public static Vector3D EdgeToPlane(Geometry g, H3.Cell.Edge edge)
{
if (g != Geometry.Hyperbolic)
{
throw new System.NotImplementedException();
}
Vector3D b1, b2;
H3Models.Ball.GeodesicIdealEndpoints(edge.Start, edge.End, out b1, out b2);
if (((b2 + b1) / 2).IsOrigin)
{
Vector3D lineNormal = b2 - b1;
lineNormal.RotateXY(Math.PI / 2);
lineNormal.Normalize();
return(new Vector3D(lineNormal.X, lineNormal.Y, lineNormal.Z, 0));
}
Vector3D center, normal;
double radius, angleTot;
H3Models.Ball.Geodesic(edge.Start, edge.End, out center, out radius, out normal, out angleTot);
Vector3D closest = H3Models.Ball.ClosestToOrigin(new Circle3D()
{
Center = center, Radius = radius, Normal = normal
});
Vector3D closestKlein = HyperbolicModels.PoincareToKlein(closest);
center.Normalize();
return(new Vector3D(center.X, center.Y, center.Z, closestKlein.Abs()));
}
// So that we can leverage Euclidean barycentric coordinates, we will first convert our simplex to the Klein model.
// We will need to take care to properly convert back to the Ball as needed.
public static Vector3D[] KleinVerts(Geometry g, Vector3D[] conformalVerts)
{
return(conformalVerts.Select(v =>
{
switch (g)
{
case Geometry.Spherical:
return SphericalModels.StereoToGnomonic(v);
case Geometry.Euclidean:
return v;
case Geometry.Hyperbolic:
return HyperbolicModels.PoincareToKlein(v);
}
throw new System.ArgumentException();
}).ToArray());
}
public static void Polarity()
{
TilingConfig config = new TilingConfig(3, 7, 50);
Tiling tiling = new Tiling();
tiling.GenerateInternal(config);
List <Vector3D> points = new List <Vector3D>();
List <H3.Cell.Edge> edges = new List <H3.Cell.Edge>();
foreach (Polygon p in tiling.Tiles.Select(t => t.Boundary))
{
foreach (Segment s in p.Segments)
{
foreach (Vector3D v in s.Subdivide(25))
{
Vector3D klein = HyperbolicModels.PoincareToKlein(v);
H3.Cell.Edge e = Dual(klein);
points.Add(klein);
edges.Add(e);
}
}
}
using (StreamWriter sw = File.CreateText("polarity.pov"))
{
double rad = 0.01;
foreach (Vector3D vert in points)
{
sw.WriteLine(PovRay.Sphere(new Sphere()
{
Center = vert, Radius = rad
}));
}
foreach (H3.Cell.Edge edge in edges)
{
sw.WriteLine(PovRay.Cylinder(edge.Start, edge.End, rad / 2));
}
}
}
public static Circle3D GetCircleForBallPoint(Vector3D p)
{
Sphere ball = new Sphere();
p = SphericalModels.GnomonicToStereo(p);
if (Tolerance.GreaterThanOrEqual(p.Abs(), 1))
{
return(null);
}
Sphere t = H3Models.Ball.OrthogonalSphereInterior(p);
//return H3Models.Ball.IdealCircle( t );
// Get the corresponding point on the exterior (our inversion).
p = HyperbolicModels.PoincareToKlein(p);
p = HyperbolicModels.PoincareToKlein(p);
p = ball.ReflectPoint(p);
return(GetCircle(p));
}
// CHEAT! (would be better to do a geometrical construction)
// We are going to iterate to the starting point that will make all edge lengths the same.
public static Vector3D IterateToStartingPoint(HoneycombDef?def, int[] activeMirrors, Simplex simplex)
{
if (activeMirrors.Length == 1)
{
return(simplex.Verts[activeMirrors[0]]);
}
// We are minimizing the output of this function,
// because we want all edge lengths to be as close as possible.
// Input vector should be in the Ball Model.
Func <Vector3D, double> diffFunc = v =>
{
List <double> lengths = new List <double>();
for (int i = 0; i < activeMirrors.Length; i++)
{
Vector3D reflected = simplex.ReflectInFacet(v, activeMirrors[i]);
lengths.Add(H3Models.Ball.HDist(v, reflected));
}
double result = 0;
double average = lengths.Average();
foreach (double length in lengths)
{
result += Math.Abs(length - average);
}
if (Infinity.IsInfinite(result))
{
result = double.PositiveInfinity;
}
return(result);
};
// So that we can leverage Euclidean barycentric coordinates, we will first convert our simplex to the Klein model.
// We will need to take care to properly convert back to the Ball as needed.
Vector3D[] kleinVerts = simplex.Verts.Select(v => HyperbolicModels.PoincareToKlein(v)).ToArray();
if (def != null)
{
HoneycombDef d = def.Value;
Geometry vertexGeometry = Geometry2D.GetGeometry(d.Q, d.R);
if (vertexGeometry == Geometry.Hyperbolic)
{
kleinVerts[3] = SimplexCalcs.VertexPointKlein(d.P, d.Q, d.R);
}
}
// Normalizing barycentric coords amounts to making sure the 4 coords add to 1.
Func <Vector3D, Vector3D> baryNormalize = b =>
{
return(b / (b.X + b.Y + b.Z + b.W));
};
// Bary Coords to Euclidean
Func <Vector3D[], Vector3D, Vector3D> baryToEuclidean = (kv, b) =>
{
Vector3D result =
kv[0] * b.X + kv[1] * b.Y + kv[2] * b.Z + kv[3] * b.W;
return(result);
};
// Our starting barycentric coords (halfway between all active mirrors).
Vector3D bary = new Vector3D();
foreach (int a in activeMirrors)
{
bary[a] = 0.5;
}
bary = baryNormalize(bary);
// For each iteration, we'll shrink this search offset.
// NOTE: The starting offset and decrease factor I'm using don't guarantee convergence,
// but it seems to be working pretty well (even when varying these parameters).
//double searchOffset = 1.0 - bary[activeMirrors[0]];
//double searchOffset = bary[activeMirrors[0]];
double factor = 1.5; // Adjusting this helps get some to converge, e.g. 4353-1111
double searchOffset = bary[activeMirrors[0]] / factor;
double min = double.MaxValue;
int iterations = 1000;
for (int i = 0; i < iterations; i++)
{
min = diffFunc(HyperbolicModels.KleinToPoincare(baryToEuclidean(kleinVerts, bary)));
foreach (int a in activeMirrors)
{
Vector3D baryTest1 = bary, baryTest2 = bary;
baryTest1[a] += searchOffset;
baryTest2[a] -= searchOffset;
baryTest1 = baryNormalize(baryTest1);
baryTest2 = baryNormalize(baryTest2);
double t1 = diffFunc(HyperbolicModels.KleinToPoincare(baryToEuclidean(kleinVerts, baryTest1)));
double t2 = diffFunc(HyperbolicModels.KleinToPoincare(baryToEuclidean(kleinVerts, baryTest2)));
if (t1 < min)
{
min = t1;
bary = baryTest1;
}
if (t2 < min)
{
min = t2;
bary = baryTest2;
}
}
if (Tolerance.Equal(min, 0.0, 1e-14))
{
System.Console.WriteLine(string.Format("Converged in {0} iterations.", i));
break;
}
searchOffset /= factor;
}
if (!Tolerance.Equal(min, 0.0, 1e-14))
{
System.Console.WriteLine("Did not converge: " + min);
// Be a little looser before thrown an exception.
if (!Tolerance.Equal(min, 0.0, 1e-12))
{
System.Console.ReadKey(true);
//throw new System.Exception( "Boo. We did not converge." );
return(Vector3D.DneVector());
}
}
Vector3D euclidean = baryToEuclidean(kleinVerts, bary);
return(HyperbolicModels.KleinToPoincare(euclidean));
}
public static H3.Cell.Edge[] OneHoneycombOrthoscheme(HoneycombDef def, int[] active, int baseHue, Settings settings = null)
{
// Setup parameters.
int numEdges = 250000;
if (settings != null)
{
active = settings.PovRay.Active;
def = new HoneycombDef(settings.P, settings.Q, settings.R);
numEdges = settings.PovRay.NumEdges;
}
CalcThickness(active);
if (settings != null)
{
H3.m_settings.AngularThickness = settings.PovRay.EdgeWidth; // ZZZ - should really stop using that settings class.
}
string baseName = BaseName(def);
string mirrorsString = ActiveMirrorsString(active);
string suffix = "-" + mirrorsString;
string fileName = baseName + suffix;
if (ViewPath != null)
{
fileName += string.Format("_{0:D4}", ViewPath.Step);
}
if (File.Exists(fileName + ".pov"))
{
File.Delete(fileName + ".pov");
//Console.WriteLine( string.Format( "Skipping {0}", fileName ) );
//return;
}
Program.Log(string.Format("Building {0}", fileName));
// The wiki mirrors are labeled in the reverse of ours.
Func <int, int> mapMirror = i => 3 - i;
active = active.Select(i => mapMirror(i)).OrderBy(i => i).ToArray();
Simplex simplex = new Simplex();
simplex.Facets = SimplexCalcs.Mirrors(def.P, def.Q, def.R);
simplex.Verts = SimplexCalcs.VertsBall(def.P, def.Q, def.R);
Vector3D startingPoint = IterateToStartingPoint(def, active, simplex);
if (startingPoint.DNE)
{
return(null);
}
List <H3.Cell.Edge> startingEdges = new List <H3.Cell.Edge>();
foreach (int a in active)
{
Vector3D reflected = simplex.ReflectInFacet(startingPoint, a);
startingEdges.Add(new H3.Cell.Edge(startingPoint, reflected));
//startingEdges.Add( new H3.Cell.Edge( simplex.Verts[0], simplex.Verts[3] ) ); // Used for Borromean Rings complement image.
}
if (false)
{
Vector3D[] kv = simplex.Verts.Select(v => HyperbolicModels.PoincareToKlein(v)).ToArray();
kv[3] = SimplexCalcs.VertexPointKlein(def.P, def.Q, def.R);
Vector3D t = (kv[3] - kv[0]) * 0.5;
Sphere gSphere = H3Models.Ball.OrthogonalSphereInterior(HyperbolicModels.KleinToPoincare(t));
gSphere = H3Models.BallToKlein(gSphere);
Vector3D t2 = Euclidean3D.IntersectionPlaneLine(gSphere.Normal, gSphere.Offset, kv[3] - kv[2], kv[2]);
//t2 = kv[2] + ( kv[3] - kv[2]) * 0.5;
t = HyperbolicModels.KleinToPoincare(t);
t2 = HyperbolicModels.KleinToPoincare(t2);
startingEdges.Add(new H3.Cell.Edge(t, t2));
startingEdges.Add(new H3.Cell.Edge(t, simplex.ReflectInFacet(t, 3)));
}
// If we are doing a view path, transform our geometry.
if (ViewPath != null)
{
//Vector3D p = new Vector3D( 0, 0, .5 );
Vector3D p = new Vector3D(0.08, 0.12, 0.07);
simplex.Facets = simplex.Facets.Select(f => H3Models.Transform_PointToOrigin(f, p)).ToArray();
simplex.Verts = simplex.Verts.Select(v => H3Models.Transform_PointToOrigin(v, p)).ToArray();
startingEdges = startingEdges.Select(e => new H3.Cell.Edge(
H3Models.Transform_PointToOrigin(e.Start, p),
H3Models.Transform_PointToOrigin(e.End, p))).ToList();
}
SetupBaseHue(fileName, mirrorsString, baseHue);
Recurse.m_background = baseHue == -1 ? new Vector3D() : new Vector3D(baseHue, 1, .1);
H3.Cell.Edge[] edges = Recurse.CalcEdgesSmart2(simplex.Facets, startingEdges.ToArray(), numEdges);
//H3.Cell.Edge[] edges = Recurse.CalcEdges( simplex.Facets, startingEdges.ToArray(),
// new Recurse.Settings() { ThreshType = Recurse.EdgeThreshType.Radial, Threshold = H3Models.Ball.FindLocationForDesiredRadius( settings.PovRay.EdgeWidth, 0.8/100 ) } );
//edges = edges.Where( e => e.Depths[0] % 2 == 1 ).ToArray(); // Used for Borromean Rings complement image.
// Shapeways truncated 436.
if (false)
{
if (true)
{
Mobius m = Mobius.Scale(1.0 / H3Models.UHS.ToE(Honeycomb.InRadius(def.P, def.Q, def.R)));
double a = -Math.PI / 2 + Math.Asin(1 / Math.Sqrt(3));
edges = edges.Select(e =>
{
Vector3D v1 = e.Start;
Vector3D v2 = e.End;
v1.RotateAboutAxis(new Vector3D(1, 0, 0), a);
v2.RotateAboutAxis(new Vector3D(1, 0, 0), a);
v1 = H3Models.Ball.ApplyMobius(m, v1);
v2 = H3Models.Ball.ApplyMobius(m, v2);
return(new H3.Cell.Edge(v1, v2));
}).ToArray();
double thresh = -.01;
Vector3D looking = new Vector3D(0, 0, -1);
edges = edges.Where(e => e.Start.Dot(looking) > thresh && e.End.Dot(looking) > thresh).ToArray();
Dictionary <H3.Cell.Edge, int> edgeDict = edges.ToDictionary(e => e, e => 1);
H3.RemoveDanglingEdgesRecursive(edgeDict);
edges = edgeDict.Keys.ToArray();
}
else
{
Mobius m = Mobius.Scale(2);
edges = edges.Select(e =>
{
Vector3D v1 = e.Start;
Vector3D v2 = e.End;
v1 = H3Models.Ball.ApplyMobius(m, v1);
v2 = H3Models.Ball.ApplyMobius(m, v2);
return(new H3.Cell.Edge(v1, v2));
}).ToArray();
Dictionary <H3.Cell.Edge, int> edgeDict = edges.ToDictionary(e => e, e => 1);
H3.RemoveDanglingEdgesRecursive(edgeDict);
edges = edgeDict.Keys.ToArray();
}
}
//H3.m_settings.Output = H3.Output.STL;
//H3.m_settings.Scale = 50;
H3.SaveToFile(fileName, edges, finite: true, append: true);
bool doCells = false;
H3.Cell[] cellsToHighlight = null;
if (doCells)
{
int[] polyMirrors = new int[] { 1, 2, 3 };
active = active.Select(i => mapMirror(i)).OrderBy(i => i).ToArray();
H3.Cell startingCell = PolyhedronToHighlight(Geometry.Hyperbolic, polyMirrors, simplex, startingPoint);
cellsToHighlight = Recurse.CalcCells(simplex.Facets, new H3.Cell[] { startingCell });
H3.AppendFacets(fileName, cellsToHighlight);
}
return(edges);
}
private static void HyperidealSquares()
{
Mobius rot = new Mobius();
rot.Isometry(Geometry.Spherical, Math.PI / 4, new Vector3D());
List <Segment> segs = new List <Segment>();
int[] qs = new int[] { 5, -1 };
foreach (int q in qs)
{
TilingConfig config = new TilingConfig(4, q, 1);
Tile t = Tiling.CreateBaseTile(config);
List <Segment> polySegs = t.Boundary.Segments;
polySegs = polySegs.Select(s => { s.Transform(rot); return(s); }).ToList();
segs.AddRange(polySegs);
}
Vector3D v1 = new Vector3D(1, 0);
v1.RotateXY(Math.PI / 6);
Vector3D v2 = v1;
v2.Y *= -1;
Vector3D cen;
double rad;
H3Models.Ball.OrthogonalCircle(v1, v2, out cen, out rad);
Segment seg = Segment.Arc(v1, v2, cen, false);
rot.Isometry(Geometry.Spherical, Math.PI / 2, new Vector3D());
for (int i = 0; i < 4; i++)
{
seg.Transform(rot);
segs.Add(seg.Clone());
}
SVG.WriteSegments("output1.svg", segs);
System.Func <Segment, Segment> PoincareToKlein = s =>
{
return(Segment.Line(
HyperbolicModels.PoincareToKlein(s.P1),
HyperbolicModels.PoincareToKlein(s.P2)));
};
segs = segs.Select(s => PoincareToKlein(s)).ToList();
Vector3D v0 = new Vector3D(v1.X, v1.X);
Vector3D v3 = v0;
v3.Y *= -1;
Segment seg1 = Segment.Line(v0, v1), seg2 = Segment.Line(v2, v3);
Segment seg3 = Segment.Line(new Vector3D(1, 1), new Vector3D(1, -1));
for (int i = 0; i < 4; i++)
{
seg1.Transform(rot);
seg2.Transform(rot);
seg3.Transform(rot);
segs.Add(seg1.Clone());
segs.Add(seg2.Clone());
segs.Add(seg3.Clone());
}
SVG.WriteSegments("output2.svg", segs);
}