/// <summary>
/// Hopf Link between two points on S^2.
/// </summary>
public static void HopfLink(StreamWriter sw, Vector3D s2_1, Vector3D s2_2, bool anti)
{
Vector3D[] circlePoints;
string circleString;
circlePoints = OneHopfCircleProjected(s2_1, anti);
circleString = PovRay.EdgeSphereSweep(circlePoints, SizeFunc);
sw.WriteLine(circleString);
circlePoints = OneHopfCircleProjected(s2_2, anti);
circleString = PovRay.EdgeSphereSweep(circlePoints, SizeFunc);
sw.WriteLine(circleString);
Mesh mesh = new Mesh();
Vector3D[] interpolated = S3.GeodesicPoints(s2_1, s2_2);
for (int i = 0; i < interpolated.Length - 1; i++)
{
Vector3D v1 = interpolated[i];
Vector3D v2 = interpolated[i + 1];
Vector3D[] p1 = OneHopfCircleProjected(v1, anti);
Vector3D[] p2 = OneHopfCircleProjected(v2, anti);
for (int j = 0; j < p1.Length - 1; j++)
{
Mesh.Triangle t1 = new Mesh.Triangle(p1[j], p1[j + 1], p2[j]);
Mesh.Triangle t2 = new Mesh.Triangle(p2[j], p1[j + 1], p2[j + 1]);
mesh.Triangles.Add(t1);
mesh.Triangles.Add(t2);
}
}
PovRay.WriteMesh(sw, mesh, append: true);
}
private void AppendMesh(List <H3.Cell.Edge> fiberList, string filename)
{
Mesh mesh = new Mesh();
for (int j = 0; j < fiberList.Count - 1; j++)
{
AddRow(mesh, fiberList[j], fiberList[j + 1]);
}
//AddRow( mesh, fiberList.Last(), fiberList.First() );
PovRay.WriteMesh(mesh, filename, append: true);
}
public static void RLD_Surface()
{
RLD_outputs outputs;
Mesh mesh = new Mesh();
SurfaceInternal(out outputs);
double scale = m_params.Scale;
// Now add in all the catenoids.
double mInc = Math.PI * 2 / m_params.M;
for (int k = 1; k < outputs.x_i.Length; k++)
{
for (int m = 0; m < m_params.M; m++)
{
Vector3D loc = SphericalCoords.SphericalToCartesian(new Vector3D(1, Math.PI / 2 - outputs.x_i[k], m * mInc));
mesh.Append(Catenoid(scale, loc, outputs.phi_i[k], outputs.t_i[k]));
}
}
PovRay.WriteMesh(mesh, "RLD.pov");
}
public static void CatenoidBasedSurface()
{
RLD_outputs outputs;
SurfaceInternal(out outputs);
double scale = m_params.Scale;
// Map a point for a given k/m from the hemihypersphere to the complex plane.
// You can also pass in -1 for k to get a point on the equator of the hemihypersphere.
double mInc = Math.PI * 2 / m_params.M;
Func <RLD_outputs, int, int, Vector3D> onPlane = (o, k, m) =>
{
double theta = k == -1 ? 0 : outputs.x_i[k];
theta += Math.PI / 2;
return
(Sterographic.SphereToPlane(
SphericalCoords.SphericalToCartesian(
new Vector3D(1, theta, m * mInc)
)
));
};
// Setup texture coords on fundamental triangle.
// We'll use a fundamental triangle in the southern hemisphere,
// with stereographically projected coords at (0,0), (1,0), and CCW on the unit circle depending on M.
Polygon p = new Polygon();
p.Segments.Add(Segment.Line(new Vector3D(), new Vector3D(1, 0)));
p.Segments.Add(Segment.Arc(new Vector3D(1, 0), onPlane(outputs, 1, 1), onPlane(outputs, -1, 1)));
p.Segments.Add(Segment.Line(onPlane(outputs, -1, 1), new Vector3D()));
int levels = 9;
TextureHelper.SetLevels(levels);
Vector3D[] coords = TextureHelper.TextureCoords(p, Geometry.Spherical, doGeodesicDome: true);
int[] elementIndices = TextureHelper.TextureElements(1, levels);
// Setup a nearTree for the catenoid locations (on the plane).
NearTree nearTree = new NearTree(Metric.Spherical);
for (int k = 1; k < outputs.x_i.Length; k++)
{
for (int m = 0; m <= 1; m++)
{
Vector3D loc = onPlane(outputs, k, m);
nearTree.InsertObject(new NearTreeObject()
{
ID = k, Location = loc
});
}
}
// Given a point on the plane, find the nearest catenoid center and calculate the height of the surface based on that.
// This also calculates the locking of the point.
Func <Vector3D, Tuple <double, Vector3D, Vector3D> > heightAndLocking = coord =>
{
NearTreeObject closest;
if (!nearTree.FindNearestNeighbor(out closest, coord, double.MaxValue))
{
throw new System.Exception();
}
Vector3D locked = new Vector3D();
if (p.Segments[0].IsPointOn(coord) ||
p.Segments[2].IsPointOn(coord))
{
locked = new Vector3D(1, 1, 0, 0);
}
//if( p.Segments[1].IsPointOn( v ) ) // Not working right for some reason, but line below will work.
if (Tolerance.Equal(coord.Abs(), 1))
{
locked = new Vector3D(1, 1, 1, 0);
}
Vector3D vSphere = Sterographic.PlaneToSphere(coord);
Vector3D cSphere = Sterographic.PlaneToSphere(closest.Location);
double dist = vSphere.AngleTo(cSphere);
int k = (int)closest.ID;
double waist = outputs.t_i[k];
double rld_height = outputs.phi_i[k];
double h = waist * 3.5 * 2; // height where catenoid will meet rld_height.
double factor = scale * rld_height * 2 / h; // Artifical scaling so we can see things.
dist /= factor;
double z = double.NaN;
if (dist >= waist)
{
z = waist * DonHatch.acosh(dist / waist);
}
else if (dist >= 0.7 * waist)
{
z = 0;
// Move the coord to the thinnest waist circle.
Mobius m = new Mobius();
m.Hyperbolic(Geometry.Spherical, coord.ToComplex(), waist / dist);
coord = m.Apply(coord);
}
if (dist < waist * 20)
{
locked = new Vector3D(1, 1, 1, 1);
}
return(new Tuple <double, Vector3D, Vector3D>(z * factor, locked, coord));
};
// Calculate all the coordinates.
Vector3D[] locks = new Vector3D[coords.Length];
for (int i = 0; i < coords.Length; i++)
{
Vector3D coord = coords[i];
var hl = heightAndLocking(coord);
locks[i] = hl.Item2;
coord = hl.Item3;
coords[i] = Normal(Sterographic.PlaneToSphere(coord), (double)hl.Item1);
}
// Relax it.
Relax(coords, elementIndices, locks);
Mesh mesh = new Mesh();
Sphere s = new Sphere();
for (int i = 0; i < elementIndices.Length; i += 3)
{
Vector3D a = coords[elementIndices[i]];
Vector3D b = coords[elementIndices[i + 1]];
Vector3D c = coords[elementIndices[i + 2]];
if (a.DNE || b.DNE || c.DNE)
{
continue;
}
for (int m = 0; m <= 0; m++)
{
mesh.Triangles.Add(new Mesh.Triangle(a, b, c));
mesh.Triangles.Add(new Mesh.Triangle(
s.ReflectPoint(a),
s.ReflectPoint(b),
s.ReflectPoint(c)));
a.RotateXY(mInc);
b.RotateXY(mInc);
c.RotateXY(mInc);
}
}
PovRay.WriteMesh(mesh, "RLD.pov");
}
