CCW, discard current point and back up Student Submission Graham Scan Algorithm GS Example to walk through � Stack of vertices x � Possible hull vertices y � z – next vertex � y – top of stack z � x – next on stack � If CCW(x, y, z) � Push(z) x � If (! CCW(x, y, z)) z y � Pop stack 5

Student submission: Give order Voronoi Diagram vertices are discarded in the scan � Given a set of points: subdivide space into the regions closest to each point p Student Submission Algorithms for Computing the Compute the Voronoi Diagram Voronoi Student Submission 6