/// <summary>
/// Calculates the size of the area that's both in the specified face and closer to the specified vertex than any other vertex.
/// </summary>
public static double AreaSharedByVertexAndFace(IPolyhedron surface, Vertex vertex, Face face)
{
var vertexPosition = vertex.Position;
var faces = surface.FacesOf(vertex);
var edges = surface.EdgesOf(vertex);
var index = faces.IndexOf(face);
if (!faces.Contains(face))
{
return(0.0);
}
var midpointOfFace = face.Center();
var previousEdge = edges.AtCyclicIndex(index - 1);
var midpointOfPreviousEdge = BisectionPoint(surface, previousEdge);
var nextEdge = edges.AtCyclicIndex(index);
var midpointOfNextEdge = BisectionPoint(surface, nextEdge);
var crossProductOfFirstSegment = Vector.CrossProduct(midpointOfPreviousEdge - vertexPosition, midpointOfFace - vertexPosition);
var areaOfFirstSegment = Vector.ScalarProduct(crossProductOfFirstSegment, midpointOfFace.Normalize()) / 2;
var crossProductOfSecondSegment = Vector.CrossProduct(midpointOfFace - vertexPosition, midpointOfNextEdge - vertexPosition);
var areaOfSecondSegment = Vector.ScalarProduct(crossProductOfSecondSegment, midpointOfFace.Normalize()) / 2;
return(areaOfFirstSegment + areaOfSecondSegment);
}
/// <summary>
/// Calculates the center of a polygonal face then projects it onto the sphere the face is embedded in.
///
/// Degenerate for faces whose centroid is the origin.
/// </summary>
public static Vector SphericalCenter(this Face face)
{
var vectors = face.Vertices.Select(v => v.Position).ToArray();
var center = face.Center();
var radius = vectors.Average(vector => vector.Norm());
return(radius * center.Normalize());
}
