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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
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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/** This class represents a region on the 2-sphere: a set of spherical polygons.
 * @since 3.3
 */
public class SphericalPolygonsSet extends AbstractRegion<Sphere2D, Sphere1D> {

    
Boundary defined as an array of closed loops start vertices. /** Boundary defined as an array of closed loops start vertices. */
    private List<Vertex> loops;

    
Build a polygons set representing the whole real 2-sphere.
Params:
tolerance – below which points are consider to be identical/** 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.
Params:
pole – pole of the hemisphere (the pole is in the inside half)
tolerance – below which points are consider to be identical/** 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<Sphere2D>(new Circle(pole, tolerance).wholeHyperplane(),
                                    new BSPTree<Sphere2D>(Boolean.FALSE),
                                    new BSPTree<Sphere2D>(Boolean.TRUE),
                                    null),
              tolerance);
    }

    
Build a polygons set representing a regular polygon.
Params:
center – center of the polygon (the center is in the inside half)
meridian – point defining the reference meridian for first polygon vertex
outsideRadius – distance of the vertices to the center
n – number of sides of the polygon
tolerance – below which points are consider to be identical/** 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 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 Boolean.TRUE and Boolean.FALSE
Params:
tree – inside/outside BSP tree representing the region
tolerance – below which points are consider to be identical/** Build a polygons set from a BSP tree.
     * <p>The leaf nodes of the BSP tree <em>must</em> 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}</p>
     * @param tree inside/outside BSP tree representing the region
     * @param tolerance below which points are consider to be identical
     */
    public SphericalPolygonsSet(final BSPTree<Sphere2D> 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 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 
checkPoint method will not be meaningful anymore.
If the boundary is empty, the region will represent the whole
space.
Params:
boundary – collection of boundary elements, as a collection of SubHyperplane objects
tolerance – below which points are consider to be identical/** Build a polygons set from a Boundary REPresentation (B-rep).
     * <p>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.</p>
     * <p>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.</p>
     * <p>If the boundary is empty, the region will represent the whole
     * space.</p>
     * @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<SubHyperplane<Sphere2D>> 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 general constructor using 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 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 hyperplaneThickness parameter. 
Params:
hyperplaneThickness – tolerance below which points are considered to
belong to the hyperplane (which is therefore more a slab)
vertices – vertices of the simple loop boundary/** Build a polygon from a simple list of vertices.
     * <p>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.</p>
     * <p>This constructor does not handle polygons with a boundary
     * forming several disconnected paths (such as polygons with holes).</p>
     * <p>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}.</p>
     * <p>If the list is empty, the region will represent the whole
     * space.</p>
     * <p>
     * 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
     * &pi; 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.
     * </p>
     * @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.
Params:
center – center of the polygon (the center is in the inside half)
meridian – point defining the reference meridian for first polygon vertex
outsideRadius – distance of the vertices to the center
n – number of sides of the polygon
Returns:
vertices array/** 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 general
constructor using subhyperplanes.
Params:
hyperplaneThickness – tolerance below which points are consider to
belong to the hyperplane (which is therefore more a slab)
vertices – vertices of the simple loop boundary
Returns:
the BSP tree of the input vertices/** Build the BSP tree of a polygons set from a simple list of vertices.
     * <p>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.</p>
     * <p>This constructor does not handle polygons with a boundary
     * forming several disconnected paths (such as polygons with holes).</p>
     * <p>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.</p>
     * <p>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}.</p>
     * @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<Sphere2D> verticesToTree(final double hyperplaneThickness,
                                                    final S2Point ... vertices) {

        final int n = vertices.length;
        if (n == 0) {
            // the tree represents the whole space
            return new BSPTree<Sphere2D>(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<Edge> edges = new ArrayList<Edge>(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<Sphere2D> tree = new BSPTree<Sphere2D>();
        insertEdges(hyperplaneThickness, tree, edges);

        return tree;

    }

    
Recursively build a tree by inserting cut sub-hyperplanes.
Params:
hyperplaneThickness – tolerance below which points are considered to
belong to the hyperplane (which is therefore more a slab)
node – current tree node (it is a leaf node at the beginning
of the call)
edges – list of edges to insert in the cell defined by this node
(excluding edges not belonging to the cell defined by this node)/** 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<Sphere2D> node,
                                    final List<Edge> 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<Sphere2D> 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<Edge> outsideList = new ArrayList<Edge>();
        final List<Edge> insideList  = new ArrayList<Edge>();
        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} /** {@inheritDoc} */
    @Override
    public SphericalPolygonsSet buildNew(final BSPTree<Sphere2D> tree) {
        return new SphericalPolygonsSet(tree, getTolerance());
    }

    
{@inheritDoc}
Throws:
MathIllegalStateException – if the tolerance setting does not allow to build
a clean non-ambiguous boundary/** {@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<Sphere2D> 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.
Throws:
MathIllegalStateException – if the tolerance setting does not allow to build
a clean non-ambiguous boundary
See Also:
Vertex
Edge
Returns:
boundary of the polygon, organized as an unmodifiable list of loops start vertices./** Get the boundary loops of the polygon.
     * <p>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.</p>
     * <p>If the polygon has no boundary at all, a zero length loop
     * array will be returned.</p>
     * <p>If the polygon is a simple one-piece polygon, then the returned
     * array will contain a single vertex.
     * </p>
     * <p>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.</p>
     * @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<Vertex> 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<Sphere2D> root = getTree(true);
                final EdgesBuilder visitor = new EdgesBuilder(root, getTolerance());
                root.visit(visitor);
                final List<Edge> edges = visitor.getEdges();


                // convert the list of all edges into a list of start vertices
                loops = new ArrayList<Vertex>();
                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<Edge> 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 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 ball.contains(point) method return always false or true. 

This method is not guaranteed to return the smallest enclosing cap.

Returns:
a spherical cap enclosing the polygon/** Get a spherical cap enclosing the polygon.
     * <p>
     * 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:
     * </p>
     * <pre>
     *   // compute region, plus an enclosing spherical cap
     *   SphericalPolygonsSet complexShape = ...;
     *   EnclosingBall<Sphere2D, S2Point> 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 ...
     *
     *   }
     * </pre>
     * <p>
     * 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.
     * </p>
     * <p>
     * This method is <em>not</em> guaranteed to return the smallest enclosing cap.
     * </p>
     * @return a spherical cap enclosing the polygon
     */
    public EnclosingBall<Sphere2D, S2Point> getEnclosingCap() {

        // handle special cases first
        if (isEmpty()) {
            return new EnclosingBall<Sphere2D, S2Point>(S2Point.PLUS_K, Double.NEGATIVE_INFINITY);
        }
        if (isFull()) {
            return new EnclosingBall<Sphere2D, S2Point>(S2Point.PLUS_K, Double.POSITIVE_INFINITY);
        }

        // as the polygons is neither empty nor full, it has some boundaries and cut hyperplanes
        final BSPTree<Sphere2D> 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<Sphere2D, S2Point>(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<Sphere2D, S2Point>(new S2Point(circle.getPole()),
                                                        0.5 * FastMath.PI);
        }

        // gather some inside points, to be used by the encloser
        final List<Vector3D> points = getInsidePoints();

        // extract points from the boundary loops, to be used by the encloser as well
        final List<Vertex> 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<Euclidean3D, Vector3D> encloser =
                new WelzlEncloser<Euclidean3D, Vector3D>(getTolerance(), generator);
        EnclosingBall<Euclidean3D, Vector3D> 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<Sphere2D, S2Point> enclosingS2 =
                    new EnclosingBall<Sphere2D, S2Point>(S2Point.PLUS_K, Double.POSITIVE_INFINITY);
            for (Vector3D outsidePoint : getOutsidePoints()) {
                final S2Point outsideS2 = new S2Point(outsidePoint);
                final BoundaryProjection<Sphere2D> projection = projectToBoundary(outsideS2);
                if (FastMath.PI - projection.getOffset() < enclosingS2.getRadius()) {
                    enclosingS2 = new EnclosingBall<Sphere2D, S2Point>(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<Sphere2D, S2Point> enclosingS2 =
                new EnclosingBall<Sphere2D, S2Point>(new S2Point(enclosing3D.getCenter()),
                                                     FastMath.acos((1 + h * h - r * r) / (2 * h)),
                                                     support);

        return enclosingS2;

    }

    
Gather some inside points.
Returns:
list of points known to be strictly in all inside convex cells/** Gather some inside points.
     * @return list of points known to be strictly in all inside convex cells
     */
    private List<Vector3D> getInsidePoints() {
        final PropertiesComputer pc = new PropertiesComputer(getTolerance());
        getTree(true).visit(pc);
        return pc.getConvexCellsInsidePoints();
    }

    
Gather some outside points.
Returns:
list of points known to be strictly in all outside convex cells/** Gather some outside points.
     * @return list of points known to be strictly in all outside convex cells
     */
    private List<Vector3D> getOutsidePoints() {
        final SphericalPolygonsSet complement =
                (SphericalPolygonsSet) new RegionFactory<Sphere2D>().getComplement(this);
        final PropertiesComputer pc = new PropertiesComputer(getTolerance());
        complement.getTree(true).visit(pc);
        return pc.getConvexCellsInsidePoints();
    }

}