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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.geometry.spherical.twod;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Collections;
import java.util.Iterator;
import java.util.List;
import org.apache.commons.math3.exception.MathIllegalStateException;
import org.apache.commons.math3.geometry.enclosing.EnclosingBall;
import org.apache.commons.math3.geometry.enclosing.WelzlEncloser;
import org.apache.commons.math3.geometry.euclidean.threed.Euclidean3D;
import org.apache.commons.math3.geometry.euclidean.threed.Rotation;
import org.apache.commons.math3.geometry.euclidean.threed.RotationConvention;
import org.apache.commons.math3.geometry.euclidean.threed.SphereGenerator;
import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
import org.apache.commons.math3.geometry.partitioning.AbstractRegion;
import org.apache.commons.math3.geometry.partitioning.BSPTree;
import org.apache.commons.math3.geometry.partitioning.BoundaryProjection;
import org.apache.commons.math3.geometry.partitioning.RegionFactory;
import org.apache.commons.math3.geometry.partitioning.SubHyperplane;
import org.apache.commons.math3.geometry.spherical.oned.Sphere1D;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;
/** This class represents a region on the 2-sphere: a set of spherical polygons.
* @since 3.3
*/
public class SphericalPolygonsSet extends AbstractRegion {
/** Boundary defined as an array of closed loops start vertices. */
private List loops;
/** Build a polygons set representing the whole real 2-sphere.
* @param tolerance below which points are consider to be identical
*/
public SphericalPolygonsSet(final double tolerance) {
super(tolerance);
}
/** Build a polygons set representing a hemisphere.
* @param pole pole of the hemisphere (the pole is in the inside half)
* @param tolerance below which points are consider to be identical
*/
public SphericalPolygonsSet(final Vector3D pole, final double tolerance) {
super(new BSPTree(new Circle(pole, tolerance).wholeHyperplane(),
new BSPTree(Boolean.FALSE),
new BSPTree(Boolean.TRUE),
null),
tolerance);
}
/** Build a polygons set representing a regular polygon.
* @param center center of the polygon (the center is in the inside half)
* @param meridian point defining the reference meridian for first polygon vertex
* @param outsideRadius distance of the vertices to the center
* @param n number of sides of the polygon
* @param tolerance below which points are consider to be identical
*/
public SphericalPolygonsSet(final Vector3D center, final Vector3D meridian,
final double outsideRadius, final int n,
final double tolerance) {
this(tolerance, createRegularPolygonVertices(center, meridian, outsideRadius, n));
}
/** Build a polygons set from a BSP tree.
* The leaf nodes of the BSP tree must have a
* {@code Boolean} attribute representing the inside status of
* the corresponding cell (true for inside cells, false for outside
* cells). In order to avoid building too many small objects, it is
* recommended to use the predefined constants
* {@code Boolean.TRUE} and {@code Boolean.FALSE}
* @param tree inside/outside BSP tree representing the region
* @param tolerance below which points are consider to be identical
*/
public SphericalPolygonsSet(final BSPTree tree, final double tolerance) {
super(tree, tolerance);
}
/** Build a polygons set from a Boundary REPresentation (B-rep).
* The boundary is provided as a collection of {@link
* SubHyperplane sub-hyperplanes}. Each sub-hyperplane has the
* interior part of the region on its minus side and the exterior on
* its plus side.
* The boundary elements can be in any order, and can form
* several non-connected sets (like for example polygons with holes
* or a set of disjoint polygons considered as a whole). In
* fact, the elements do not even need to be connected together
* (their topological connections are not used here). However, if the
* boundary does not really separate an inside open from an outside
* open (open having here its topological meaning), then subsequent
* calls to the {@link
* org.apache.commons.math3.geometry.partitioning.Region#checkPoint(org.apache.commons.math3.geometry.Point)
* checkPoint} method will not be meaningful anymore.
* If the boundary is empty, the region will represent the whole
* space.
* @param boundary collection of boundary elements, as a
* collection of {@link SubHyperplane SubHyperplane} objects
* @param tolerance below which points are consider to be identical
*/
public SphericalPolygonsSet(final Collection> boundary, final double tolerance) {
super(boundary, tolerance);
}
/** Build a polygon from a simple list of vertices.
* The boundary is provided as a list of points considering to
* represent the vertices of a simple loop. The interior part of the
* region is on the left side of this path and the exterior is on its
* right side.
* This constructor does not handle polygons with a boundary
* forming several disconnected paths (such as polygons with holes).
* For cases where this simple constructor applies, it is expected to
* be numerically more robust than the {@link #SphericalPolygonsSet(Collection,
* double) general constructor} using {@link SubHyperplane subhyperplanes}.
* If the list is empty, the region will represent the whole
* space.
*
* Polygons with thin pikes or dents are inherently difficult to handle because
* they involve circles with almost opposite directions at some vertices. Polygons
* whose vertices come from some physical measurement with noise are also
* difficult because an edge that should be straight may be broken in lots of
* different pieces with almost equal directions. In both cases, computing the
* circles intersections is not numerically robust due to the almost 0 or almost
* π angle. Such cases need to carefully adjust the {@code hyperplaneThickness}
* parameter. A too small value would often lead to completely wrong polygons
* with large area wrongly identified as inside or outside. Large values are
* often much safer. As a rule of thumb, a value slightly below the size of the
* most accurate detail needed is a good value for the {@code hyperplaneThickness}
* parameter.
*
* @param hyperplaneThickness tolerance below which points are considered to
* belong to the hyperplane (which is therefore more a slab)
* @param vertices vertices of the simple loop boundary
*/
public SphericalPolygonsSet(final double hyperplaneThickness, final S2Point ... vertices) {
super(verticesToTree(hyperplaneThickness, vertices), hyperplaneThickness);
}
/** Build the vertices representing a regular polygon.
* @param center center of the polygon (the center is in the inside half)
* @param meridian point defining the reference meridian for first polygon vertex
* @param outsideRadius distance of the vertices to the center
* @param n number of sides of the polygon
* @return vertices array
*/
private static S2Point[] createRegularPolygonVertices(final Vector3D center, final Vector3D meridian,
final double outsideRadius, final int n) {
final S2Point[] array = new S2Point[n];
final Rotation r0 = new Rotation(Vector3D.crossProduct(center, meridian),
outsideRadius, RotationConvention.VECTOR_OPERATOR);
array[0] = new S2Point(r0.applyTo(center));
final Rotation r = new Rotation(center, MathUtils.TWO_PI / n, RotationConvention.VECTOR_OPERATOR);
for (int i = 1; i < n; ++i) {
array[i] = new S2Point(r.applyTo(array[i - 1].getVector()));
}
return array;
}
/** Build the BSP tree of a polygons set from a simple list of vertices.
* The boundary is provided as a list of points considering to
* represent the vertices of a simple loop. The interior part of the
* region is on the left side of this path and the exterior is on its
* right side.
* This constructor does not handle polygons with a boundary
* forming several disconnected paths (such as polygons with holes).
* This constructor handles only polygons with edges strictly shorter
* than \( \pi \). If longer edges are needed, they need to be broken up
* in smaller sub-edges so this constraint holds.
* For cases where this simple constructor applies, it is expected to
* be numerically more robust than the {@link #PolygonsSet(Collection) general
* constructor} using {@link SubHyperplane subhyperplanes}.
* @param hyperplaneThickness tolerance below which points are consider to
* belong to the hyperplane (which is therefore more a slab)
* @param vertices vertices of the simple loop boundary
* @return the BSP tree of the input vertices
*/
private static BSPTree verticesToTree(final double hyperplaneThickness,
final S2Point ... vertices) {
final int n = vertices.length;
if (n == 0) {
// the tree represents the whole space
return new BSPTree(Boolean.TRUE);
}
// build the vertices
final Vertex[] vArray = new Vertex[n];
for (int i = 0; i < n; ++i) {
vArray[i] = new Vertex(vertices[i]);
}
// build the edges
List edges = new ArrayList(n);
Vertex end = vArray[n - 1];
for (int i = 0; i < n; ++i) {
// get the endpoints of the edge
final Vertex start = end;
end = vArray[i];
// get the circle supporting the edge, taking care not to recreate it
// if it was already created earlier due to another edge being aligned
// with the current one
Circle circle = start.sharedCircleWith(end);
if (circle == null) {
circle = new Circle(start.getLocation(), end.getLocation(), hyperplaneThickness);
}
// create the edge and store it
edges.add(new Edge(start, end,
Vector3D.angle(start.getLocation().getVector(),
end.getLocation().getVector()),
circle));
// check if another vertex also happens to be on this circle
for (final Vertex vertex : vArray) {
if (vertex != start && vertex != end &&
FastMath.abs(circle.getOffset(vertex.getLocation())) <= hyperplaneThickness) {
vertex.bindWith(circle);
}
}
}
// build the tree top-down
final BSPTree tree = new BSPTree();
insertEdges(hyperplaneThickness, tree, edges);
return tree;
}
/** Recursively build a tree by inserting cut sub-hyperplanes.
* @param hyperplaneThickness tolerance below which points are considered to
* belong to the hyperplane (which is therefore more a slab)
* @param node current tree node (it is a leaf node at the beginning
* of the call)
* @param edges list of edges to insert in the cell defined by this node
* (excluding edges not belonging to the cell defined by this node)
*/
private static void insertEdges(final double hyperplaneThickness,
final BSPTree node,
final List edges) {
// find an edge with an hyperplane that can be inserted in the node
int index = 0;
Edge inserted = null;
while (inserted == null && index < edges.size()) {
inserted = edges.get(index++);
if (!node.insertCut(inserted.getCircle())) {
inserted = null;
}
}
if (inserted == null) {
// no suitable edge was found, the node remains a leaf node
// we need to set its inside/outside boolean indicator
final BSPTree parent = node.getParent();
if (parent == null || node == parent.getMinus()) {
node.setAttribute(Boolean.TRUE);
} else {
node.setAttribute(Boolean.FALSE);
}
return;
}
// we have split the node by inserting an edge as a cut sub-hyperplane
// distribute the remaining edges in the two sub-trees
final List outsideList = new ArrayList();
final List insideList = new ArrayList();
for (final Edge edge : edges) {
if (edge != inserted) {
edge.split(inserted.getCircle(), outsideList, insideList);
}
}
// recurse through lower levels
if (!outsideList.isEmpty()) {
insertEdges(hyperplaneThickness, node.getPlus(), outsideList);
} else {
node.getPlus().setAttribute(Boolean.FALSE);
}
if (!insideList.isEmpty()) {
insertEdges(hyperplaneThickness, node.getMinus(), insideList);
} else {
node.getMinus().setAttribute(Boolean.TRUE);
}
}
/** {@inheritDoc} */
@Override
public SphericalPolygonsSet buildNew(final BSPTree tree) {
return new SphericalPolygonsSet(tree, getTolerance());
}
/** {@inheritDoc}
* @exception MathIllegalStateException if the tolerance setting does not allow to build
* a clean non-ambiguous boundary
*/
@Override
protected void computeGeometricalProperties() throws MathIllegalStateException {
final BSPTree tree = getTree(true);
if (tree.getCut() == null) {
// the instance has a single cell without any boundaries
if (tree.getCut() == null && (Boolean) tree.getAttribute()) {
// the instance covers the whole space
setSize(4 * FastMath.PI);
setBarycenter(new S2Point(0, 0));
} else {
setSize(0);
setBarycenter(S2Point.NaN);
}
} else {
// the instance has a boundary
final PropertiesComputer pc = new PropertiesComputer(getTolerance());
tree.visit(pc);
setSize(pc.getArea());
setBarycenter(pc.getBarycenter());
}
}
/** Get the boundary loops of the polygon.
* The polygon boundary can be represented as a list of closed loops,
* each loop being given by exactly one of its vertices. From each loop
* start vertex, one can follow the loop by finding the outgoing edge,
* then the end vertex, then the next outgoing edge ... until the start
* vertex of the loop (exactly the same instance) is found again once
* the full loop has been visited.
* If the polygon has no boundary at all, a zero length loop
* array will be returned.
* If the polygon is a simple one-piece polygon, then the returned
* array will contain a single vertex.
*
* All edges in the various loops have the inside of the region on
* their left side (i.e. toward their pole) and the outside on their
* right side (i.e. away from their pole) when moving in the underlying
* circle direction. This means that the closed loops obey the direct
* trigonometric orientation.
* @return boundary of the polygon, organized as an unmodifiable list of loops start vertices.
* @exception MathIllegalStateException if the tolerance setting does not allow to build
* a clean non-ambiguous boundary
* @see Vertex
* @see Edge
*/
public List getBoundaryLoops() throws MathIllegalStateException {
if (loops == null) {
if (getTree(false).getCut() == null) {
loops = Collections.emptyList();
} else {
// sort the arcs according to their start point
final BSPTree root = getTree(true);
final EdgesBuilder visitor = new EdgesBuilder(root, getTolerance());
root.visit(visitor);
final List edges = visitor.getEdges();
// convert the list of all edges into a list of start vertices
loops = new ArrayList();
while (!edges.isEmpty()) {
// this is an edge belonging to a new loop, store it
Edge edge = edges.get(0);
final Vertex startVertex = edge.getStart();
loops.add(startVertex);
// remove all remaining edges in the same loop
do {
// remove one edge
for (final Iterator iterator = edges.iterator(); iterator.hasNext();) {
if (iterator.next() == edge) {
iterator.remove();
break;
}
}
// go to next edge following the boundary loop
edge = edge.getEnd().getOutgoing();
} while (edge.getStart() != startVertex);
}
}
}
return Collections.unmodifiableList(loops);
}
/** Get a spherical cap enclosing the polygon.
*
* This method is intended as a first test to quickly identify points
* that are guaranteed to be outside of the region, hence performing a full
* {@link #checkPoint(org.apache.commons.math3.geometry.Vector) checkPoint}
* only if the point status remains undecided after the quick check. It is
* is therefore mostly useful to speed up computation for small polygons with
* complex shapes (say a country boundary on Earth), as the spherical cap will
* be small and hence will reliably identify a large part of the sphere as outside,
* whereas the full check can be more computing intensive. A typical use case is
* therefore:
*
*
* // compute region, plus an enclosing spherical cap
* SphericalPolygonsSet complexShape = ...;
* EnclosingBall cap = complexShape.getEnclosingCap();
*
* // check lots of points
* for (Vector3D p : points) {
*
* final Location l;
* if (cap.contains(p)) {
* // we cannot be sure where the point is
* // we need to perform the full computation
* l = complexShape.checkPoint(v);
* } else {
* // no need to do further computation,
* // we already know the point is outside
* l = Location.OUTSIDE;
* }
*
* // use l ...
*
* }
*
*
* In the special cases of empty or whole sphere polygons, special
* spherical caps are returned, with angular radius set to negative
* or positive infinity so the {@link
* EnclosingBall#contains(org.apache.commons.math3.geometry.Point) ball.contains(point)}
* method return always false or true.
*
*
* This method is not guaranteed to return the smallest enclosing cap.
*
* @return a spherical cap enclosing the polygon
*/
public EnclosingBall getEnclosingCap() {
// handle special cases first
if (isEmpty()) {
return new EnclosingBall(S2Point.PLUS_K, Double.NEGATIVE_INFINITY);
}
if (isFull()) {
return new EnclosingBall(S2Point.PLUS_K, Double.POSITIVE_INFINITY);
}
// as the polygons is neither empty nor full, it has some boundaries and cut hyperplanes
final BSPTree root = getTree(false);
if (isEmpty(root.getMinus()) && isFull(root.getPlus())) {
// the polygon covers an hemisphere, and its boundary is one 2π long edge
final Circle circle = (Circle) root.getCut().getHyperplane();
return new EnclosingBall(new S2Point(circle.getPole()).negate(),
0.5 * FastMath.PI);
}
if (isFull(root.getMinus()) && isEmpty(root.getPlus())) {
// the polygon covers an hemisphere, and its boundary is one 2π long edge
final Circle circle = (Circle) root.getCut().getHyperplane();
return new EnclosingBall(new S2Point(circle.getPole()),
0.5 * FastMath.PI);
}
// gather some inside points, to be used by the encloser
final List points = getInsidePoints();
// extract points from the boundary loops, to be used by the encloser as well
final List boundary = getBoundaryLoops();
for (final Vertex loopStart : boundary) {
int count = 0;
for (Vertex v = loopStart; count == 0 || v != loopStart; v = v.getOutgoing().getEnd()) {
++count;
points.add(v.getLocation().getVector());
}
}
// find the smallest enclosing 3D sphere
final SphereGenerator generator = new SphereGenerator();
final WelzlEncloser encloser =
new WelzlEncloser(getTolerance(), generator);
EnclosingBall enclosing3D = encloser.enclose(points);
final Vector3D[] support3D = enclosing3D.getSupport();
// convert to 3D sphere to spherical cap
final double r = enclosing3D.getRadius();
final double h = enclosing3D.getCenter().getNorm();
if (h < getTolerance()) {
// the 3D sphere is centered on the unit sphere and covers it
// fall back to a crude approximation, based only on outside convex cells
EnclosingBall enclosingS2 =
new EnclosingBall(S2Point.PLUS_K, Double.POSITIVE_INFINITY);
for (Vector3D outsidePoint : getOutsidePoints()) {
final S2Point outsideS2 = new S2Point(outsidePoint);
final BoundaryProjection projection = projectToBoundary(outsideS2);
if (FastMath.PI - projection.getOffset() < enclosingS2.getRadius()) {
enclosingS2 = new EnclosingBall(outsideS2.negate(),
FastMath.PI - projection.getOffset(),
(S2Point) projection.getProjected());
}
}
return enclosingS2;
}
final S2Point[] support = new S2Point[support3D.length];
for (int i = 0; i < support3D.length; ++i) {
support[i] = new S2Point(support3D[i]);
}
final EnclosingBall enclosingS2 =
new EnclosingBall(new S2Point(enclosing3D.getCenter()),
FastMath.acos((1 + h * h - r * r) / (2 * h)),
support);
return enclosingS2;
}
/** Gather some inside points.
* @return list of points known to be strictly in all inside convex cells
*/
private List getInsidePoints() {
final PropertiesComputer pc = new PropertiesComputer(getTolerance());
getTree(true).visit(pc);
return pc.getConvexCellsInsidePoints();
}
/** Gather some outside points.
* @return list of points known to be strictly in all outside convex cells
*/
private List getOutsidePoints() {
final SphericalPolygonsSet complement =
(SphericalPolygonsSet) new RegionFactory().getComplement(this);
final PropertiesComputer pc = new PropertiesComputer(getTolerance());
complement.getTree(true).visit(pc);
return pc.getConvexCellsInsidePoints();
}
}