public static void HopfOrbit()
{
List <Vector3D> s2Points = new List <Vector3D>();
for (double theta = Math.PI * .1; theta <= Math.PI * .9; theta += Math.PI * .2)
{
for (double lon = -Math.PI; lon <= Math.PI; lon += Math.PI / 10)
{
s2Points.Add(SphericalCoords.SphericalToCartesian(new Vector3D(1.0, theta, lon)));
}
}
using (StreamWriter sw = File.CreateText(@".\out.pov"))
{
System.Func <Vector3D, Sphere> sizeFunc = v => new Sphere()
{
Center = v, Radius = 0.01
};
foreach (Vector3D s2Point in s2Points)
{
Vector3D[] circlePoints = OneHopfCircle(s2Point);
//for( int i = 0; i < circlePoints.Length; i++ )
// circlePoints[i] = circlePoints[i].ProjectTo3DSafe( 1.0 );
// Note: effectively orthogonal projects here because EdgeSphereSweep doesn't write W coord.
string circleString = PovRay.EdgeSphereSweep(circlePoints, sizeFunc);
sw.WriteLine(circleString);
}
}
}
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>
/// 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 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");
}
/// <summary>
/// This allows us to change the model we have on the plane.
/// We usually want UHS, but for Pov-Ray mapping these images to a sphere, we need to have it be an equirectangular projection
/// NOTE: The bounds should be set to 1.0 for this to work! v.X and v.Y must be in-between -1 and 1. (also, don't rotate simplex mirrors, for POV-Ray anyway)
/// </summary>
private Vector3D PlaneModelToBall(Vector3D v, double t = 0.0)
{
bool equirectangular = false;
if (!equirectangular)
{
// Normal UHS (sterographic projection).
return(H3Models.UHSToBall(v));
}
else
{
// If you want output to have twice the width.
double xScale = 2;
v.X /= xScale;
// 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(ApplyTransformationToSphere(onBall, t));
}
}
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");
}
/// <summary>
/// Converts to cartesian coordinates.
/// </summary>
public Vector3 ToCartesian()
{
return(SphericalCoords.SphericalToCartesian(this));
}
