org.kynosarges.tektosyne.geometry

Class MultiLineIntersection

java.lang.Object

org.kynosarges.tektosyne.geometry.MultiLineIntersection

public final class MultiLineIntersection
extends java.lang.Object

Provides algorithms to find all intersections between multiple line segments. Provides two algorithms to find all intersections between multiple line segments, specified as arrays of LineD instances:

• find(org.kynosarges.tektosyne.geometry.LineD[]) provides a sweep line algorithm which is very efficient if the number of intersections is much smaller than the number of lines.

• findSimple(org.kynosarges.tektosyne.geometry.LineD[]) provides a brute force algorithm which is better suited for a small number of lines or a large number of intersections.

Both algorithms sort their results lexicographically on the points of intersection, first by increasing y-coordinates and then, in case of equality, by increasing x-coordinates. The MultiLinePoint instance created for each point identifies all line segments that contain the point. Please refer to the two methods for further details.

The sweep line algorithm implemented by find(org.kynosarges.tektosyne.geometry.LineD[]) was first described by J. L. Bentley and T. A. Ottmann, Algorithms for reporting and counting geometric intersections, IEEE Transactions on Computers C-28 (1979), p.643-647. An implementation outline is given by Mark de Berg et al., Computational Geometry (3rd ed.), Springer-Verlag 2008, p.20-29, and a limited C++ implementation by Michael J. Laszlo, Computational Geometry and Computer Graphics in C++, Prentice Hall 1996, p.173-181.

Our implementation supports an unlimited number of line segments meeting at any intersection point, and uses an improved sweep line comparer that raises the algorithm’s stability to the same level as that of the brute force algorithm. A comment block at the top of the source code file gives a detailed description of this comparer.

Method Summary

Modifier and Type	Method and Description
static MultiLinePoint[]	find(LineD[] lines) Finds all intersections between the specified line segments, using a sweep line algorithm.
static MultiLinePoint[]	findSimple(LineD[] lines) Finds all intersections between the specified line segments, using a brute force algorithm.
static MultiLinePoint[]	findSimple(LineD[] lines, double epsilon) Finds all intersections between the specified line segments, using a brute force algorithm with the specified epsilon for coordinate comparisons.
static LineD[]	split(LineD[] lines, MultiLinePoint[] crossings) Splits the specified line segments on the specified intersection points.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

find

public static MultiLinePoint[] find(LineD[] lines)

Finds all intersections between the specified line segments, using a sweep line algorithm. Moves a horizontal sweep line across the specified lines, testing only those elements for intersection which are adjacent along the sweep line. The runtime is O((n + k) log n) where k is the number of discovered intersections.

find is very efficient when there are no or few intersections, achieving its best performance if the specified lines are horizontal parallels. However, the event point schedule and sweep line structure impose a large overhead if there are many candidate lines to consider. The worst case of O(n^2) intersections is much slower than the brute force algorithm implemented by findSimple(org.kynosarges.tektosyne.geometry.LineD[]).

find always uses exact coordinate comparisons. Epsilon comparisons would corrupt the search structures due to the merging of nearby event points. Call findSimple(org.kynosarges.tektosyne.geometry.LineD[]) with an epsilon argument to use epsilon comparisons.

Parameters:

lines - a LineD array containing the line segments to intersect

Returns:

a lexicographically sorted MultiLinePoint array describing all points of intersection between the specified lines

Throws:

java.lang.IllegalArgumentException - if lines contains a LineD whose LineD.start and LineD.end points are equal

java.lang.IllegalStateException - if the search structure was corrupted

java.lang.NullPointerException - if lines or any of its elements is null

findSimple

public static MultiLinePoint[] findSimple(LineD[] lines)

Finds all intersections between the specified line segments, using a brute force algorithm. Performs a pairwise intersection of every lines element with every other element. The runtime is therefore always O(n^2), regardless of the number of intersections found.

However, the constant factor is low and O(n^2) intersections are found in optimal time because findSimple performs no additional work to avoid testing for possible intersections. For a small number of lines (n < 50), findSimple usually beats the sweep line algorithm implemented by find(org.kynosarges.tektosyne.geometry.LineD[]) regardless of the number of intersections.

Parameters:

lines - a LineD array containing the line segments to intersect

Returns:

a lexicographically sorted MultiLinePoint array describing all points of intersection between the specified lines

Throws:

java.lang.NullPointerException - if lines or any of its elements is null

findSimple

public static MultiLinePoint[] findSimple(LineD[] lines,
                                          double epsilon)

Finds all intersections between the specified line segments, using a brute force algorithm with the specified epsilon for coordinate comparisons. Identical with the basic findSimple(LineD[]) overload, but uses the specified epsilon to determine intersections between the specified lines and to combine nearby intersections.

Parameters:

lines - a LineD array containing the line segments to intersect

epsilon - the maximum absolute difference at which coordinates should be considered equal

Returns:

a lexicographically sorted MultiLinePoint array describing all points of intersection between the specified lines

Throws:

java.lang.IllegalArgumentException - if epsilon is less than zero

java.lang.NullPointerException - if lines or any of its elements is null

split

public static LineD[] split(LineD[] lines,
                            MultiLinePoint[] crossings)

Splits the specified line segments on the specified intersection points. Returns a collection of LineD instances that are guaranteed not to intersect, except at their LineD.start or LineD.end points. The specified crossings are usually the result of find(org.kynosarges.tektosyne.geometry.LineD[]) or findSimple(org.kynosarges.tektosyne.geometry.LineD[]) for the specified lines.

split sets the LineD.start or LineD.end point of any LineD that participates in a LineLocation.START or LineLocation.END intersection to the MultiLinePoint.shared point of that intersection, so as to preserve coordinate identities that were established by a positive comparison epsilon.

Parameters:

lines - a LineD array containing the line segments to split

crossings - a MultiLinePoint array describing all points of intersection between lines

Returns:

a LineD array containing the line segments resulting from splitting all lines on the matching crossings

Throws:

java.lang.ArrayIndexOutOfBoundsException - if crossings contains any MultiLinePoint.Line.index values that are invalid for lines

java.lang.IllegalArgumentException - if crossings contains any MultiLinePoint.Line.location other than LineLocation.START, LineLocation.BETWEEN, or LineLocation.END

java.lang.NullPointerException - if lines or crossings is null or contains any null elements