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package org.apache.commons.math3.geometry.euclidean.twod.hull;
import java.util.ArrayList;
import java.util.Collection;
import java.util.List;
import org.apache.commons.math3.geometry.euclidean.twod.Vector2D;
A simple heuristic to improve the performance of convex hull algorithms.
The heuristic is based on the idea of a convex quadrilateral, which is formed by
four points with the lowest and highest x / y coordinates. Any point that lies inside
this quadrilateral can not be part of the convex hull and can thus be safely discarded
before generating the convex hull itself.
The complexity of the operation is O(n), and may greatly improve the time it takes to
construct the convex hull afterwards, depending on the point distribution.
See Also: Since: 3.3
/**
* A simple heuristic to improve the performance of convex hull algorithms.
* <p>
* The heuristic is based on the idea of a convex quadrilateral, which is formed by
* four points with the lowest and highest x / y coordinates. Any point that lies inside
* this quadrilateral can not be part of the convex hull and can thus be safely discarded
* before generating the convex hull itself.
* <p>
* The complexity of the operation is O(n), and may greatly improve the time it takes to
* construct the convex hull afterwards, depending on the point distribution.
*
* @see <a href="http://en.wikipedia.org/wiki/Convex_hull_algorithms#Akl-Toussaint_heuristic">
* Akl-Toussaint heuristic (Wikipedia)</a>
* @since 3.3
*/
public final class AklToussaintHeuristic {
Hide utility constructor. /** Hide utility constructor. */
private AklToussaintHeuristic() {
}
Returns a point set that is reduced by all points for which it is safe to assume
that they are not part of the convex hull.
Params: - points – the original point set
Returns: a reduced point set, useful as input for convex hull algorithms
/**
* Returns a point set that is reduced by all points for which it is safe to assume
* that they are not part of the convex hull.
*
* @param points the original point set
* @return a reduced point set, useful as input for convex hull algorithms
*/
public static Collection<Vector2D> reducePoints(final Collection<Vector2D> points) {
// find the leftmost point
int size = 0;
Vector2D minX = null;
Vector2D maxX = null;
Vector2D minY = null;
Vector2D maxY = null;
for (Vector2D p : points) {
if (minX == null || p.getX() < minX.getX()) {
minX = p;
}
if (maxX == null || p.getX() > maxX.getX()) {
maxX = p;
}
if (minY == null || p.getY() < minY.getY()) {
minY = p;
}
if (maxY == null || p.getY() > maxY.getY()) {
maxY = p;
}
size++;
}
if (size < 4) {
return points;
}
final List<Vector2D> quadrilateral = buildQuadrilateral(minY, maxX, maxY, minX);
// if the quadrilateral is not well formed, e.g. only 2 points, do not attempt to reduce
if (quadrilateral.size() < 3) {
return points;
}
final List<Vector2D> reducedPoints = new ArrayList<Vector2D>(quadrilateral);
for (final Vector2D p : points) {
// check all points if they are within the quadrilateral
// in which case they can not be part of the convex hull
if (!insideQuadrilateral(p, quadrilateral)) {
reducedPoints.add(p);
}
}
return reducedPoints;
}
Build the convex quadrilateral with the found corner points (with min/max x/y coordinates).
Params: - points – the respective points with min/max x/y coordinate
Returns: the quadrilateral
/**
* Build the convex quadrilateral with the found corner points (with min/max x/y coordinates).
*
* @param points the respective points with min/max x/y coordinate
* @return the quadrilateral
*/
private static List<Vector2D> buildQuadrilateral(final Vector2D... points) {
List<Vector2D> quadrilateral = new ArrayList<Vector2D>();
for (Vector2D p : points) {
if (!quadrilateral.contains(p)) {
quadrilateral.add(p);
}
}
return quadrilateral;
}
Checks if the given point is located within the convex quadrilateral.
Params: - point – the point to check
- quadrilateralPoints – the convex quadrilateral, represented by 4 points
Returns: true if the point is inside the quadrilateral, false otherwise
/**
* Checks if the given point is located within the convex quadrilateral.
* @param point the point to check
* @param quadrilateralPoints the convex quadrilateral, represented by 4 points
* @return {@code true} if the point is inside the quadrilateral, {@code false} otherwise
*/
private static boolean insideQuadrilateral(final Vector2D point,
final List<Vector2D> quadrilateralPoints) {
Vector2D p1 = quadrilateralPoints.get(0);
Vector2D p2 = quadrilateralPoints.get(1);
if (point.equals(p1) || point.equals(p2)) {
return true;
}
// get the location of the point relative to the first two vertices
final double last = point.crossProduct(p1, p2);
final int size = quadrilateralPoints.size();
// loop through the rest of the vertices
for (int i = 1; i < size; i++) {
p1 = p2;
p2 = quadrilateralPoints.get((i + 1) == size ? 0 : i + 1);
if (point.equals(p1) || point.equals(p2)) {
return true;
}
// do side of line test: multiply the last location with this location
// if they are the same sign then the operation will yield a positive result
// -x * -y = +xy, x * y = +xy, -x * y = -xy, x * -y = -xy
if (last * point.crossProduct(p1, p2) < 0) {
return false;
}
}
return true;
}
}