public static double GetNormalizedCircumRadius(int p, double q)
{
double hypot = GetTriangleHypotenuse(p, q);
switch (Geometry2D.GetGeometry(p, q))
{
case Geometry.Spherical:
return(Spherical2D.s2eNorm(hypot) * DiskRadius);
case Geometry.Euclidean:
return(EuclideanHypotenuse);
case Geometry.Hyperbolic:
{
if (Infinity.IsInfinite(hypot))
{
return(DiskRadius);
}
return(DonHatch.h2eNorm(hypot) * DiskRadius);
}
}
Debug.Assert(false);
return(1);
}
/// <summary>
/// Calculates the point of our simplex that is at the middle of a face.
/// </summary>
private static Vector3D FaceCenterBall(int p, int q, int r)
{
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
switch (cellGeometry)
{
case Geometry.Spherical:
{
Vector3D cellCenter = CellCenterBall(p, q, r);
Sphere[] mirrors = Mirrors(p, q, r, moveToBall: true);
return(mirrors[0].ProjectToSurface(cellCenter));
}
case Geometry.Euclidean:
{
Sphere[] mirrors = Mirrors(p, q, r, moveToBall: false);
Vector3D faceCenterUHS = mirrors[0].Center;
faceCenterUHS.Z += mirrors[0].Radius;
return(H3Models.UHSToBall(faceCenterUHS));
}
case Geometry.Hyperbolic:
{
throw new System.NotImplementedException();
}
}
throw new System.ArgumentException();
}
/// <summary>
/// Calculates the point of our simplex that is at the center of a cell.
/// </summary>
private static Vector3D CellCenterBall(int p, int q, int r)
{
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
switch (cellGeometry)
{
case Geometry.Spherical:
{
return(new Vector3D());
}
case Geometry.Euclidean:
{
return(new Vector3D(0, 0, 1));
}
case Geometry.Hyperbolic:
{
//throw new System.NotImplementedException();
return(new Vector3D(0, 0, 1));
}
}
throw new System.ArgumentException();
}
/// <summary>
/// Calculates the point of our simplex that is at a vertex.
/// </summary>
public static Vector3D VertexPointBall(int p, int q, int r)
{
Geometry vertexGeometry = Geometry2D.GetGeometry(q, r);
if (vertexGeometry == Geometry.Hyperbolic)
{
// Outside the ball, and not in a good way. Use the Klein version.
//throw new System.NotImplementedException();
return(new Vector3D(0, 0, -1));
}
// Get in UHS first.
Sphere cellFacet = Mirrors(p, q, r, moveToBall: false).First();
double rSquared = Math.Pow(cellFacet.Radius, 2);
double cSquared = Math.Pow(cellFacet.Center.Abs(), 2);
Vector3D uhs;
if (Tolerance.Equal(rSquared, cSquared)) // e.g. 363
{
uhs = new Vector3D();
}
else
{
double height = Math.Sqrt(rSquared - cSquared);
uhs = new Vector3D(0, 0, height);
}
return(H3Models.UHSToBall(uhs));
}
public TilingConfig(int p, int q) : this()
{
if (Geometry2D.GetGeometry(p, q) != Geometry.Spherical)
{
throw new System.ArgumentException();
}
SetupConfig(p, q, PlatonicSolids.NumFacets(p, q));
}
/// <summary>
/// Get the side length opposite angle PI/P,
/// In the induced geometry.
/// </summary>
public static double GetTrianglePSide(int p, int q)
{
Geometry g = Geometry2D.GetGeometry(p, q);
double alpha = Math.PI / 2;
double beta = Math.PI / q;
double gamma = Math.PI / p; // The one we want.
if (g == Geometry.Euclidean)
{
return(EuclideanHypotenuse * Math.Sin(gamma));
}
return(GetTriangleSide(g, gamma, beta, alpha));
}
public static Mobius FCOrientMobius(int p, int q)
{
Vector3D p1, p2, p3;
Segment seg = null;
TilePoints(p, q, out p1, out p2, out p3, out seg);
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
p1.RotateXY(Math.PI / 2); // XXX - repeated from above. Implement a better way to sequence transformations.
Mobius m = new Mobius();
m.Isometry(cellGeometry, 0, -p1);
return(m);
}
/// <summary>
/// Get the side length opposite angle PI/Q,
/// In the induced geometry.
/// </summary>
public static double GetTriangleQSide(int p, int q)
{
Geometry g = Geometry2D.GetGeometry(p, q);
double alpha = Math.PI / 2;
double beta = PiOverNSafe(q); // The one we want.
double gamma = PiOverNSafe(p);
if (g == Geometry.Euclidean)
{
return(EuclideanHypotenuse * Math.Sin(beta));
}
return(GetTriangleSide(g, beta, gamma, alpha));
}
public static Vector3D VertexPointKlein(int p, int q, int r)
{
Geometry vertexGeometry = Geometry2D.GetGeometry(q, r);
if (vertexGeometry != Geometry.Hyperbolic)
{
throw new System.NotImplementedException();
}
Sphere[] mirrors = Mirrors(p, q, r);
Sphere klein = H3Models.BallToKlein(mirrors[0]);
Vector3D off = klein.Offset;
double h = off.Abs() / Math.Cos(off.AngleTo(new Vector3D(0, 0, -1)));
return(new Vector3D(0, 0, -h));
}
public static Mobius VertexCenteredMobius(int p, int q)
{
double angle = Math.PI / q;
if (Utils.Even(q))
{
angle *= 2;
}
Vector3D offset = new Vector3D(-1 * Geometry2D.GetNormalizedCircumRadius(p, q), 0, 0);
offset.RotateXY(angle);
Mobius m = new Mobius();
m.Isometry(Geometry2D.GetGeometry(p, q), angle, offset.ToComplex());
return(m);
}
/// <summary>
/// In the induced geometry.
/// </summary>
public static double GetTriangleHypotenuse(int p, int q)
{
Geometry g = Geometry2D.GetGeometry(p, q);
if (g == Geometry.Euclidean)
{
return(EuclideanHypotenuse);
}
// We have a 2,q,p triangle, where the right angle alpha
// is opposite the hypotenuse (the length we want).
double alpha = Math.PI / 2;
double beta = Math.PI / q;
double gamma = Math.PI / p;
return(GetTriangleSide(g, alpha, beta, gamma));
}
/// <summary>
/// This Mobius transform will center the tiling on an edge.
/// </summary>
public Mobius EdgeMobius()
{
Geometry g = Geometry2D.GetGeometry(this.P, this.Q);
Polygon poly = new Polygon();
poly.CreateRegular(this.P, this.Q);
Segment seg = poly.Segments[0];
Vector3D offset = seg.Midpoint;
double angle = Math.PI / this.P;
offset.RotateXY(-angle);
Mobius m = new Mobius();
m.Isometry(g, -angle, -offset);
return(m);
}
public static double GetNormalizedCircumRadius(int p, int q)
{
Geometry g = Geometry2D.GetGeometry(p, q);
double hypot = GetTriangleHypotenuse(p, q);
switch (g)
{
case Geometry.Spherical:
return(Spherical2D.s2eNorm(hypot) * DiskRadius);
case Geometry.Euclidean:
return(EuclideanHypotenuse);
case Geometry.Hyperbolic:
return(DonHatch.h2eNorm(hypot) * DiskRadius);
}
Debug.Assert(false);
return(1);
}
/// <summary>
/// Calculates the point of our simplex that is at the middle of an edge.
/// </summary>
private static Vector3D MidEdgePointBall(int p, int q, int r)
{
// We need the mid-radius, but we have to do the calculation
// with our Euclidean simplex mirrors (to avoid infinities that happen in the formulas).
Circle3D edge = HoneycombEdgeUHS(p, q, r);
if (edge.Radius == 0)
{
return(edge.Center);
}
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
switch (cellGeometry)
{
case Geometry.Spherical:
{
Sphere sphereInBall = H3Models.UHSToBall(new Sphere()
{
Center = edge.Center, Radius = edge.Radius
});
Vector3D mid = sphereInBall.ProjectToSurface(new Vector3D()); // Project origin to sphere.
return(mid);
}
case Geometry.Euclidean:
{
Vector3D mid = H3Models.UHSToBall(edge.Center + new Vector3D(0, 0, edge.Radius));
return(mid);
}
case Geometry.Hyperbolic:
{
throw new System.NotImplementedException();
}
}
throw new System.ArgumentException();
}
/// <summary>
/// Calculates the 3 mirrors connected to the cell center.
/// This works in all geometries and returns results in the UHS model (or the appropriate analogue).
/// </summary>
private static Sphere[] InteriorMirrors(int p, int q)
{
// Some construction points we need.
Vector3D p1, p2, p3;
Segment seg = null;
TilePoints(p, q, out p1, out p2, out p3, out seg);
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
// XZ-plane
Sphere s1 = new Sphere()
{
Center = new Vector3D(0, 1, 0),
Radius = double.PositiveInfinity
};
Sphere s2 = null;
if (cellGeometry == Geometry.Euclidean)
{
s2 = new Sphere()
{
Center = -p2,
Offset = p2,
Radius = double.PositiveInfinity
};
}
else if (
cellGeometry == Geometry.Spherical ||
cellGeometry == Geometry.Hyperbolic)
{
s2 = new Sphere()
{
Center = seg.Center,
Radius = seg.Radius,
//Invert = true // XXX - maybe we don't invert in the case of spherical?
};
// j34
/*double off = seg.Center.Abs() - seg.Radius;
* off *= 1.05;
* Vector3D vOff = seg.Center;
* vOff.Normalize();
* vOff *= off;
* s2 = H3Models.Ball.OrthogonalSphereInterior( vOff );*/
// 4j3
/*double off = seg.Center.Abs() - seg.Radius;
* off = DonHatch.h2eNorm( DonHatch.e2hNorm( off ) + m_jOffset );
* Vector3D vOff = seg.Center;
* vOff.Normalize();
* vOff *= off;
* s2 = H3Models.Ball.OrthogonalSphereInterior( vOff );*/
}
Sphere s3;
if (Infinite(p) && Infinite(q) /*&& FiniteOrInfinite( r )*/)
{
Vector3D tempCenter = seg.Center;
tempCenter.X *= -1;
s3 = new Sphere()
{
Center = tempCenter,
Radius = seg.Radius,
//Invert = true,
};
}
else
{
s3 = new Sphere()
{
Center = p3,
Radius = double.PositiveInfinity
};
}
// The reason for the special case ordering is to make sure
// the last mirror is always the cell-reflecting mirror of the dual honeycomb.
// The order we want are the mirrors opposite: cell, face, edge, vertex
// NOTE: This also puts the mirrors in the same order as in standard presentations,
// e.g. from Coxeter's 57-cell paper.
Sphere[] surfaces;
if (!Infinite(p) && Infinite(q))
{
surfaces = new Sphere[] { s2, s1, s3 }
}
;
else
{
surfaces = new Sphere[] { s3, s1, s2 }
};
return(surfaces);
}
public void CreateRegular(int numSides, int q)
{
int p = numSides;
Segments.Clear();
List <Vector3D> points = new List <Vector3D>();
Geometry g = Geometry2D.GetGeometry(p, q);
double circumRadius = Geometry2D.GetNormalizedCircumRadius(p, q);
double angle = 0;
for (int i = 0; i < p; i++)
{
Vector3D point = new Vector3D();
point.X = (circumRadius * Math.Cos(angle));
point.Y = (circumRadius * Math.Sin(angle));
points.Add(point);
angle += Utils.DegreesToRadians(360.0 / p);
}
// Turn this into segments.
for (int i = 0; i < points.Count; i++)
{
int idx1 = i;
int idx2 = i == points.Count - 1 ? 0 : i + 1;
Segment newSegment = new Segment();
newSegment.P1 = points[idx1];
newSegment.P2 = points[idx2];
if (g != Geometry.Euclidean)
{
newSegment.Type = SegmentType.Arc;
if (2 == p)
{
// Our magic formula below breaks down for digons.
double factor = Math.Tan(Math.PI / 6);
newSegment.Center = newSegment.P1.X > 0 ?
new Vector3D(0, -circumRadius, 0) * factor :
new Vector3D(0, circumRadius, 0) * factor;
}
else
{
// Our segments are arcs in Non-Euclidean geometries.
// Magically, the same formula turned out to work for both.
// (Maybe this is because the Poincare Disc model of the
// hyperbolic plane is stereographic projection as well).
double piq = q == -1 ? 0 : Math.PI / q; // Handle q infinite.
double t1 = Math.PI / p;
double t2 = Math.PI / 2 - piq - t1;
double factor = (Math.Tan(t1) / Math.Tan(t2) + 1) / 2;
newSegment.Center = (newSegment.P1 + newSegment.P2) * factor;
}
newSegment.Clockwise = Geometry.Spherical == g ? false : true;
}
// XXX - Make this configurable?
// This is the color of cell boundary lines.
//newSegment.m_color = CColor( 1, 1, 0, 1 );
Segments.Add(newSegment);
}
}
public static Sphere[] Mirrors(int p, int q, int r, ref Vector3D cellCenter, bool moveToBall = true, double scaling = -1)
{
Geometry g = Util.GetGeometry(p, q, r);
if (g == Geometry.Spherical)
{
// These are in the ball model.
Sphere[] result = SimplexCalcs.MirrorsSpherical(p, q, r);
return(result);
}
else if (g == Geometry.Euclidean)
{
return(SimplexCalcs.MirrorsEuclidean());
}
// This is a rotation we'll apply to the mirrors at the end.
// This is to try to make our image outputs have vertical bi-lateral symmetry and the most consistent in all cases.
// NOTE: + is CW, not CCW. (Because the way I did things, our images have been reflected vertically, and I'm too lazy to go change this.)
double rotation = Math.PI / 2;
// Some construction points we need.
Vector3D p1, p2, p3;
Segment seg = null;
TilePoints(p, q, out p1, out p2, out p3, out seg);
//
// Construct in UHS
//
Geometry cellGeometry = Geometry2D.GetGeometry(p, q);
Vector3D center = new Vector3D();
double radius = 0;
if (cellGeometry == Geometry.Spherical)
{
// Finite or Infinite r
// Spherical trig
double halfSide = Geometry2D.GetTrianglePSide(q, p);
double mag = Math.Sin(halfSide) / Math.Cos(Util.PiOverNSafe(r));
mag = Math.Asin(mag);
// e.g. 43j
//mag *= 0.95;
// Move mag to p1.
mag = Spherical2D.s2eNorm(mag);
H3Models.Ball.DupinCyclideSphere(p1, mag, Geometry.Spherical, out center, out radius);
}
else if (cellGeometry == Geometry.Euclidean)
{
center = p1;
radius = p1.Dist(p2) / Math.Cos(Util.PiOverNSafe(r));
}
else if (cellGeometry == Geometry.Hyperbolic)
{
if (Infinite(p) && Infinite(q) && FiniteOrInfinite(r))
{
//double iiiCellRadius = 2 - Math.Sqrt( 2 );
//Circle3D iiiCircle = new Circle3D() { Center = new Vector3D( 1 - iiiCellRadius, 0, 0 ), Radius = iiiCellRadius };
//radius = iiiCellRadius; // infinite r
//center = new Vector3D( 1 - radius, 0, 0 );
// For finite r, it was easier to calculate cell facet in a more symmetric position,
// then move into position with the other mirrors via a Mobius transformation.
double rTemp = 1 / (Math.Cos(Util.PiOverNSafe(r)) + 1);
Mobius m = new Mobius();
m.Isometry(Geometry.Hyperbolic, -Math.PI / 4, new Vector3D(0, Math.Sqrt(2) - 1));
Vector3D c1 = m.Apply(new Vector3D(1 - 2 * rTemp, 0, 0));
Vector3D c2 = c1;
c2.Y *= -1;
Vector3D c3 = new Vector3D(1, 0);
Circle3D c = new Circle3D(c1, c2, c3);
radius = c.Radius;
center = c.Center;
}
else if (Infinite(p) && Finite(q) && FiniteOrInfinite(r))
{
// http://www.wolframalpha.com/input/?i=r%2Bx+%3D+1%2C+sin%28pi%2Fp%29+%3D+r%2Fx%2C+solve+for+r
// radius = 2 * Math.Sqrt( 3 ) - 3; // Appolonian gasket wiki page
//radius = Math.Sin( Math.PI / q ) / ( Math.Sin( Math.PI / q ) + 1 );
//center = new Vector3D( 1 - radius, 0, 0 );
// For finite r, it was easier to calculate cell facet in a more symmetric position,
// then move into position with the other mirrors via a Mobius transformation.
double rTemp = 1 / (Math.Cos(Util.PiOverNSafe(r)) + 1);
Mobius m = new Mobius();
m.Isometry(Geometry.Hyperbolic, 0, p2);
Vector3D findingAngle = m.Inverse().Apply(new Vector3D(1, 0));
double angle = Math.Atan2(findingAngle.Y, findingAngle.X);
m.Isometry(Geometry.Hyperbolic, angle, p2);
Vector3D c1 = m.Apply(new Vector3D(1 - 2 * rTemp, 0, 0));
Vector3D c2 = c1;
c2.Y *= -1;
Vector3D c3 = new Vector3D(1, 0);
Circle3D c = new Circle3D(c1, c2, c3);
radius = c.Radius;
center = c.Center;
}
else if (Finite(p) && Infinite(q) && FiniteOrInfinite(r))
{
radius = p2.Abs(); // infinite r
radius = DonHatch.asinh(Math.Sinh(DonHatch.e2hNorm(p2.Abs())) / Math.Cos(Util.PiOverNSafe(r))); // hyperbolic trig
// 4j3
//m_jOffset = radius * 0.02;
//radius += m_jOffset ;
radius = DonHatch.h2eNorm(radius);
center = new Vector3D();
rotation *= -1;
}
else if (/*Finite( p ) &&*/ Finite(q))
{
// Infinite r
//double mag = Geometry2D.GetTrianglePSide( q, p );
// Finite or Infinite r
double halfSide = Geometry2D.GetTrianglePSide(q, p);
double mag = DonHatch.asinh(Math.Sinh(halfSide) / Math.Cos(Util.PiOverNSafe(r))); // hyperbolic trig
H3Models.Ball.DupinCyclideSphere(p1, DonHatch.h2eNorm(mag), out center, out radius);
}
else
{
throw new System.NotImplementedException();
}
}
Sphere cellBoundary = new Sphere()
{
Center = center,
Radius = radius
};
Sphere[] interior = InteriorMirrors(p, q);
Sphere[] surfaces = new Sphere[] { cellBoundary, interior[0], interior[1], interior[2] };
// Apply rotations.
bool applyRotations = true;
if (applyRotations)
{
foreach (Sphere s in surfaces)
{
RotateSphere(s, rotation);
}
p1.RotateXY(rotation);
}
// Apply scaling
bool applyScaling = scaling != -1;
if (applyScaling)
{
//double scale = 1.0/0.34390660467269524;
//scale = 0.58643550768408892;
foreach (Sphere s in surfaces)
{
Sphere.ScaleSphere(s, scaling);
}
}
bool facetCentered = false;
if (facetCentered)
{
PrepForFacetCentering(p, q, surfaces, ref cellCenter);
}
// Move to ball if needed.
if (moveToBall)
{
surfaces = MoveToBall(surfaces, ref cellCenter);
}
return(surfaces);
}
/// <summary>
/// Returns the 6 simplex edges in the UHS model.
/// </summary>
public static H3.Cell.Edge[] SimplexEdgesUHS(int p, int q, int r)
{
// Only implemented for honeycombs with hyperideal cells right now.
if (!(Geometry2D.GetGeometry(p, q) == Geometry.Hyperbolic))
{
throw new System.NotImplementedException();
}
Sphere[] simplex = SimplexCalcs.Mirrors(p, q, r, moveToBall: false);
Circle[] circles = simplex.Select(s => H3Models.UHS.IdealCircle(s)).ToArray();
Vector3D[] defPoints = new Vector3D[6];
Vector3D dummy;
Euclidean2D.IntersectionLineCircle(circles[1].P1, circles[1].P2, circles[0], out defPoints[0], out dummy);
Euclidean2D.IntersectionLineCircle(circles[2].P1, circles[2].P2, circles[0], out defPoints[1], out dummy);
Euclidean2D.IntersectionLineCircle(circles[1].P1, circles[1].P2, circles[3], out defPoints[2], out dummy);
Euclidean2D.IntersectionLineCircle(circles[2].P1, circles[2].P2, circles[3], out defPoints[3], out dummy);
Circle3D c = simplex[0].Intersection(simplex[3]);
Vector3D normal = c.Normal;
normal.RotateXY(Math.PI / 2);
Vector3D intersection;
double height, off;
Euclidean2D.IntersectionLineLine(c.Center, c.Center + normal, circles[1].P1, circles[1].P2, out intersection);
off = (intersection - c.Center).Abs();
height = Math.Sqrt(c.Radius * c.Radius - off * off);
intersection.Z = height;
defPoints[4] = intersection;
Euclidean2D.IntersectionLineLine(c.Center, c.Center + normal, circles[2].P1, circles[2].P2, out intersection);
off = (intersection - c.Center).Abs();
height = Math.Sqrt(c.Radius * c.Radius - off * off);
intersection.Z = height;
defPoints[5] = intersection;
// Hyperideal vertex too?
bool order = false;
H3.Cell.Edge[] edges = null;
if (Geometry2D.GetGeometry(q, r) == Geometry.Hyperbolic)
{
edges = new H3.Cell.Edge[]
{
new H3.Cell.Edge(new Vector3D(), new Vector3D(0, 0, 10)),
new H3.Cell.Edge(defPoints[4], defPoints[5], order),
new H3.Cell.Edge(defPoints[0], defPoints[4], order),
new H3.Cell.Edge(defPoints[1], defPoints[5], order),
new H3.Cell.Edge(defPoints[2], defPoints[4], order),
new H3.Cell.Edge(defPoints[3], defPoints[5], order),
};
}
else
{
Vector3D vPointUHS = H3Models.BallToUHS(VertexPointBall(p, q, r));
defPoints[0] = defPoints[1] = vPointUHS;
edges = new H3.Cell.Edge[]
{
new H3.Cell.Edge(vPointUHS, new Vector3D(0, 0, 10)),
new H3.Cell.Edge(defPoints[4], defPoints[5], order),
new H3.Cell.Edge(defPoints[0], defPoints[4], order),
new H3.Cell.Edge(defPoints[1], defPoints[5], order),
new H3.Cell.Edge(defPoints[2], defPoints[4], order),
new H3.Cell.Edge(defPoints[3], defPoints[5], order),
};
}
return(edges);
}