Computational Geometry Piyush Kumar (Lecture 3: Convexity and Convex hulls) Welcome to CIS5930.
-
Upload
adelia-patterson -
Category
Documents
-
view
227 -
download
0
Transcript of Computational Geometry Piyush Kumar (Lecture 3: Convexity and Convex hulls) Welcome to CIS5930.
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
p0
p1p2
p4
p5
p6
p7
p8
p9
p11
p12
Extreme pointInt angle < pi
Extreme edgeSupports 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(n3) in the plane (How?) Anyone with an O(n2) 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: Lec 1-4, 6Assignment2.cpp and ch_2.cpp Will talk more about it towards the
end of class.
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 EUEU : 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 sweep O((n+k)log n): 1979.Edelsbrunner Chazelle 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?
Intersection point?
Solve for s, t. Beware of degenerate cases.
Compute_intersection_point primitive is one of the most time consuming parts of segment intersection algorithms.For speed: Floating point filters on rational arithmetic is used.
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 ParadigmInitialization: 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.
Pretend these never happened!
No line segment is verticalIf two segments intersect, then they intersect in a single point (that is, they are not collinear).No three line segments intersect in a common point.
Useful lemma Given si,sj intersecting in p, there is a placement of the sweepline prior to this event such that si,sj 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 – AlgorithmEvent 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
Point containment in simple polygon Area of a simple polygon Convex hull of a simple polygon Triangulation of a simple polygon Fast preprocessing of a convex polygon to do in/out queries.
Convex hull of poygons
Melkman’s Algorithm.Uses a deque: head-tail linked list, for which elements can be added to or removed from the front (head) or back (tail). Considered to be the best CH algorithm for simple polygons.
Partition plane in 4 colors
Points in the deque are circled in blue. The point N is special point stored separately. F and B are front and back vertices of deque.
Partition plane in 4 colors
Next point on the polygon can only lie in the colored regions.Invariant : N followed by the vertices in the deque (front to back) form a convex polygon.
Partition plane in 4 colors
If p falls into region I, push N onto the front of the deque, then overwrite N by p. To restore the invariant, backtrack/delete vertices from the front of the deque until a convex turn is encountered.
Partition plane in 4 colors
If p falls into region II, push N onto the back of the deque, then overwrite N by p. Restore the invariant by backtracking/deleting vertices from the back of the deque until a convex turn is encountered.
Partition plane in 4 colors
If p falls into region III, simply overwrite N by p, and restore the invariants as in both cases I and II.
Partition plane in 4 colors
If p falls into region IV , ignore this and all following vertices until one emerges into another region.