cyberpython / computationalgeometry

Java implementations of Graham's Scan, Jarvis' March and the Bentley-Ottmann line-segments intersection algorithm

ComputationalGeometry - Computational Geometry Algorithms in Java

Copyright (C) 2010 Georgios Migdos

This software was developed as an assignment for the "Special Topics On Algorithms" course at the Athens Univesity Of Economics And Business (Spring 2010).

Available under the MIT License

Screenshots

![Screenshot1][screenshot1] [screenshot1]: ComputationalGeometry/raw/master/screenshots/screenshot_th.png "Screenshot"

Todo

Javadoc

Implemented algorithms

###Segment Intersections:

• Bentley-Ottmann

###Convex Hull:

• Graham's Scan
• Jarvis' March (Gift Wrapping)

API

####aueb.geom.algorithms.Intersections public static List bentleyOttmann(List segments, List eventLog);

####aueb.geom.algorithms.ConvexHull public static List grahamsScan(List points, List events); public static List jarvisMarch(List points, List events);

The last parameter of all 3 methods is a List that is filled with events(algorithm steps) as the method executes. MUST NOT be null.

The Bentley-Ottmann implementation uses an ArrayList instead of a self-balancing binary search tree (e.g. a Red-Black tree) so its complexity is O((k*n+n^2)*log(n)) instead of O((k+n)*log(n)).

Algorithms

###Bentley-Ottmann: ------------------------------------ BEGIN ------------------------------------- 1) Initialize a priority queue Q of potential future events, each associated with a point in the plane and prioritized by the x-coordinate of the point. Initially, Q contains an event for each of the endpoints of the input segments. 2) Initialize a binary search tree T of the line segments that cross the sweep line L, ordered by the y-coordinates of the crossing points. Initially, T is empty. 3) While Q is nonempty, find and remove the event from Q associated with a point p with minimum x-coordinate. Determine what type of event this is and process it according to the following case analysis:

	*	If p is the left endpoint of a line segment s, insert s into T.
		Find the segments r and t that are immediately below and above s in T (if they exist)
		and if their crossing forms a potential future event in the event queue, remove it.
		If s crosses r or t, add those crossing points as potential future events in the event queue.
	
	*	If p is the right endpoint of a line segment s, remove s from T.
		Find the segments r and t that were (prior to the removal of s)
		immediately above and below itin T (if they exist).
		If r and t cross, add that crossing point as a potential future event in the event queue.
	
	*	If p is the crossing point of two segments s and t (with s below t to the left of the crossing),
		swap the positions of s and t in T. Find the segments r and u (if they exist) that are immediately
		below and above s and t respectively. Remove any crossing points rs and tu from the event queue, and,
		if r and t cross or s and u cross, add those crossing points to the event queue.
------------------------------------  END  -------------------------------------

###Graham's Scan: ------------------------------------ BEGIN ------------------------------------- Three points are a counter-clockwise turn if ccw > 0, clockwise if ccw < 0, and collinear if ccw = 0 because ccw is a determinant that gives the signed area of the triangle formed by p1, p2, and p3. function ccw(p1, p2, p3): return (p2.x - p1.x)(p3.y - p1.y) - (p2.y - p1.y)(p3.x - p1.x)

	p0 = the point with min y‐coordinate;
	<p1,p2,...,pm> : the remaining vertices in <Q>;
	PUSH(p0,S);
	PUSH(p1,S);
	PUSH(p2,S);
	for i=3 to m do
	{ while ccw(N‐TOP(S),TOP(S),pi) <= 0
		  do POP(S);
	  PUSH(pi,S);
	}
	return S;
------------------------------------  END  -------------------------------------

###Jarvis' March ------------------------------------ BEGIN ------------------------------------- Start from the lowest point p0 i=1; // for Right Chain REPEAT { find the point pi with the smallest polar angle with pi‐1; i=i+1; } UNTIL pi is the highest point // for Left Chain REPEAT { find the point pi with the smallest polar angle with pi‐1 for the negative axis; i=i+1; } UNTIL pi is the lowest point

------------------------------------  END  -------------------------------------