Computational Geometry Piyush Kumar (Lecture 3: Convexity and Convex hulls) Welcome to CIS5930

Convexity A set S is convex if for any pair of points p,q  S we have pq  S. p q non-convex q p convex

Convex Hulls : Equivalent definitions The intersection of all covex sets that contains P The intersection of all halfspaces that contains P. The union of all triangles determined by points in P. All convex combinations of points in P. P here is a set of input points

Convex Hulls Applications  Collision Detection  Fitting shapes  Shape Approximation  NN-Searching  Useful as a preprocessing step to many other algorithms in computational geometry.  The most ubiquitous structure in computational geometry.

Convex hulls p0p0 p1p1 p2p2 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 p 11 p 12 Extreme point Int angle < pi Extreme edge Supports the point set

Convex hull : Representation We will represent the convex hull by an enumeration of the vertices of the CH(P) in counterclockwise order. Naïve Algorithm to compute convex hulls can be implemented in O(n 3 ) in the plane (How?) Anyone with an O(n 2 ) algorithm?

Convex hull has a lower bound equivalent to sorting has many similar algorithms to sorting. We will see today  Graham Scan  Incremental (one point at a time)  D&C  Qhull ( similar to Quick Sort)  Jarvis March  Chan’s Algorithm

Assignments for next week Notes of Dr. Mount: Upto Page 20 D&C notes Assignment2.cpp and ch_2.cpp  Will talk more about it towards the end of class.

Today (Jan 16 th ) Chan’s Algorithm Line Segment intersection Homeworks / Projects

Line Segment Intersection Applications  VLSI (Intel uses it a lot)  Map Overlay  Clipping in Graphics  CSG Problem : Given a set of line segments in the plane, compute all the intersection point.

Line Segment Intersection Lower Bound from EU EU : Given a list of n numbers, are all these numbers unique? [Y / N]?  Lower bound is Ω(nlogn)  How do we use this fact to prove a Ω(nlogn) on Line segment intersection problem?  Does this imply a lower bound of Ω(nlogn+k)? Tell me a naïve algorithm to solve this problem.

Line Segment intersection Naïve O(n^2) Bentley Ottman O((n+k)log n) Edelsbrunner Chazeele 92  O(nlogn +k) : Supercomplicated O(nlogn) space Clarkson and Shor  O(nlogn +k) Randmized O(n) space Balaban : Deterministic O(nlogn + k) in O(n space. Solved a long open problem.

Segment Intersection How do we intersect two segments? How do we implement such a primitive? CG FAQ 1.3 Any special cases?

Today ( Jan 18 th ) Line Segment intersection Algorithm Project discussion Polygons

Line Segment intersection Sweep line paradigm  Main idea is to sweep the entire plane with a line and compute what we want to, as we sweep past the input. Event scheduling and updates  Carefully schedule the computation so that it does not take too much time to compute the output.

Line Segment Intersection A Sorted sequence data structure  Insert  Delete  Successor/Predecessor  All in O(log n) X-structure (or the event queue) Y-structure (or the sweep line)

Plane Sweep Paradigm Initialization:  Add all segment endpoints to the X-structure or event queue (O(n log n)).  Sweep line status is empty. Algorithm proceeds by inserting and deleting discrete events from the queue until it is empty.

Useful lemma Given s1,s2 intersecting in p, there is a placement of the sweepline prior to this event such that s1,s2 are adjacent along the sweepline. Just before an intersection occurs, the two relevant segments are adjacent to each other in the sweep line status.

Plane Sweep Event A: Segment left endpoint  Insert segment in sweep line or the Y-structure.  Test for intersection to the right of the sweep line with the segments immediately above and below it. Insert intersection points (if found) into X-structure or event queue. Complexity: ? Worst case?

Plane Sweep – Algorithm Event B: Segment right endpoint  Delete segment from sweep line status.  Test for intersection to the right of the sweep line between the segments immediately above and below. (can you do any optimization for this case? ) Insert point (if found) into event queue. Complexity: ?

Plane Sweep – Algorithm Event C: Intersection point  Report the point.  Swap the two line relevant segments in the sweep line status.  For the new upper segment – test it against its predecessor for an intersection. Insert point (if found) into event queue.  Similar for new lower segment (with successor). Complexity: O(klogn)

The Simplified Algorithm Construct the X-structure scan thru the X-structure (or the event queue) from left to right processing:  Segment Left endpoint  Segment right endpoint  Intersection points

Polygons Definition of a simple polygon Point containment in simple polygon Area of a simple polygon Triangulation of a simple polygon Convex hull of a simple polygon Fast preprocessing of a convex polygon to do in/out queries.