/*
    * 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  package org.hipparchus.geometry.spherical.twod;
  
  import java.util.ArrayList;
  import java.util.Arrays;
  import java.util.Collection;
  import java.util.Collections;
  import java.util.List;
  import java.util.function.IntPredicate;
  
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathIllegalStateException;
  import org.hipparchus.geometry.LocalizedGeometryFormats;
  import org.hipparchus.geometry.enclosing.EnclosingBall;
  import org.hipparchus.geometry.enclosing.WelzlEncloser;
  import org.hipparchus.geometry.euclidean.threed.Euclidean3D;
  import org.hipparchus.geometry.euclidean.threed.Rotation;
  import org.hipparchus.geometry.euclidean.threed.RotationConvention;
  import org.hipparchus.geometry.euclidean.threed.SphereGenerator;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.geometry.partitioning.AbstractRegion;
  import org.hipparchus.geometry.partitioning.BSPTree;
  import org.hipparchus.geometry.partitioning.BoundaryProjection;
  import org.hipparchus.geometry.partitioning.RegionFactory;
  import org.hipparchus.geometry.partitioning.SubHyperplane;
  import org.hipparchus.geometry.spherical.oned.Arc;
  import org.hipparchus.geometry.spherical.oned.Sphere1D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.MathUtils;
  import org.hipparchus.util.Precision;
  
  /** This class represents a region on the 2-sphere: a set of spherical polygons.
   */
  public class SphericalPolygonsSet extends AbstractRegion<Sphere2D, Sphere1D> {
  
      /** Boundary defined as an array of closed loops start vertices. */
      private List<Vertex> loops;
  
      /** Build a polygons set representing the whole real 2-sphere.
       * @param tolerance below which points are consider to be identical
       * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
       */
      public SphericalPolygonsSet(final double tolerance) throws MathIllegalArgumentException {
          super(tolerance);
          Sphere2D.checkTolerance(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
       * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
       */
      public SphericalPolygonsSet(final Vector3D pole, final double tolerance)
          throws MathIllegalArgumentException {
          super(new BSPTree<Sphere2D>(new Circle(pole, tolerance).wholeHyperplane(),
                                      new BSPTree<Sphere2D>(Boolean.FALSE),
                                      new BSPTree<Sphere2D>(Boolean.TRUE),
                                      null),
                tolerance);
          Sphere2D.checkTolerance(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
       * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
       */
      public SphericalPolygonsSet(final Vector3D center, final Vector3D meridian,
                                  final double outsideRadius, final int n,
                                  final double tolerance)
          throws MathIllegalArgumentException {
          this(tolerance, createRegularPolygonVertices(center, meridian, outsideRadius, n));
      }
  
      /** 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
      * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
      */
     public SphericalPolygonsSet(final BSPTree<Sphere2D> tree, final double tolerance)
         throws MathIllegalArgumentException {
         super(tree, tolerance);
         Sphere2D.checkTolerance(tolerance);
     }
 
     /** 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.hipparchus.geometry.partitioning.Region#checkPoint(org.hipparchus.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
      * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
      */
     public SphericalPolygonsSet(final Collection<SubHyperplane<Sphere2D>> boundary, final double tolerance)
         throws MathIllegalArgumentException {
         super(boundary, tolerance);
         Sphere2D.checkTolerance(tolerance);
     }
 
     /** 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>This constructor assumes that edges between {@code vertices}, including the edge
      * between the last and the first vertex, are shorter than pi. If edges longer than pi
      * are used it may produce unintuitive results, such as reversing the direction of the
      * edge. This implies using a {@code vertices} array of length 1 or 2 in this
      * constructor produces an ill-defined region. Use one of the other constructors or
      * {@link RegionFactory} instead.</p>
      * <p>The list of {@code vertices} is reduced by selecting a sub-set of vertices
      * before creating the boundary set. Every point in {@code vertices} will be on the
      * {@link #checkPoint(org.hipparchus.geometry.Point) boundary} of the constructed polygon set, but not
      * necessarily the center-line of the boundary.</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). Should be greater than
      * {@code FastMath.ulp(4 * FastMath.PI)} for meaningful results.
      * @param vertices vertices of the simple loop boundary
      * @exception MathIllegalArgumentException if tolerance is smaller than {@link Sphere1D#SMALLEST_TOLERANCE}
      * @exception org.hipparchus.exception.MathRuntimeException if {@code vertices}
      * contains only a single vertex or repeated vertices.
      */
     public SphericalPolygonsSet(final double hyperplaneThickness, final S2Point ... vertices)
         throws MathIllegalArgumentException {
         super(verticesToTree(hyperplaneThickness, vertices), hyperplaneThickness);
         Sphere2D.checkTolerance(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.
      * <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 #SphericalPolygonsSet(Collection, double) 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,
                                                     S2Point ... vertices) {
         // thin vertices to those that define distinct circles
         vertices = reduce(hyperplaneThickness, vertices).toArray(new S2Point[0]);
         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
         final List<Edge> 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
             final Circle 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));
 
         }
 
         // build the tree top-down
         final BSPTree<Sphere2D> tree = new BSPTree<>();
         insertEdges(hyperplaneThickness, tree, edges);
 
         return tree;
 
     }
 
     /**
      * Compute a subset of vertices that define the boundary to within the given
      * tolerance. This method partitions {@code vertices} into segments that all lie same
      * arc to within {@code hyperplaneThickness}, and then returns the end points of the
      * arcs. Combined arcs are limited to length of pi. If the input vertices has arcs
      * longer than pi these will be preserved in the returned data.
      *
      * @param hyperplaneThickness of each circle in radians.
      * @param vertices            to decimate.
      * @return a subset of {@code vertices}.
      */
     private static List<S2Point> reduce(final double hyperplaneThickness,
                                         final S2Point[] vertices) {
         final int n = vertices.length;
         if (n <= 3) {
             // can't reduce to fewer than three points
             return Arrays.asList(vertices.clone());
         }
         final List<S2Point> points = new ArrayList<>();
         /* Use a simple greedy search to add points to a circle s.t. all intermediate
          * points are within the thickness. Running time is O(n lg n) worst case.
          * Since the first vertex may be the middle of a straight edge, look backward
          * and forward to establish the first edge.
          * Uses the property that any two points define a circle, so don't check
          * circles that just span two points.
          */
         // first look backward
         final IntPredicate onCircleBackward = j -> {
             final int i = n - 2 - j;
             // circle spanning considered points
             final Circle circle = new Circle(vertices[0], vertices[i], hyperplaneThickness);
             final Arc arc = circle.getArc(vertices[0], vertices[i]);
             if (arc.getSize() >= FastMath.PI) {
                 return false;
             }
             for (int k = i + 1; k < n; k++) {
                 final S2Point vertex = vertices[k];
                 if (FastMath.abs(circle.getOffset(vertex)) > hyperplaneThickness ||
                         arc.getOffset(circle.toSubSpace(vertex)) > 0) {
                     // point is not within the thickness or arc, start new edge
                     return false;
                 }
             }
             return true;
         };
         // last index in vertices of last entry added to points
         int last = n - 2 - searchHelper(onCircleBackward, 0, n - 2);
         if (last > 1) {
             points.add(vertices[last]);
         } else {
             // all points lie on one semi-circle, distance from 0 to 1 is > pi
             // ill-defined case, just return three points from the list
             return Arrays.asList(Arrays.copyOfRange(vertices, 0, 3));
         }
         final int first = last;
         // then build edges forward
         for (int j = 1; ; j += 2) {
             final int lastFinal = last;
             final IntPredicate onCircle = i -> {
                 // circle spanning considered points
                 final Circle circle = new Circle(vertices[lastFinal], vertices[i], hyperplaneThickness);
                 final Arc arc = circle.getArc(vertices[lastFinal], vertices[i]);
                 if (arc.getSize() >= FastMath.PI) {
                     return false;
                 }
                 final int end = lastFinal < i ? i : i + n;
                 for (int k = lastFinal + 1; k < end; k++) {
                     final S2Point vertex = vertices[k % n];
                     if (FastMath.abs(circle.getOffset(vertex)) > hyperplaneThickness ||
                             arc.getOffset(circle.toSubSpace(vertex)) > 0) {
                         // point is not within the thickness or arc, start new edge
                         return false;
                     }
                 }
                 return true;
             };
             j = searchHelper(onCircle, j, first + 1);
             if (j >= first) {
                 break;
             }
             last = j;
             points.add(vertices[last]);
         }
         // put first point last
         final S2Point swap = points.remove(0);
         points.add(swap);
         return points;
     }
 
     /**
      * Search {@code items} for the first item where {@code predicate} is false between
      * {@code a} and {@code b}. Assumes that predicate switches from true to false at
      * exactly one location in [a, b]. Similar to {@link Arrays#binarySearch(int[], int,
      * int, int)} except that 1. it operates on indices, not elements, 2. there is not a
      * shortcut for equality, and 3. it is optimized for cases where the return value is
      * close to a.
      *
      * <p> This method achieves O(lg n) performance in the worst case, where n = b - a.
      * Performance improves to O(lg(i-a)) when i is close to a, where i is the return
      * value.
      *
      * @param predicate to apply.
      * @param a         start, inclusive.
      * @param b         end, exclusive.
      * @return a if a==b, a-1 if predicate.test(a) == false, b - 1 if predicate.test(b-1),
      * otherwise i s.t. predicate.test(i) == true && predicate.test(i + 1) == false.
      * @throws MathIllegalArgumentException if a > b.
      */
     private static int searchHelper(final IntPredicate predicate,
                                     final int a,
                                     final int b) {
         if (a > b) {
             throw new MathIllegalArgumentException(
                     LocalizedCoreFormats.LOWER_ENDPOINT_ABOVE_UPPER_ENDPOINT, a, b);
         }
         // Argument checks and special cases
         if (a == b) {
             return a;
         }
         if (!predicate.test(a)) {
             return a - 1;
         }
 
         // start with exponential search
         int start = a;
         int end = b;
         for (int i = 2; a + i < b; i *= 2) {
             if (predicate.test(a + i)) {
                 // update lower bound of search
                 start = a + i;
             } else {
                 // found upper bound of search
                 end = a + i;
                 break;
             }
         }
 
         // next binary search
         // copied from Arrays.binarySearch() and modified to work on indices alone
         int low = start;
         int high = end - 1;
         while (low <= high) {
             final int mid = (low + high) >>> 1;
             if (predicate.test(mid)) {
                 low = mid + 1;
             } else {
                 high = mid - 1;
             }
         }
         // low is now insertion point, according to Arrays.binarySearch()
         return low - 1;
     }
 
     /** 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<>();
         final List<Edge> 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<Sphere2D> 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<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.
      * <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 EdgesWithNodeInfoBuilder visitor = new EdgesWithNodeInfoBuilder(getTolerance());
                 getTree(true).visit(visitor);
                 final List<EdgeWithNodeInfo> edges = visitor.getEdges();
 
                 // connect all edges, using topological criteria first
                 // and using Euclidean distance only as a last resort
                 int pending = edges.size();
                 pending -= naturalFollowerConnections(edges);
                 if (pending > 0) {
                     pending -= splitEdgeConnections(edges);
                 }
                 if (pending > 0) {
                     closeVerticesConnections(edges);
                 }
 
                 // extract the edges loops
                 loops = new ArrayList<>();
                 for (EdgeWithNodeInfo s = getUnprocessed(edges); s != null; s = getUnprocessed(edges)) {
                     loops.add(s.getStart());
                     followLoop(s);
                 }
 
             }
         }
 
         return Collections.unmodifiableList(loops);
 
     }
 
     /** Connect the edges using only natural follower information.
      * @param edges edges complete edges list
      * @return number of connections performed
      */
     private int naturalFollowerConnections(final List<EdgeWithNodeInfo> edges) {
         int connected = 0;
         for (final EdgeWithNodeInfo edge : edges) {
             if (edge.getEnd().getOutgoing() == null) {
                 for (final EdgeWithNodeInfo candidateNext : edges) {
                     if (EdgeWithNodeInfo.areNaturalFollowers(edge, candidateNext)) {
                         // connect the two edges
                         edge.setNextEdge(candidateNext);
                         ++connected;
                         break;
                     }
                 }
             }
         }
         return connected;
     }
 
     /** Connect the edges resulting from a circle splitting a circular edge.
      * @param edges edges complete edges list
      * @return number of connections performed
      */
     private int splitEdgeConnections(final List<EdgeWithNodeInfo> edges) {
         int connected = 0;
         for (final EdgeWithNodeInfo edge : edges) {
             if (edge.getEnd().getOutgoing() == null) {
                 for (final EdgeWithNodeInfo candidateNext : edges) {
                     if (EdgeWithNodeInfo.resultFromASplit(edge, candidateNext)) {
                         // connect the two edges
                         edge.setNextEdge(candidateNext);
                         ++connected;
                         break;
                     }
                 }
             }
         }
         return connected;
     }
 
     /** Connect the edges using spherical distance.
      * <p>
      * This connection heuristic should be used last, as it relies
      * only on a fuzzy distance criterion.
      * </p>
      * @param edges edges complete edges list
      * @return number of connections performed
      */
     private int closeVerticesConnections(final List<EdgeWithNodeInfo> edges) {
         int connected = 0;
         for (final EdgeWithNodeInfo edge : edges) {
             if (edge.getEnd().getOutgoing() == null && edge.getEnd() != null) {
                 final Vector3D end = edge.getEnd().getLocation().getVector();
                 EdgeWithNodeInfo selectedNext = null;
                 double min = Double.POSITIVE_INFINITY;
                 for (final EdgeWithNodeInfo candidateNext : edges) {
                     if (candidateNext.getStart().getIncoming() == null) {
                         final double distance = Vector3D.distance(end, candidateNext.getStart().getLocation().getVector());
                         if (distance < min) {
                             selectedNext = candidateNext;
                             min          = distance;
                         }
                     }
                 }
                 if (min <= getTolerance()) {
                     // connect the two edges
                     edge.setNextEdge(selectedNext);
                     ++connected;
                 }
             }
         }
         return connected;
     }
 
     /** Get first unprocessed edge from a list.
      * @param edges edges list
      * @return first edge that has not been processed yet
      * or null if all edges have been processed
      */
     private EdgeWithNodeInfo getUnprocessed(final List<EdgeWithNodeInfo> edges) {
         for (final EdgeWithNodeInfo edge : edges) {
             if (!edge.isProcessed()) {
                 return edge;
             }
         }
         return null;
     }
 
     /** Build the loop containing a edge.
      * <p>
      * All edges put in the loop will be marked as processed.
      * </p>
      * @param defining edge used to define the loop
      */
     private void followLoop(final EdgeWithNodeInfo defining) {
 
         defining.setProcessed(true);
 
         // process edges in connection order
         EdgeWithNodeInfo previous = defining;
         EdgeWithNodeInfo next     = (EdgeWithNodeInfo) defining.getEnd().getOutgoing();
         while (next != defining) {
             if (next == null) {
                 // this should not happen
                 throw new MathIllegalStateException(LocalizedGeometryFormats.OUTLINE_BOUNDARY_LOOP_OPEN);
             }
             next.setProcessed(true);
 
             // filter out spurious vertices
             if (Vector3D.angle(previous.getCircle().getPole(), next.getCircle().getPole()) <= Precision.EPSILON) {
                 // the vertex between the two edges is a spurious one
                 // replace the two edges by a single one
                 previous.setNextEdge(next.getEnd().getOutgoing());
                 previous.setLength(previous.getLength() + next.getLength());
             }
 
             previous = next;
             next     = (EdgeWithNodeInfo) next.getEnd().getOutgoing();
 
         }
 
     }
 
     /** 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.hipparchus.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&lt;Sphere2D, S2Point&gt; 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.hipparchus.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<>(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<>(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<>(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]);
         }
 
         return new EnclosingBall<>(new S2Point(enclosing3D.getCenter()),
                                    FastMath.acos((1 + h * h - r * r) / (2 * h)),
                                    support);
 
 
     }
 
     /** 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.
      * @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();
     }
 
 }