private Tile TemplateTile()
{
double inRadiusHyp = InRadius;
double inRadiusEuclidean = DonHatch.h2eNorm(inRadiusHyp);
double faceRadius = FaceRadius(inRadiusEuclidean);
// Calc the midpoint, and project to plane.
Vector3D midPoint = MidPoint(inRadiusEuclidean, faceRadius);
midPoint.Z *= -1;
midPoint = Sterographic.SphereToPlane(midPoint);
//double midPointSpherical = MidPoint( inRadiusEuclidean, faceRadius );
//double midPoint = Spherical2D.s2eNorm( midPointSpherical );
// Create and scale based on our midpoint.
Polygon poly = new Polygon();
poly.CreateRegular(Q, R);
double standardMidpointAbs = poly.Segments[0].Midpoint.Abs();
m_shrink = midPoint.Abs() / standardMidpointAbs;
poly.Scale(m_shrink);
Matrix4D m = Matrix4D.MatrixToRotateinCoordinatePlane(-Math.PI / Q, 0, 1);
poly.Rotate(m);
return(new Tile(poly, poly.Clone(), Geometry.Hyperbolic));
}
public static void EdgesToStl(H3.Cell.Edge[] edges)
{
Shapeways mesh = new Shapeways();
int divisions = 25;
foreach (H3.Cell.Edge edge in edges)
{
Segment seg = Segment.Line(
Sterographic.R3toS3(edge.Start),
Sterographic.R3toS3(edge.End));
Vector3D[] points = seg.Subdivide(divisions);
ProjectAndAddS3Points(mesh, points);
}
for (int i = 0; i < mesh.Mesh.Triangles.Count; i++)
{
mesh.Mesh.Triangles[i] = new Mesh.Triangle(
SphericalModels.StereoToEqualVolume(mesh.Mesh.Triangles[i].a),
SphericalModels.StereoToEqualVolume(mesh.Mesh.Triangles[i].b),
SphericalModels.StereoToEqualVolume(mesh.Mesh.Triangles[i].c));
}
STL.SaveMeshToSTL(mesh.Mesh, @"output.stl");
}
public static Vector3D SinusoidalToStereo(Vector3D v)
{
double lat = Math.PI / 2 * (1 - v.Y);
Vector3D spherical = new Vector3D(1, lat, Math.PI * v.X / Math.Cos(lat - Math.PI / 2));
Vector3D onBall = SphericalCoords.SphericalToCartesian(spherical);
return(Sterographic.SphereToPlane(onBall));
}
/// <summary>
/// 2-dimensional function.
/// http://archive.bridgesmathart.org/2013/bridges2013-217.pdf
/// </summary>
public static Vector3D MercatorToStereo(Vector3D v)
{
v *= Math.PI; // Input is [-1,1]
double lat = 2 * Math.Atan(Math.Exp(v.Y)) - Math.PI / 2;
double inclination = lat + Math.PI / 2;
Vector3D spherical = new Vector3D(1, inclination, v.X);
Vector3D onBall = SphericalCoords.SphericalToCartesian(spherical);
return(Sterographic.SphereToPlane(onBall));
}
public static Vector3D EquirectangularToStereo(Vector3D v)
{
// http://mathworld.wolfram.com/EquirectangularProjection.html
// y is the latitude
// x is the longitude
// Assume inputs go from -1 to 1.
Vector3D spherical = new Vector3D(1, Math.PI / 2 * (1 - v.Y), v.X * Math.PI);
Vector3D onBall = SphericalCoords.SphericalToCartesian(spherical);
return(Sterographic.SphereToPlane(onBall));
}
/// <summary>
/// 2-dimensional function.
/// ZZZ - Should make this general.
/// </summary>
public static Vector3D OrthographicToStereo(Vector3D v)
{
// We can only do the projection for half of the sphere.
double t = v.X * v.X + v.Y * v.Y;
if (t > 1)
{
t = 1;
}
v.Z = Math.Sqrt(1 - t);
return(Sterographic.SphereToPlane(v));
}
private static double EquidistantToStereo(double dist)
{
if (dist > 1)
{
throw new System.ArgumentException();
}
Vector3D v = new Vector3D(0, -1);
v.RotateXY(dist * Math.PI);
v = Sterographic.SphereToPlane(new Vector3D(v.X, 0, v.Y));
return(v.Abs());
}
/// <summary>
/// https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection
/// </summary>
private static double EqualAreaToStereo(double dist)
{
if (dist > 1)
{
throw new System.ArgumentException();
}
// We have dist normalized between 0 and 1, so this formula is slightly
// different than on Wikipedia, where dist ranges up to 2.
Vector3D v = new Vector3D(1, 2 * Math.Acos(dist), 0);
v = Sterographic.SphereToPlane(SphericalCoords.SphericalToCartesian(v));
return(v.Abs());
}
public static void Hypercube()
{
List <Vector3D> vertices = new List <Vector3D>();
vertices.Add(new Vector3D(1, 1, 1, 1));
vertices.Add(new Vector3D(1, 1, 1, -1));
vertices.Add(new Vector3D(1, 1, -1, 1));
vertices.Add(new Vector3D(1, 1, -1, -1));
vertices.Add(new Vector3D(1, -1, 1, 1));
vertices.Add(new Vector3D(1, -1, 1, -1));
vertices.Add(new Vector3D(1, -1, -1, 1));
vertices.Add(new Vector3D(1, -1, -1, -1));
vertices.Add(new Vector3D(-1, 1, 1, 1));
vertices.Add(new Vector3D(-1, 1, 1, -1));
vertices.Add(new Vector3D(-1, 1, -1, 1));
vertices.Add(new Vector3D(-1, 1, -1, -1));
vertices.Add(new Vector3D(-1, -1, 1, 1));
vertices.Add(new Vector3D(-1, -1, 1, -1));
vertices.Add(new Vector3D(-1, -1, -1, 1));
vertices.Add(new Vector3D(-1, -1, -1, -1));
HashSet <H3.Cell.Edge> edges = new HashSet <H3.Cell.Edge>(new H3.Cell.EdgeEqualityComparer());
foreach (Vector3D v1 in vertices)
{
foreach (Vector3D v2 in vertices)
{
if (v1.Dist(v2) == 2)
{
edges.Add(new H3.Cell.Edge(v1, v2));
}
}
}
// Radial project to S3, then stereographic to R3.
foreach (H3.Cell.Edge edge in edges)
{
edge.Start.Normalize();
edge.Start = Sterographic.S3toR3(edge.Start);
edge.End.Normalize();
edge.End = Sterographic.S3toR3(edge.End);
}
PovRay.WriteEdges(new PovRay.Parameters()
{
AngularThickness = .05
}, Geometry.Spherical, edges.ToArray(), "433.pov", append: false);
}
public static Vector3D StereoToGnomonic(Vector3D p)
{
Vector3D sphere = Sterographic.PlaneToSphere(p);
// We can't only represent the lower hemisphere.
if (sphere.Z >= 0)
{
sphere.Z = 0;
sphere.Normalize();
sphere *= Infinity.FiniteScale;
return(sphere);
}
double z = sphere.Z;
sphere.Z = 0;
return(-sphere * m_gScale / z);
}
public static void GenPolyhedron()
{
Tiling tiling;
int p = 3;
int q = 6;
GetAssociatedTiling(p, q, 5000, out tiling);
double overallScale = 12.5; // 2.5 cm = 1 in diameter
Shapeways mesh = new Shapeways();
foreach (Tile tile in tiling.Tiles)
{
foreach (Segment seg in tile.Boundary.Segments)
{
double tilingScale = 0.75;
seg.Scale(new Vector3D(), tilingScale);
Vector3D v1 = Sterographic.PlaneToSphereSafe(seg.P1);
Vector3D v2 = Sterographic.PlaneToSphereSafe(seg.P2);
//if( v1.Dist( v2 ) < 0.01 )
// continue;
if (SphericalCoords.CartesianToSpherical(v1).Y < Math.PI / 12 &&
SphericalCoords.CartesianToSpherical(v2).Y < Math.PI / 12)
{
continue;
}
double dist = v1.Dist(v2);
int divisions = 2 + (int)(dist * 20);
Vector3D[] points = seg.Subdivide(divisions);
points = points.Select(v => Sterographic.PlaneToSphereSafe(v)).ToArray();
mesh.AddCurve(points, v => SizeFunc(v, overallScale));
}
}
mesh.Mesh.Scale(overallScale);
string outputFileName = @"d:\temp\" + p + q + ".stl";
STL.SaveMeshToSTL(mesh.Mesh, outputFileName);
}
// Maps unit disk to Boy's surface.
public static Vector3D MapToBoys(Vector3D v)
{
// https://en.wikipedia.org/wiki/Boy%27s_surface#Parametrization_of_Boy.27s_surface
Complex w = v.ToComplex();
Complex t0 = (Complex.Pow(w, 6) + Complex.Pow(w, 3) * Math.Sqrt(5) - 1);
Complex t1 = w * (1 - Complex.Pow(w, 4)) / t0;
Complex t2 = w * (1 + Complex.Pow(w, 4)) / t0;
Complex t3 = (1 + Complex.Pow(w, 6)) / t0;
double g1 = -3 / 2 * t1.Imaginary;
double g2 = -3 / 2 * t2.Real;
double g3 = t3.Imaginary - 0.5;
double denom = g1 * g1 + g2 * g2 + g3 * g3;
v = new Vector3D(g1, g2, g3);
v /= denom;
// The above is actually a map to R^3, but we want a surface in S^3
return(Sterographic.R3toS3(v));
}
/// <summary>
/// Inputs and Outputs are in R3 (stereographically projected).
/// </summary>
public static Vector3D[] GeodesicPoints(Vector3D v1, Vector3D v2)
{
Vector3D start = Sterographic.R3toS3(v1);
Vector3D end = Sterographic.R3toS3(v2);
AvoidNorthPole(ref start, end);
AvoidNorthPole(ref end, start);
int div = 42;
//int div = 56; // 343
//int div = 50; // 333
Segment seg = Segment.Line(start, end);
Vector3D[] result = seg.Subdivide(div);
for (int i = 0; i < result.Length; i++)
{
result[i].Normalize();
result[i] = Sterographic.S3toR3(result[i]);
}
return(result);
}
public static void DrawElements(HyperbolicModel model, Vector3D[] textureCoords, Vector3D[] textureVerts, int[] elements,
Isometry mouseIsometry, double textureScale)
{
////////////////////// ZZZ - Use VBOs
GL.Begin(BeginMode.Triangles);
{
double factor = textureScale;
int skipped = 0;
for (int i = 0; i < elements.Length; i++)
{
int idx = elements[i];
// In Poincare model.
GL.TexCoord2((textureCoords[idx].X * factor + 1) / 2, (textureCoords[idx].Y * factor + 1) / 2);
Complex transformed = textureVerts[idx].ToComplex();
if (mouseIsometry != null)
{
transformed = mouseIsometry.Apply(transformed);
}
switch (model)
{
case HyperbolicModel.Poincare:
{
Vertex(transformed);
break;
}
case HyperbolicModel.Klein:
{
Vector3D temp = Vector3D.FromComplex(transformed);
Vertex(PoincareToKlein(temp));
break;
}
case HyperbolicModel.Pseudosphere:
{
Mobius m = new Mobius();
m.UpperHalfPlane();
Complex u = m.Apply(transformed);
double x = u.Real;
double y = u.Imaginary;
double max = 1 * System.Math.PI;
double min = -1 * System.Math.PI;
if (0 == i % 3 && (x < min - 1 || x > max + 1 || y < 0))
{
skipped = 1;
continue;
}
if (skipped > 0 && skipped < 3)
{
skipped++;
continue;
}
skipped = 0;
GL.TexCoord2((textureCoords[idx].X * factor + 1) / 2, (textureCoords[idx].Y * factor + 1) / 2);
// Pseudosphere
Func <double, Complex> tractrix = new Func <double, Complex>(
(t) =>
{
return(new Complex(t - Math.Tanh(t), 1.0 / Math.Cosh(t)));
});
//Vector3D temp1 = Vector3D.FromComplex( u );
if (x < min)
{
x = min;
}
if (x > max)
{
x = max;
}
if (y < 1)
{
y = 1;
}
Vector3D temp1 = new Vector3D(x, y);
double logy = Math.Log(temp1.Y);
Complex tract = tractrix(logy);
Vector3D temp2 = new Vector3D(
Math.Cos(temp1.X) * tract.Imaginary,
Math.Sin(temp1.X) * tract.Imaginary,
tract.Real);
GL.Vertex3(temp2.X, temp2.Y, temp2.Z);
//temp1 = m.Inverse().Apply( temp1 );
//GL.Vertex3( temp1.X, temp1.Y, temp1.Z );
//Vertex( temp1 );
break;
}
case HyperbolicModel.Hyperboloid:
{
Vector3D hyper = Sterographic.PlaneToHyperboloid(Vector3D.FromComplex(transformed)); // Hyperboloid
GL.Vertex3(hyper.X, hyper.Y, hyper.Z);
break;
}
default:
{
System.Diagnostics.Debug.Assert(false);
break;
}
}
/* // PETALS
* int petals = 7;
* double newMag = transformed.Magnitude * ( 1 + 0.5 * Math.Sin( transformed.Phase * petals ) );
* double newPhase = transformed.Phase + ( -0.2 * newMag * Math.Pow( Math.Sin( newMag * 3 ), 1 ) * Math.Cos( transformed.Phase * petals ) );
* transformed = Complex.FromPolarCoordinates( newMag, newPhase );
*
* Vertex( transformed );
* */
//double mag = System.Math.Pow( transformed.Magnitude, 3 ) / transformed.Magnitude; // nice
//double mag = System.Math.Pow( transformed.Magnitude - 3, 2 ) + .0; // looks spherical
//double mag = transformed.Magnitude + 0.1* System.Math.Sin( transformed.Magnitude * 15 ); // Fun warping (20 is cool too)
//Vertex( transformed * mag );
/*double xmag = 1;
* double ymag = transformed.Imaginary + 0.1 * System.Math.Sin( transformed.Imaginary * 15 );
* xmag = System.Math.Abs( xmag );
* ymag = System.Math.Abs( ymag );
* Vertex( new Complex( transformed.Real * xmag, transformed.Imaginary * ymag ) ); */
//Vertex( 2 / System.Math.PI * Complex.Log( ( 1 + transformed ) / ( 1 - transformed ) ) ); // Band model
//Vertex( Complex.Pow( transformed, 3 ) / transformed.Magnitude ); // Spikey
// Spiral
//Complex band = 2 / System.Math.PI * Complex.Log( ( 1 + transformed ) / ( 1 - transformed ) );
//band = new Complex( band.Real, band.Imaginary + 0.3 * System.Math.Sin( band.Real * 2 ) );
//band = new Complex( band.Real * .5, band.Imaginary );
//band += new Complex( 0, .5 );
//Vertex( band );
/*
* double x = band.Real;
* double y = band.Imaginary;
*
* double r = System.Math.Exp( x );
* double theta = 3*( x + y/1.75 ); */
//Vertex( new Complex( r * System.Math.Sin( theta ), r * System.Math.Cos( theta ) ) ); // Spiral
}
}
GL.End();
}
