/// <summary>
/// Sets the tags of the DCEL mesh component and linked components to the given tag value.
/// </summary>
/// <param name="vertex">The vertex. (Can be <see langword="null"/>.)</param>
/// <param name="tag">The tag.</param>
/// <remarks>
/// This method does nothing if <paramref name="vertex"/> is already tagged with the given
/// value.
/// </remarks>
private static void TagLinkedComponents(DcelVertex vertex, int tag)
{
if (vertex != null && vertex.Tag != tag)
{
vertex.Tag = tag;
TagLinkedComponents(vertex.Edge, tag);
}
}
/// <summary>
/// Sets the tags of the DCEL mesh component and linked components to the given tag value.
/// </summary>
/// <param name="vertex">The vertex. (Can be <see langword="null"/>.)</param>
/// <param name="tag">The tag.</param>
/// <remarks>
/// This method does nothing if <paramref name="vertex"/> is already tagged with the given
/// value.
/// </remarks>
private static void TagLinkedComponents(DcelVertex vertex, int tag)
{
if (vertex != null && vertex.Tag != tag)
{
vertex.Tag = tag;
TagLinkedComponents(vertex.Edge, tag);
}
}
/// <summary>
/// Converts this mesh to a <see cref="TriangleMesh"/>.
/// </summary>
/// <returns>A triangle mesh that represents the same mesh.</returns>
/// <remarks>
/// This mesh must consist only of faces that are convex polygons (triangles, quads, etc.).
/// Concave faces will not be triangulated correctly.
/// </remarks>
/// <exception cref="GeometryException">The DCEL mesh is invalid.</exception>
public TriangleMesh ToTriangleMesh()
{
TriangleMesh triMesh = new TriangleMesh();
UpdateCache();
int numberOfVertices = Vertices.Count;
int numberOfFaces = Faces.Count;
for (int i = 0; i < numberOfVertices; i++)
{
DcelVertex vertex = Vertices[i];
// Store the vertex indices in the InternalTags of the vertices.
vertex.InternalTag = i;
// Add all vertices to the triangle mesh.
triMesh.Vertices.Add(vertex.Position);
}
for (int i = 0; i < numberOfFaces; i++)
{
var face = Faces[i];
var startEdge = face.Boundary;
var nextEdge = startEdge.Next;
// Tag edges to see if they have been visited.
startEdge.InternalTag = 1;
nextEdge.InternalTag = 1;
// Triangulate polygon.
var v0 = startEdge.Origin;
var v1 = nextEdge.Origin;
nextEdge = nextEdge.Next;
while (nextEdge != startEdge)
{
if (nextEdge == null || nextEdge.InternalTag == 1)
{
throw new GeometryException("DCEL mesh is invalid.");
}
nextEdge.InternalTag = 1;
var v2 = nextEdge.Origin;
// The tags of the vertices store their index number.
triMesh.Indices.Add(v0.InternalTag);
triMesh.Indices.Add(v1.InternalTag);
triMesh.Indices.Add(v2.InternalTag);
v1 = v2;
nextEdge = nextEdge.Next;
}
}
ResetInternalTags();
return(triMesh);
}
/// <summary>
/// Initializes a new instance of the <see cref="VertexAdjacency"/> class.
/// </summary>
/// <param name="mesh">The mesh for which the adjacency information is built.</param>
/// <exception cref="NotSupportedException">
/// Too many vertices in convex hull. Max. 65534 vertices in convex hull are supported.
/// </exception>
public VertexAdjacency(DcelMesh mesh)
{
if (mesh == null)
{
throw new ArgumentNullException("mesh");
}
int numberOfVertices = mesh.Vertices.Count;
if (numberOfVertices >= ushort.MaxValue)
{
throw new NotSupportedException("Too many vertices in convex hull. Max. 65534 vertices in convex hull are supported.");
}
Debug.Assert(mesh.Vertices.All(v => v.Tag == 0), "Tags of DcelMesh should be cleared.");
// Store the index of each vertex in its tag for fast lookup.
for (int i = 0; i < numberOfVertices; i++)
{
mesh.Vertices[i].Tag = i;
}
ListIndices = new ushort[numberOfVertices + 1];
List <ushort> adjacencyLists = new List <ushort>(numberOfVertices * 4);
for (int i = 0; i < numberOfVertices; i++)
{
DcelVertex vertex = mesh.Vertices[i];
ListIndices[i] = (ushort)adjacencyLists.Count;
// Gather all adjacent vertices.
DcelEdge startEdge = vertex.Edge;
DcelEdge edge = startEdge;
do
{
DcelVertex adjacentVertex = edge.Next.Origin;
int adjacentIndex = adjacentVertex.Tag;
Debug.Assert(mesh.Vertices[adjacentIndex] == adjacentVertex, "Indices stored in DcelVertex.Tag are invalid.");
adjacencyLists.Add((ushort)adjacentIndex);
edge = edge.Twin.Next;
} while (edge != startEdge);
}
// Add one additional entry which determines the end of the last adjacency list.
ListIndices[numberOfVertices] = (ushort)adjacencyLists.Count;
// Copy adjacency lists into static array.
Lists = adjacencyLists.ToArray();
}
internal void MergeCoplanarFaces()
{
UpdateCache();
foreach (DcelEdge edge in _edges)
{
if (edge.InternalTag == 1)
{
// Edge has already been visited.
continue;
}
DcelEdge twin = edge.Twin;
DcelFace face0 = edge.Face;
DcelFace face1 = twin.Face;
if (face0 != null && face1 != null)
{
// Compare face normals.
Vector3F normal0 = face0.Normal;
Vector3F normal1 = face1.Normal;
float cosĪ± = Vector3F.Dot(normal0, normal1) / (normal0.Length * normal1.Length);
if (Numeric.AreEqual(cosĪ±, 1))
{
// Faces are coplanar:
// --> Merge faces and remove edge.
// Get vertices and edges.
DcelVertex vertex0 = edge.Origin; // Start vertex of edge.
DcelVertex vertex1 = twin.Origin; // End vertex of edge.
DcelEdge vertex0Incoming = edge.Previous;
DcelEdge vertex0Outgoing = twin.Next;
DcelEdge vertex1Incoming = twin.Previous;
DcelEdge vertex1Outgoing = edge.Next;
// Update vertices.
vertex0.Edge = vertex0Outgoing;
vertex1.Edge = vertex1Outgoing;
// Update edges.
vertex0Incoming.Next = vertex0Outgoing;
vertex0Outgoing.Previous = vertex0Incoming;
vertex1Incoming.Next = vertex1Outgoing;
vertex1Outgoing.Previous = vertex1Incoming;
// Throw away face1. Make sure that all edges point to face0.
DcelEdge current = vertex0Outgoing;
do
{
current.Face = face0;
current = current.Next;
} while (current != vertex0Outgoing);
Dirty = true;
}
// Mark edges as visited.
edge.InternalTag = 1;
twin.InternalTag = 1;
}
}
ResetTags();
}
/// <summary>
/// Creates a mesh for unit cube.
/// </summary>
/// <returns>The DCEL mesh that represent a unit cube.</returns>
/// <remarks>
/// The cube is centered at the origin and has 6 faces. The edge length is 2.
/// </remarks>
public static DcelMesh CreateCube()
{
#region ----- Vertices -----
var vertex0 = new DcelVertex(new Vector3F(-1, -1, -1), null);
var vertex1 = new DcelVertex(new Vector3F(1, -1, -1), null);
var vertex2 = new DcelVertex(new Vector3F(-1, 1, -1), null);
var vertex3 = new DcelVertex(new Vector3F(1, 1, -1), null);
var vertex4 = new DcelVertex(new Vector3F(-1, -1, 1), null);
var vertex5 = new DcelVertex(new Vector3F(1, -1, 1), null);
var vertex6 = new DcelVertex(new Vector3F(-1, 1, 1), null);
var vertex7 = new DcelVertex(new Vector3F(1, 1, 1), null);
#endregion
#region ----- Faces -----
var near = new DcelFace(); // +z face
var far = new DcelFace(); // -z face
var top = new DcelFace(); // +y face
var bottom = new DcelFace(); // -y face
var left = new DcelFace(); // -x face
var right = new DcelFace(); // +x face
#endregion
#region ----- Edges -----
var edge01 = new DcelEdge {
Origin = vertex0, Face = bottom
};
var edge10 = new DcelEdge {
Origin = vertex1, Face = far
};
var edge13 = new DcelEdge {
Origin = vertex1, Face = right
};
var edge31 = new DcelEdge {
Origin = vertex3, Face = far
};
var edge23 = new DcelEdge {
Origin = vertex2, Face = far
};
var edge32 = new DcelEdge {
Origin = vertex3, Face = top
};
var edge02 = new DcelEdge {
Origin = vertex0, Face = far
};
var edge20 = new DcelEdge {
Origin = vertex2, Face = left
};
var edge26 = new DcelEdge {
Origin = vertex2, Face = top
};
var edge62 = new DcelEdge {
Origin = vertex6, Face = left
};
var edge37 = new DcelEdge {
Origin = vertex3, Face = right
};
var edge73 = new DcelEdge {
Origin = vertex7, Face = top
};
var edge04 = new DcelEdge {
Origin = vertex0, Face = left
};
var edge40 = new DcelEdge {
Origin = vertex4, Face = bottom
};
var edge15 = new DcelEdge {
Origin = vertex1, Face = bottom
};
var edge51 = new DcelEdge {
Origin = vertex5, Face = right
};
var edge45 = new DcelEdge {
Origin = vertex4, Face = near
};
var edge54 = new DcelEdge {
Origin = vertex5, Face = bottom
};
var edge57 = new DcelEdge {
Origin = vertex5, Face = near
};
var edge75 = new DcelEdge {
Origin = vertex7, Face = right
};
var edge46 = new DcelEdge {
Origin = vertex4, Face = left
};
var edge64 = new DcelEdge {
Origin = vertex6, Face = near
};
var edge67 = new DcelEdge {
Origin = vertex6, Face = top
};
var edge76 = new DcelEdge {
Origin = vertex7, Face = near
};
#endregion
#region ----- Set DcelVertex.Edge -----
vertex0.Edge = edge01;
vertex1.Edge = edge10;
vertex2.Edge = edge20;
vertex3.Edge = edge31;
vertex4.Edge = edge40;
vertex5.Edge = edge57;
vertex6.Edge = edge67;
vertex7.Edge = edge76;
#endregion
#region ----- Set DcelFace.Boundary -----
near.Boundary = edge57;
far.Boundary = edge10;
left.Boundary = edge62;
right.Boundary = edge51;
top.Boundary = edge67;
bottom.Boundary = edge01;
#endregion
#region ----- Set DcelEdge.Twin -----
edge01.Twin = edge10; edge10.Twin = edge01;
edge13.Twin = edge31; edge31.Twin = edge13;
edge23.Twin = edge32; edge32.Twin = edge23;
edge02.Twin = edge20; edge20.Twin = edge02;
edge26.Twin = edge62; edge62.Twin = edge26;
edge37.Twin = edge73; edge73.Twin = edge37;
edge04.Twin = edge40; edge40.Twin = edge04;
edge15.Twin = edge51; edge51.Twin = edge15;
edge45.Twin = edge54; edge54.Twin = edge45;
edge57.Twin = edge75; edge75.Twin = edge57;
edge46.Twin = edge64; edge64.Twin = edge46;
edge67.Twin = edge76; edge76.Twin = edge67;
#endregion
#region ----- Set DcelEdge.Next/Previous -----
edge10.Next = edge02; edge02.Next = edge23; edge23.Next = edge31; edge31.Next = edge10; // far
edge02.Previous = edge10; edge23.Previous = edge02; edge31.Previous = edge23; edge10.Previous = edge31;
edge45.Next = edge57; edge57.Next = edge76; edge76.Next = edge64; edge64.Next = edge45; // near
edge57.Previous = edge45; edge76.Previous = edge57; edge64.Previous = edge76; edge45.Previous = edge64;
edge62.Next = edge20; edge20.Next = edge04; edge04.Next = edge46; edge46.Next = edge62; // left
edge20.Previous = edge62; edge04.Previous = edge20; edge46.Previous = edge04; edge62.Previous = edge46;
edge51.Next = edge13; edge13.Next = edge37; edge37.Next = edge75; edge75.Next = edge51; // right
edge13.Previous = edge51; edge37.Previous = edge13; edge75.Previous = edge37; edge51.Previous = edge75;
edge67.Next = edge73; edge73.Next = edge32; edge32.Next = edge26; edge26.Next = edge67; // top
edge73.Previous = edge67; edge32.Previous = edge73; edge26.Previous = edge32; edge67.Previous = edge26;
edge01.Next = edge15; edge15.Next = edge54; edge54.Next = edge40; edge40.Next = edge01; // bottom
edge15.Previous = edge01; edge54.Previous = edge15; edge40.Previous = edge54; edge01.Previous = edge40;
#endregion
return(new DcelMesh {
Vertex = vertex0
});
}
/// <summary>
/// Creates a mesh for unit cube.
/// </summary>
/// <returns>The DCEL mesh that represent a unit cube.</returns>
/// <remarks>
/// The cube is centered at the origin and has 6 faces. The edge length is 2.
/// </remarks>
public static DcelMesh CreateCube()
{
#region ----- Vertices -----
var vertex0 = new DcelVertex(new Vector3F(-1, -1, -1), null);
var vertex1 = new DcelVertex(new Vector3F( 1, -1, -1), null);
var vertex2 = new DcelVertex(new Vector3F(-1, 1, -1), null);
var vertex3 = new DcelVertex(new Vector3F( 1, 1, -1), null);
var vertex4 = new DcelVertex(new Vector3F(-1, -1, 1), null);
var vertex5 = new DcelVertex(new Vector3F( 1, -1, 1), null);
var vertex6 = new DcelVertex(new Vector3F(-1, 1, 1), null);
var vertex7 = new DcelVertex(new Vector3F( 1, 1, 1), null);
#endregion
#region ----- Faces -----
var near = new DcelFace(); // +z face
var far = new DcelFace(); // -z face
var top = new DcelFace(); // +y face
var bottom = new DcelFace(); // -y face
var left = new DcelFace(); // -x face
var right = new DcelFace(); // +x face
#endregion
#region ----- Edges -----
var edge01 = new DcelEdge { Origin = vertex0, Face = bottom };
var edge10 = new DcelEdge { Origin = vertex1, Face = far };
var edge13 = new DcelEdge { Origin = vertex1, Face = right };
var edge31 = new DcelEdge { Origin = vertex3, Face = far };
var edge23 = new DcelEdge { Origin = vertex2, Face = far };
var edge32 = new DcelEdge { Origin = vertex3, Face = top };
var edge02 = new DcelEdge { Origin = vertex0, Face = far };
var edge20 = new DcelEdge { Origin = vertex2, Face = left };
var edge26 = new DcelEdge { Origin = vertex2, Face = top };
var edge62 = new DcelEdge { Origin = vertex6, Face = left };
var edge37 = new DcelEdge { Origin = vertex3, Face = right };
var edge73 = new DcelEdge { Origin = vertex7, Face = top };
var edge04 = new DcelEdge { Origin = vertex0, Face = left };
var edge40 = new DcelEdge { Origin = vertex4, Face = bottom };
var edge15 = new DcelEdge { Origin = vertex1, Face = bottom };
var edge51 = new DcelEdge { Origin = vertex5, Face = right };
var edge45 = new DcelEdge { Origin = vertex4, Face = near };
var edge54 = new DcelEdge { Origin = vertex5, Face = bottom };
var edge57 = new DcelEdge { Origin = vertex5, Face = near };
var edge75 = new DcelEdge { Origin = vertex7, Face = right };
var edge46 = new DcelEdge { Origin = vertex4, Face = left };
var edge64 = new DcelEdge { Origin = vertex6, Face = near };
var edge67 = new DcelEdge { Origin = vertex6, Face = top };
var edge76 = new DcelEdge { Origin = vertex7, Face = near };
#endregion
#region ----- Set DcelVertex.Edge -----
vertex0.Edge = edge01;
vertex1.Edge = edge10;
vertex2.Edge = edge20;
vertex3.Edge = edge31;
vertex4.Edge = edge40;
vertex5.Edge = edge57;
vertex6.Edge = edge67;
vertex7.Edge = edge76;
#endregion
#region ----- Set DcelFace.Boundary -----
near.Boundary = edge57;
far.Boundary = edge10;
left.Boundary = edge62;
right.Boundary = edge51;
top.Boundary = edge67;
bottom.Boundary = edge01;
#endregion
#region ----- Set DcelEdge.Twin -----
edge01.Twin = edge10; edge10.Twin = edge01;
edge13.Twin = edge31; edge31.Twin = edge13;
edge23.Twin = edge32; edge32.Twin = edge23;
edge02.Twin = edge20; edge20.Twin = edge02;
edge26.Twin = edge62; edge62.Twin = edge26;
edge37.Twin = edge73; edge73.Twin = edge37;
edge04.Twin = edge40; edge40.Twin = edge04;
edge15.Twin = edge51; edge51.Twin = edge15;
edge45.Twin = edge54; edge54.Twin = edge45;
edge57.Twin = edge75; edge75.Twin = edge57;
edge46.Twin = edge64; edge64.Twin = edge46;
edge67.Twin = edge76; edge76.Twin = edge67;
#endregion
#region ----- Set DcelEdge.Next/Previous -----
edge10.Next = edge02; edge02.Next = edge23; edge23.Next = edge31; edge31.Next = edge10; // far
edge02.Previous = edge10; edge23.Previous = edge02; edge31.Previous = edge23; edge10.Previous = edge31;
edge45.Next = edge57; edge57.Next = edge76; edge76.Next = edge64; edge64.Next = edge45; // near
edge57.Previous = edge45; edge76.Previous = edge57; edge64.Previous = edge76; edge45.Previous = edge64;
edge62.Next = edge20; edge20.Next = edge04; edge04.Next = edge46; edge46.Next = edge62; // left
edge20.Previous = edge62; edge04.Previous = edge20; edge46.Previous = edge04; edge62.Previous = edge46;
edge51.Next = edge13; edge13.Next = edge37; edge37.Next = edge75; edge75.Next = edge51; // right
edge13.Previous = edge51; edge37.Previous = edge13; edge75.Previous = edge37; edge51.Previous = edge75;
edge67.Next = edge73; edge73.Next = edge32; edge32.Next = edge26; edge26.Next = edge67; // top
edge73.Previous = edge67; edge32.Previous = edge73; edge26.Previous = edge32; edge67.Previous = edge26;
edge01.Next = edge15; edge15.Next = edge54; edge54.Next = edge40; edge40.Next = edge01; // bottom
edge15.Previous = edge01; edge54.Previous = edge15; edge40.Previous = edge54; edge01.Previous = edge40;
#endregion
return new DcelMesh { Vertex = vertex0 };
}
/// <summary>
/// Cuts mesh with a plane.
/// </summary>
/// <param name="plane">The plane.</param>
/// <returns>
/// <see langword="true"/> if the mesh was cut; otherwise, <see langword="false"/> if the mesh
/// was not modified.
/// </returns>
/// <remarks>
/// <para>
/// The mesh is cut with the given plane. If any parts are in front of the plane, they are
/// removed. Edges and faces that go through the plane are cut. The resulting mesh is closed
/// with a new face in the cut plane.
/// </para>
/// <para>
/// If the whole mesh is in front of the plane, the whole mesh is removed (<see cref="Vertex"/>
/// is set to <see langword="null"/>). If the whole mesh is behind the plane, this method does
/// nothing.
/// </para>
/// <para>
/// <strong>Prerequisites:</strong> This operation uses <see cref="DcelVertex.UserData"/> of the
/// vertices and it assumes that the mesh is valid, convex and closed. All faces must be
/// triangles or convex polygons.
/// </para>
/// </remarks>
public bool CutConvex(Plane plane)
{
// Cuts the mesh with the given plane. The part over the plane is removed.
// Prerequisites:
// - Mesh is valid, convex and closed.
// - Faces are convex polygons.
// - Vertex user data is null.
// Notes:
// - Vertex user data is used.
// Get AABB of whole mesh.
var aabb = GetAabb();
// Create an epsilon relative to the mesh size.
float epsilon = Numeric.EpsilonF * (1 + aabb.Extent.Length);
// Store distance d to plane in all vertices. d is negative if vertex is below the plane.
float minDistance = Single.PositiveInfinity;
float maxDistance = Single.NegativeInfinity;
foreach (var vertex in Vertices)
{
float d = Vector3F.Dot(vertex.Position, plane.Normal) - plane.DistanceFromOrigin;
vertex.UserData = d;
minDistance = Math.Min(d, minDistance);
maxDistance = Math.Max(d, maxDistance);
}
if (minDistance > -epsilon)
{
// All vertices are in or above the plane. - The whole mesh is cut away.
Vertex = null;
Dirty = true;
return true;
}
if (maxDistance < epsilon)
{
// All vertices are in or under the plane. - The mesh is not cut.
return false;
}
// Now, we know that the mesh must be cut.
// Find a first edge that starts under/in the plane and must be cut.
DcelEdge first = null;
foreach (var edge in Edges)
{
var startDistance = (float)edge.Origin.UserData;
var endDistance = (float)edge.Twin.Origin.UserData;
if (startDistance < -epsilon // The edge starts below the plane and the start vertex is definitely not on the plane.
&& endDistance > -epsilon) // The edge ends in or above the plane.
{
first = edge;
break;
}
}
Debug.Assert(first != null, "The mesh is cut by the plane, but could not find an edge that is cut!?");
// The cut will be sealed with this face.
DcelFace cap = new DcelFace();
var outEdge = first; // Edge that goes out of the plane and has to be cut.
DcelVertex newVertex = new DcelVertex();
newVertex.Position = GetCutPosition(outEdge);
newVertex.Edge = first.Twin;
do
{
// Find inEdge (edge that goes into the plane and must be cut).
var inEdge = outEdge.Next;
while ((float)inEdge.Twin.Origin.UserData > -epsilon) // Loop until the edge end is definitely under the plane.
inEdge = inEdge.Next;
// The face will be cut between outEdge and inEdge.
// Two new half edges:
var newEdge = new DcelEdge(); // Edge for the cut face.
var capEdge = new DcelEdge(); // Twin edge on the cap.
// Update outEdge.
outEdge.Next = newEdge;
// Init newEdge.
newEdge.Face = outEdge.Face;
newEdge.Origin = newVertex;
newEdge.Previous = outEdge;
newEdge.Next = inEdge;
newEdge.Twin = capEdge;
// Find next newVertex on inEdge.
if (inEdge.Twin != first)
{
// Find cut on inEdge.
newVertex = new DcelVertex();
newVertex.Position = GetCutPosition(inEdge);
newVertex.Edge = inEdge;
}
else
{
// We have come full circle. The inEdge is the twin of the first outEdge.
newVertex = first.Next.Origin;
}
// Init capEdge.
capEdge.Face = cap;
capEdge.Origin = newVertex;
capEdge.Twin = newEdge;
if (cap.Boundary == null)
{
cap.Boundary = capEdge;
}
else
{
capEdge.Next = outEdge.Twin.Previous.Twin;
capEdge.Next.Previous = capEdge;
}
// Update inEdge.
inEdge.Origin = newVertex;
inEdge.Previous = newEdge;
// Make sure the cut face does not link to a removed edge.
outEdge.Face.Boundary = outEdge;
// The next outEdge is the twin of the inEdge.
outEdge = inEdge.Twin;
} while (outEdge != first);
// We still have to link the first and the last cap edge.
var firstCapEdge = cap.Boundary;
var lastCapEdge = first.Twin.Previous.Twin;
firstCapEdge.Next = lastCapEdge;
lastCapEdge.Previous = firstCapEdge;
// Make sure DcelMesh.Vertex is not one of the removed vertices.
Vertex = newVertex;
Dirty = true;
// Clear vertex user data.
foreach (var vertex in Vertices)
vertex.UserData = null;
return true;
}
/// <summary>
/// Cuts mesh with a plane.
/// </summary>
/// <param name="plane">The plane.</param>
/// <returns>
/// <see langword="true"/> if the mesh was cut; otherwise, <see langword="false"/> if the mesh
/// was not modified.
/// </returns>
/// <remarks>
/// <para>
/// The mesh is cut with the given plane. If any parts are in front of the plane, they are
/// removed. Edges and faces that go through the plane are cut. The resulting mesh is closed
/// with a new face in the cut plane.
/// </para>
/// <para>
/// If the whole mesh is in front of the plane, the whole mesh is removed (<see cref="Vertex"/>
/// is set to <see langword="null"/>). If the whole mesh is behind the plane, this method does
/// nothing.
/// </para>
/// <para>
/// <strong>Prerequisites:</strong> This operation uses <see cref="DcelVertex.UserData"/> of the
/// vertices and it assumes that the mesh is valid, convex and closed. All faces must be
/// triangles or convex polygons.
/// </para>
/// </remarks>
public bool CutConvex(Plane plane)
{
// Cuts the mesh with the given plane. The part over the plane is removed.
// Prerequisites:
// - Mesh is valid, convex and closed.
// - Faces are convex polygons.
// - Vertex user data is null.
// Notes:
// - Vertex user data is used.
// Get AABB of whole mesh.
var aabb = GetAabb();
// Create an epsilon relative to the mesh size.
float epsilon = Numeric.EpsilonF * (1 + aabb.Extent.Length);
// Store distance d to plane in all vertices. d is negative if vertex is below the plane.
float minDistance = Single.PositiveInfinity;
float maxDistance = Single.NegativeInfinity;
foreach (var vertex in Vertices)
{
float d = Vector3F.Dot(vertex.Position, plane.Normal) - plane.DistanceFromOrigin;
vertex.UserData = d;
minDistance = Math.Min(d, minDistance);
maxDistance = Math.Max(d, maxDistance);
}
if (minDistance > -epsilon)
{
// All vertices are in or above the plane. - The whole mesh is cut away.
Vertex = null;
Dirty = true;
return(true);
}
if (maxDistance < epsilon)
{
// All vertices are in or under the plane. - The mesh is not cut.
return(false);
}
// Now, we know that the mesh must be cut.
// Find a first edge that starts under/in the plane and must be cut.
DcelEdge first = null;
foreach (var edge in Edges)
{
var startDistance = (float)edge.Origin.UserData;
var endDistance = (float)edge.Twin.Origin.UserData;
if (startDistance < -epsilon && // The edge starts below the plane and the start vertex is definitely not on the plane.
endDistance > -epsilon) // The edge ends in or above the plane.
{
first = edge;
break;
}
}
Debug.Assert(first != null, "The mesh is cut by the plane, but could not find an edge that is cut!?");
// The cut will be sealed with this face.
DcelFace cap = new DcelFace();
var outEdge = first; // Edge that goes out of the plane and has to be cut.
DcelVertex newVertex = new DcelVertex();
newVertex.Position = GetCutPosition(outEdge);
newVertex.Edge = first.Twin;
do
{
// Find inEdge (edge that goes into the plane and must be cut).
var inEdge = outEdge.Next;
while ((float)inEdge.Twin.Origin.UserData > -epsilon) // Loop until the edge end is definitely under the plane.
{
inEdge = inEdge.Next;
}
// The face will be cut between outEdge and inEdge.
// Two new half edges:
var newEdge = new DcelEdge(); // Edge for the cut face.
var capEdge = new DcelEdge(); // Twin edge on the cap.
// Update outEdge.
outEdge.Next = newEdge;
// Init newEdge.
newEdge.Face = outEdge.Face;
newEdge.Origin = newVertex;
newEdge.Previous = outEdge;
newEdge.Next = inEdge;
newEdge.Twin = capEdge;
// Find next newVertex on inEdge.
if (inEdge.Twin != first)
{
// Find cut on inEdge.
newVertex = new DcelVertex();
newVertex.Position = GetCutPosition(inEdge);
newVertex.Edge = inEdge;
}
else
{
// We have come full circle. The inEdge is the twin of the first outEdge.
newVertex = first.Next.Origin;
}
// Init capEdge.
capEdge.Face = cap;
capEdge.Origin = newVertex;
capEdge.Twin = newEdge;
if (cap.Boundary == null)
{
cap.Boundary = capEdge;
}
else
{
capEdge.Next = outEdge.Twin.Previous.Twin;
capEdge.Next.Previous = capEdge;
}
// Update inEdge.
inEdge.Origin = newVertex;
inEdge.Previous = newEdge;
// Make sure the cut face does not link to a removed edge.
outEdge.Face.Boundary = outEdge;
// The next outEdge is the twin of the inEdge.
outEdge = inEdge.Twin;
} while (outEdge != first);
// We still have to link the first and the last cap edge.
var firstCapEdge = cap.Boundary;
var lastCapEdge = first.Twin.Previous.Twin;
firstCapEdge.Next = lastCapEdge;
lastCapEdge.Previous = firstCapEdge;
// Make sure DcelMesh.Vertex is not one of the removed vertices.
Vertex = newVertex;
Dirty = true;
// Clear vertex user data.
foreach (var vertex in Vertices)
{
vertex.UserData = null;
}
return(true);
}
/// <summary>
/// Merges a point to a convex hull that is a polygon.
/// </summary>
/// <param name="point">The point.</param>
private void GrowPlanarConvex(Vector3F point)
{
Debug.Assert(_mesh != null);
Debug.Assert(_mesh.Vertices.Count > 2, "DCEL mesh with 3 or more vertices expected.");
Debug.Assert(_type == ConvexHullType.Planar, "DCEL mesh with 2 faces expected.");
DcelFace frontFace = _mesh.Vertex.Edge.Face;
DcelFace backFace = frontFace.Boundary.Twin.Face;
// This check: if (!_isPlanar)
// is important! Because for planar input points we could not find an initial
// tetrahedron. All points are in a plane. But sometimes GetCollinearity detects that
// a point is not in the plane and this creates very degenerate spatial convex hulls.
// For real spatial input points we have no problem because we have a good initial
// tetrahedron.
if (!_isPlanar)
{
var collinearity = DcelMesh.GetCollinearity(point, frontFace);
if (collinearity == Collinearity.NotCollinearInFront)
{
// We build a pyramid where the front face is removed.
GrowPlanarConvexToSpatial(frontFace, point);
return;
}
if (collinearity == Collinearity.NotCollinearBehind)
{
// We build a pyramid where the back face is removed.
GrowPlanarConvexToSpatial(backFace, point);
return;
}
}
// ----- The point is in the plane.
// If we get to here, the initial tetrahedron is flat, so we create a flat hull.
_isPlanar = true;
// Get normal of front face.
Vector3F faceNormal = frontFace.Normal / frontFace.Normal.Length;
// Find a visible edge.
DcelEdge currentEdge = _mesh.Vertex.Edge;
DcelEdge visibleEdge = null;
do
{
float distance;
DcelMesh.GetCollinearity(point, currentEdge, faceNormal, out distance);
if (distance >= 0)
{
// Current is visible.
visibleEdge = currentEdge;
break;
}
// Get next edge.
currentEdge = currentEdge.Next;
} while (currentEdge != _mesh.Vertex.Edge);
if (visibleEdge == null)
{
// Point is in the convex hull.
return;
}
// Move the point a bit away to avoid numerical problems.
Vector3F v0 = visibleEdge.Origin.Position;
Vector3F v1 = visibleEdge.Twin.Origin.Position;
Vector3F segment = v1 - v0;
Vector3F normal = Vector3F.Cross(segment, faceNormal);
point += Numeric.EpsilonF * normal;
// Now we need to find the lower vertex and upper vertex between which
// the edges will be removed.
// Two new edges will be added: (lowerVertex, Point) and (Point, upperVertex).
DcelVertex lowerVertex = null;
DcelVertex upperVertex = null;
// Find upperVertex.
currentEdge = visibleEdge.Next;
do
{
float distance;
DcelMesh.GetCollinearity(point, currentEdge, faceNormal, out distance);
if (distance < 0)
{
// Edge is not visible. We have found the upper vertex.
upperVertex = currentEdge.Origin;
break;
}
currentEdge = currentEdge.Next;
} while (currentEdge != visibleEdge.Next);
// upperVertex == null should not happen. But numerical problems can always occur.
// In that case we do nothing.
Debug.Assert(upperVertex != null, "Could not find upperVertex.");
if (upperVertex == null)
return;
// Find lowerVertex
currentEdge = visibleEdge.Previous;
do
{
float distance;
DcelMesh.GetCollinearity(point, currentEdge, faceNormal, out distance);
if (distance < 0)
{
// Edge is not visible. The end point of the edge is the lower vertex.
lowerVertex = currentEdge.Next.Origin;
break;
}
currentEdge = currentEdge.Previous;
} while (currentEdge != visibleEdge.Previous);
// lowerVertex == null should not happen. But numerical problems can always occur.
// In that case we do nothing.
Debug.Assert(lowerVertex != null, "Could not find lowerVertex.");
if (lowerVertex == null)
return;
Debug.Assert(lowerVertex != upperVertex, "upperVertex and lowerVertex should not be identical.");
#if DEBUG
// Get the first visible edge of front face. This edge starts at lowerVertex.
var firstVisibleEdge = lowerVertex.Edge;
if (firstVisibleEdge.Face != frontFace)
firstVisibleEdge = lowerVertex.Edge.Previous.Twin;
// Check if all edges between lowerVertex and upperVertex are visible.
var edge = firstVisibleEdge;
while (edge.Origin != upperVertex)
{
float distance;
DcelMesh.GetCollinearity(point, edge, faceNormal, out distance);
Debug.Assert(distance >= 0);
edge = edge.Next;
}
// Check if all edges between upperVertex and lowerVertex are not visible.
while (edge.Origin != lowerVertex)
{
float distance;
DcelMesh.GetCollinearity(point, edge, faceNormal, out distance);
Debug.Assert(distance < 0);
edge = edge.Next;
}
#endif
// Create a new vertex for the point.
DcelVertex newVertex = new DcelVertex(point, null);
newVertex.Edge = new DcelEdge();
// Add two new edges between lowerVertex and upperVertex.
DcelEdge lowerEdge = new DcelEdge();
DcelEdge lowerEdgeTwin = new DcelEdge();
DcelEdge upperEdge = newVertex.Edge;
DcelEdge upperEdgeTwin = new DcelEdge();
lowerEdge.Origin = lowerVertex;
lowerEdge.Face = frontFace;
lowerEdge.Next = upperEdge;
lowerEdge.Twin = lowerEdgeTwin;
lowerEdgeTwin.Origin = newVertex;
lowerEdgeTwin.Face = backFace;
lowerEdgeTwin.Previous = upperEdgeTwin;
lowerEdgeTwin.Twin = lowerEdge;
upperEdge.Origin = newVertex;
upperEdge.Face = frontFace;
upperEdge.Previous = lowerEdge;
upperEdge.Twin = upperEdgeTwin;
upperEdgeTwin.Origin = upperVertex;
upperEdgeTwin.Face = backFace;
upperEdgeTwin.Next = lowerEdgeTwin;
upperEdgeTwin.Twin = upperEdge;
lowerEdge.Previous = (lowerVertex.Edge.Face == frontFace) ? lowerVertex.Edge.Previous : lowerVertex.Edge.Twin;
upperEdge.Next = (upperVertex.Edge.Face == frontFace) ? upperVertex.Edge : upperVertex.Edge.Twin.Next;
lowerEdgeTwin.Next = lowerEdge.Previous.Twin;
upperEdgeTwin.Previous = upperEdge.Next.Twin;
// Correct faces.
frontFace.Boundary = currentEdge; // The last currentEdge was not visible, so it will stay.
backFace.Boundary = currentEdge.Twin;
// Correct lower and upper vertex.
lowerVertex.Edge = lowerEdge;
upperVertex.Edge = upperEdge.Next;
// Correct edges before lowerEdge and after upperEdge.
lowerEdge.Previous.Next = lowerEdge;
lowerEdgeTwin.Next.Previous = lowerEdgeTwin;
upperEdge.Next.Previous = upperEdge;
upperEdgeTwin.Previous.Next = upperEdgeTwin;
// _mesh.Vertex could be one of the removed vertices. --> Set a vertex that is in the hull.
_mesh.Vertex = lowerVertex;
_mesh.Dirty = true;
Debug.Assert(_mesh.IsTwoSidedPolygon(), "DCEL mesh should be a two-sided polygon.");
}
/// <summary>
/// Merges a point to a convex hull that is a line segment.
/// </summary>
/// <param name="point">The point.</param>
private void GrowLinearConvex(Vector3F point)
{
Debug.Assert(_mesh != null);
Debug.Assert(_mesh.Vertices.Count == 2, "DCEL mesh with 2 vertices was expected.");
Debug.Assert(_type == ConvexHullType.Linear);
Debug.Assert(!Vector3F.AreNumericallyEqual(_mesh.Vertex.Position, point));
Debug.Assert(!Vector3F.AreNumericallyEqual(_mesh.Vertices[1].Position, point));
// Abort if the point is equal to an existing point.
if (Vector3F.AreNumericallyEqual(point, _mesh.Vertex.Position)
|| Vector3F.AreNumericallyEqual(point, _mesh.Vertex.Edge.Twin.Origin.Position))
return;
var edge = _mesh.Vertex.Edge;
var collinearity = DcelMesh.GetCollinearity(point, _mesh.Vertex.Edge);
switch (collinearity)
{
case Collinearity.CollinearBefore: // The point is the new start of the line segment.
edge.Origin.Position = point;
_mesh.Dirty = true;
return;
case Collinearity.CollinearAfter: // The point is the new end of the line segment.
edge.Twin.Origin.Position = point;
_mesh.Dirty = true;
return;
case Collinearity.CollinearContained: // The point is in the line segment.
// Point is in convex hull.
return;
}
Debug.Assert(collinearity == Collinearity.NotCollinear
|| collinearity == Collinearity.NotCollinearBehind
|| collinearity == Collinearity.NotCollinearInFront);
// Not Collinear --> Make triangle.
_type = ConvexHullType.Planar;
// Create new components.
DcelVertex v0 = _mesh.Vertex;
DcelVertex v1 = edge.Twin.Origin;
DcelVertex v2 = new DcelVertex(point, null);
DcelEdge e01 = edge;
DcelEdge e10 = e01.Twin;
DcelEdge e12 = new DcelEdge();
DcelEdge e21 = new DcelEdge();
DcelEdge e02 = new DcelEdge();
DcelEdge e20 = new DcelEdge();
DcelFace f012 = new DcelFace();
DcelFace f210 = new DcelFace();
// Link with existing components.
e01.Next = e12; e01.Previous = e20;
e10.Next = e02; e10.Previous = e21;
// Link new components.
v2.Edge = e20;
e12.Origin = v1;
e21.Origin = v2;
e20.Origin = v2;
e02.Origin = v0;
e12.Twin = e21;
e21.Twin = e12;
e20.Twin = e02;
e02.Twin = e20;
e12.Previous = e01;
e21.Previous = e02;
e20.Previous = e12;
e02.Previous = e10;
e12.Next = e20;
e21.Next = e10;
e20.Next = e01;
e02.Next = e21;
// Link face.
f012.Boundary = e01;
f210.Boundary = e21;
e01.Face = e12.Face = e20.Face = f012;
e21.Face = e10.Face = e02.Face = f210;
// Update _faces.
_faces.Add(f012);
_faces.Add(f210);
_mesh.Dirty = true;
Debug.Assert(_mesh.Faces.Count == _faces.Count, "Internal error in ConvexHullBuilder.");
Debug.Assert(_mesh.IsTriangleMesh(), "DCEL mesh should be a triangle mesh.");
Debug.Assert(_mesh.IsTwoSidedPolygon(), "DCEL mesh should be a two-sided polygon.");
}
/// <summary>
/// Merges a point to a 3D convex hull.
/// </summary>
/// <param name="point">The point.</param>
/// <param name="startFace">
/// A face which is visible from the given point. Can be <see langword="null"/>.
/// </param>
/// <returns><see langword="true"/> if the convex hull was grown.</returns>
private bool GrowSpatialConvex(Vector3F point, DcelFace startFace)
{
Debug.Assert(_mesh != null);
Debug.Assert(_mesh.Vertices.Count > 3, "DCEL mesh with more than 3 vertices expected.");
Debug.Assert(_mesh.Faces.Count > 2, "DCEL mesh with more than 2 face expected.");
// ----- General convex hull growth.
// Find a face which is visible from point.
DcelFace visibleFace = null;
// TODO: It can happen that startFace is not in _mesh.Faces.
// That means, _faces might contain more faces than _mesh.Faces!
//Debug.Assert(startFace == null || _mesh.Faces.Contains(startFace), "Internal error in ConvexHullBuilder.");
// Test the given face first. It should be visible, but the caller might have
// made a different collinearity check which is not robust.
if (startFace != null)
{
float distance;
DcelMesh.GetCollinearity(point, startFace, out distance);
if (distance > 0)
visibleFace = startFace;
}
// startFace was not robust. Find another visible face.
if (visibleFace == null)
{
int numberOfFaces = _faces.Count;
for (int i = 0; i < numberOfFaces; i++)
{
DcelFace face = _faces[i];
if (face.Tag < 0) // No need to check faces which are on the final hull.
continue;
float distance;
DcelMesh.GetCollinearity(point, face, out distance);
if (distance > 0)
{
visibleFace = face;
break;
}
}
}
if (visibleFace == null)
{
// Nothing to do because point is in the hull.
return false;
}
// Check if tags are all <= 0.
// Temporarily removed because this occurs with the Sponza model:
// Debug.Assert(_mesh.Faces.All(f => f.Tag <= 0), "All tags should be <= 0.");
// ----- Find and remove visible faces.
// Use of tags:
// - Face.Tag = -1 ... face on final hull. (Only set by of Create()).
// - Face.Tag = 0 ... default
// - Face.Tag = 1 ... face is visible and on stack; this face will be removed from the mesh.
// - Edge.Tag = 0 ... default
// - Edge.Tag = 1 ... edge is on boundary of visible area, visible area will be replaced
// so this edge is on the boundary of a temporary hole.
// - Edge.Tag = 2 ... hole edge that was handled
_taggedEdges.Clear();
// Push not handled visible faces on a stack.
Stack<DcelFace> visibleFaces = new Stack<DcelFace>();
visibleFaces.Push(visibleFace);
visibleFace.Tag = 1; // Mark as "on the stack".
_faces.Remove(visibleFace);
// We store the hole edge where the neighbor triangle faces away most extreme from point.
double minInFrontValue = double.PositiveInfinity;
DcelEdge startEdge = null;
// Search for all visible faces.
while (visibleFaces.Count > 0)
{
visibleFace = visibleFaces.Pop();
// Get the 3 triangle edges of the visible face.
DcelEdge[] edges = new DcelEdge[3];
edges[0] = visibleFace.Boundary;
edges[1] = edges[0].Next;
edges[2] = edges[1].Next;
// Add visible neighbors to the stack.
for (int i = 0; i < 3; i++)
{
DcelFace neighbor = edges[i].Twin.Face;
// Maybe neighbor was already removed?
if (neighbor == null)
continue;
// Get a measure for how visible the face is (> 0 means visible).
float inFrontValue;
//var collinearity =
DcelMesh.GetCollinearity(point, neighbor, out inFrontValue);
// If the point is in front of the face or in the face plane, then we can remove
// the face.
if (inFrontValue >= 0) // Using >= 0 without epsilon is important for numerical stability!
{
// Neighbor is visible or collinear. Push to stack (unless it is already on the stack).
if (neighbor.Tag != 1)
{
_faces.Remove(neighbor);
visibleFaces.Push(neighbor);
neighbor.Tag = 1;
}
}
else
{
if (minInFrontValue > inFrontValue)
{
startEdge = edges[i];
minInFrontValue = inFrontValue;
}
// Neighbor is not visible. This edge is part of the boundary of the hole.
// Set tag to 1 to mark the edge.
edges[i].Tag = 1;
_taggedEdges.Add(edges[i]);
}
}
}
// Temporarily removed because this occurs with the Sponza model:
// Debug.Assert(startEdge != null);
if (startEdge == null)
throw new GeometryException("Could not generate convex hull.");
// Create a new vertex for the point.
DcelVertex newVertex = new DcelVertex(point, null);
// Traverse the boundary of the hole beginning at startEdge.
DcelEdge currentEdge = startEdge;
DcelEdge previousEdge = null;
do
{
// Mark currentEdge as handled.
currentEdge.Tag = 2;
// Update the vertex-edge link because the link could point to a removed edge.
currentEdge.Origin.Edge = currentEdge;
// The next edge is part of the hole boundary and has not been handled yet.
// Create new face.
DcelFace face = new DcelFace();
face.Boundary = currentEdge;
currentEdge.Face = face;
_faces.Add(face);
// Create 2 half-edges.
currentEdge.Next = new DcelEdge();
currentEdge.Previous = new DcelEdge();
_taggedEdges.Add(currentEdge.Next);
_taggedEdges.Add(currentEdge.Previous);
currentEdge.Next.Face = face;
currentEdge.Next.Origin = currentEdge.Twin.Origin;
currentEdge.Next.Previous = currentEdge;
currentEdge.Next.Next = currentEdge.Previous;
currentEdge.Next.Tag = 2;
currentEdge.Previous.Face = face;
currentEdge.Previous.Origin = newVertex;
currentEdge.Previous.Previous = currentEdge.Next;
currentEdge.Previous.Next = currentEdge;
currentEdge.Previous.Tag = 2;
// Link new face with previous new face.
if (previousEdge != null)
{
previousEdge.Next.Twin = currentEdge.Previous;
currentEdge.Previous.Twin = previousEdge.Next;
}
// Search for next edge that is marked as a hole boundary edge.
previousEdge = currentEdge;
currentEdge = currentEdge.Twin.Previous.Twin;
int i = 0;
while (currentEdge.Tag == 0)
{
currentEdge = currentEdge.Previous.Twin;
i++;
if (i == 100)
throw new GeometryException("Could not generate convex hull.");
}
} while (currentEdge.Tag == 1);
Debug.Assert(previousEdge != null);
Debug.Assert(currentEdge == startEdge);
// Link the last new face with the first new face.
previousEdge.Next.Twin = currentEdge.Previous;
currentEdge.Previous.Twin = previousEdge.Next;
// Set the data for the new vertex.
newVertex.Edge = startEdge.Previous;
// Set mesh.Vertex to a vertex that is part of the hull.
_mesh.Vertex = startEdge.Origin;
// Reset edge tags. No need to reset face tags because marked faces are not
// part of the mesh anymore.
int numberOfEdges = _taggedEdges.Count;
for (int i = 0; i < numberOfEdges; i++)
_taggedEdges[i].Tag = 0;
_taggedEdges.Clear();
_mesh.Dirty = true;
return true;
}
/// <summary>
/// Merges a point to a convex hull that is also a point.
/// </summary>
/// <param name="point">The point.</param>
private void GrowPointConvex(Vector3F point)
{
Debug.Assert(_mesh != null);
Debug.Assert(_mesh.Vertices.Count == 1, "DCEL mesh with 1 vertex was expected.");
Debug.Assert(_type == ConvexHullType.Point);
// Abort if the point is equal to the existing point in the mesh.
if (Vector3F.AreNumericallyEqual(_mesh.Vertex.Position, point))
return;
DcelVertex newVertex = new DcelVertex(point, null);
DcelEdge edge = new DcelEdge();
DcelEdge twin = new DcelEdge();
// Link everything.
edge.Origin = _mesh.Vertex;
twin.Origin = newVertex;
edge.Twin = twin;
twin.Twin = edge;
_mesh.Vertex.Edge = edge;
newVertex.Edge = twin;
_mesh.Dirty = true;
_type = ConvexHullType.Linear;
}
/// <summary>
/// Removes the given face and connects the face edges to the given point.
/// </summary>
/// <param name="face">The face.</param>
/// <param name="point">The point.</param>
private void GrowPlanarConvexToSpatial(DcelFace face, Vector3F point)
{
_faces.Remove(face);
// Create a new vertex for the point.
DcelVertex newVertex = new DcelVertex(point, null);
// Traverse the boundary of the face.
DcelEdge currentEdge = face.Boundary;
DcelEdge previousEdge = null;
do
{
// Create new face.
DcelFace newface = new DcelFace();
newface.Boundary = currentEdge;
currentEdge.Face = newface;
_faces.Add(newface);
// Create 2 half-edges.
currentEdge.Next = new DcelEdge();
currentEdge.Previous = new DcelEdge();
currentEdge.Next.Face = newface;
currentEdge.Next.Origin = currentEdge.Twin.Origin;
currentEdge.Next.Previous = currentEdge;
currentEdge.Next.Next = currentEdge.Previous;
currentEdge.Previous.Face = newface;
currentEdge.Previous.Origin = newVertex;
currentEdge.Previous.Previous = currentEdge.Next;
currentEdge.Previous.Next = currentEdge;
// Link new face with previous new face.
if (previousEdge != null)
{
previousEdge.Next.Twin = currentEdge.Previous;
currentEdge.Previous.Twin = previousEdge.Next;
}
// Go to next edge.
previousEdge = currentEdge;
currentEdge = currentEdge.Twin.Previous.Twin;
// Go on until no edge points to face anymore.
} while (currentEdge.Face == face);
// Link the last new face with the first new face.
previousEdge.Next.Twin = currentEdge.Previous;
currentEdge.Previous.Twin = previousEdge.Next;
Debug.Assert(previousEdge != null);
Debug.Assert(currentEdge == face.Boundary);
Debug.Assert(currentEdge.Face != face);
// Set the data for the new vertex.
newVertex.Edge = currentEdge.Previous;
_mesh.Dirty = true;
_type = ConvexHullType.Spatial;
Debug.Assert(_mesh.Faces.Count == _faces.Count, "Internal error in ConvexHullBuilder.");
}