private static void HopfFibration(Tiling tiling)
{
int segDivisions = 10;
Shapeways mesh = new Shapeways();
HashSet <Vector3D> done = new HashSet <Vector3D>();
foreach (Tile tile in tiling.Tiles)
{
foreach (Segment seg in tile.Boundary.Segments)
{
if (done.Contains(seg.Midpoint))
{
continue;
}
// Subdivide the segment, and project points to S2.
Vector3D[] points = seg.Subdivide(segDivisions).Select(v => Spherical2D.PlaneToSphere(v)).ToArray();
foreach (Vector3D point in points)
{
Vector3D[] circlePoints = OneHopfCircle(point);
ProjectAndAddS3Points(mesh, circlePoints, shrink: false);
}
done.Add(seg.Midpoint);
}
}
STL.SaveMeshToSTL(mesh.Mesh, @"D:\p4\R3\sample\out1.stl");
}
public static void EdgesToStl( H3.Cell.Edge[] edges )
{
Shapeways mesh = new Shapeways();
int divisions = 10;
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.GnomonicToStereo( mesh.Mesh.Triangles[i].a ),
SphericalModels.GnomonicToStereo( mesh.Mesh.Triangles[i].b ),
SphericalModels.GnomonicToStereo( mesh.Mesh.Triangles[i].c ) );
}
STL.SaveMeshToSTL( mesh.Mesh, @"output.stl" );
}
public static void GenEuclidean()
{
Shapeways mesh = new Shapeways();
HashSet <H3.Cell.Edge> completed = new HashSet <H3.Cell.Edge>();
int count = 5;
for (int i = -count; i < count; i++)
{
for (int j = -count; j < count; j++)
{
for (int k = -count; k < count; k++)
{
// Offset
double io = i + 0.5;
double jo = j + 0.5;
double ko = k + 0.5;
// Do every edge emanating from this point.
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io + 1, jo, ko));
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io - 1, jo, ko));
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io, jo + 1, ko));
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io, jo - 1, ko));
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io, jo, ko + 1));
AddEuclideanEdge(mesh, completed, new Vector3D(io, jo, ko), new Vector3D(io, jo, ko - 1));
}
}
}
STL.SaveMeshToSTL(mesh.Mesh, "d:\\temp\\434.stl");
}
private static void HopfFibration(Tiling tiling)
{
int segDivisions = 10;
int circleDivisions = 125;
Shapeways mesh = new Shapeways();
HashSet <Vector3D> done = new HashSet <Vector3D>();
foreach (Tile tile in tiling.Tiles)
{
foreach (Segment seg in tile.Boundary.Segments)
{
if (done.Contains(seg.Midpoint))
{
continue;
}
// Subdivide the segment, and project points to S2.
Vector3D[] points = seg.Subdivide(segDivisions).Select(v => Spherical2D.PlaneToSphere(v)).ToArray();
foreach (Vector3D point in points)
{
// Get the hopf circle and add to mesh.
// http://en.wikipedia.org/wiki/Hopf_fibration#Explicit_formulae
double a = point.X;
double b = point.Y;
double c = point.Z;
double factor = 1 / (Math.Sqrt(1 + c));
if (Tolerance.Equal(c, -1))
{
continue;
}
List <Vector3D> circlePoints = new List <Vector3D>();
double angleInc = 2 * Math.PI / circleDivisions;
double angle = 0;
for (int i = 0; i <= circleDivisions; i++)
{
double sinTheta = Math.Sin(angle);
double cosTheta = Math.Cos(angle);
circlePoints.Add(new Vector3D(
(1 + c) * cosTheta,
a * sinTheta - b * cosTheta,
a * cosTheta + b * sinTheta,
(1 + c) * sinTheta));
angle += angleInc;
}
bool shrink = false;
ProjectAndAddS3Points(mesh, circlePoints.ToArray(), shrink);
}
done.Add(seg.Midpoint);
}
}
STL.SaveMeshToSTL(mesh.Mesh, @"D:\p4\R3\sample\out1.stl");
}
/// <summary>
/// Add a finite (truncated) banana to our mesh. Passed in edge should be in Ball model.
/// </summary>
public static void AddBanana( Shapeways mesh, Vector3D e1, Vector3D e2, H3.Settings settings )
{
Vector3D e1UHS = H3Models.BallToUHS( e1 );
Vector3D e2UHS = H3Models.BallToUHS( e2 );
// Endpoints of the goedesic on the z=0 plane.
Vector3D z1, z2;
H3Models.UHS.GeodesicIdealEndpoints( e1UHS, e2UHS, out z1, out z2 );
// XXX - Do we want to do a better job worrying about rotation here?
// (multiply by complex number with certain imaginary part as well)
//Vector3D z3 = ( z1 + z2 ) / 2;
//if( Infinity.IsInfinite( z3 ) )
// z3 = new Vector3D( 1, 0 );
Vector3D z3 = new Vector3D( Math.E, Math.PI ); // This should vary the rotations a bunch.
// Find the Mobius we need.
// We'll do this in two steps.
// (1) Find a mobius taking z1,z2 to origin,inf
// (2) Deal with scaling e1 to a height of 1.
Mobius m1 = new Mobius( z1, z3, z2 );
Vector3D e1UHS_transformed = m1.ApplyToQuaternion( e1UHS );
double scale = 1.0 / e1UHS_transformed.Z;
Mobius m2 = Mobius.Scale( scale );
Mobius m = m2 * m1; // Compose them (multiply in reverse order).
Vector3D e2UHS_transformed = m.ApplyToQuaternion( e2UHS );
// Make our truncated cone.
// For regular tilings, we really would only need to do this once for a given LOD.
List<Vector3D> points = new List<Vector3D>();
double logHeight = Math.Log( e2UHS_transformed.Z );
if( logHeight < 0 )
throw new System.Exception( "impl issue" );
int div1, div2;
H3Models.Ball.LOD_Finite( e1, e2, out div1, out div2, settings );
double increment = logHeight / div1;
for( int i=0; i<=div1; i++ )
{
double h = increment * i;
// This is to keep different bananas from sharing exactly coincident vertices.
double tinyOffset = 0.001;
if( i == 0 )
h -= tinyOffset;
if( i == div1 )
h += tinyOffset;
Vector3D point = new Vector3D( 0, 0, Math.Exp( h ) );
points.Add( point );
}
Shapeways tempMesh = new Shapeways();
tempMesh.Div = div2;
tempMesh.AddCurve( points.ToArray(), v => H3Models.UHS.SizeFunc( v, settings.AngularThickness ) );
// Unwind the transforms.
TakePointsBack( tempMesh.Mesh, m.Inverse(), settings );
mesh.Mesh.Triangles.AddRange( tempMesh.Mesh.Triangles );
}
/// <summary>
/// Function I was using for lamp project, probably a bit out of date.
/// </summary>
public static void AddToMeshLamp( Shapeways mesh, Vector3D v1, Vector3D v2 )
{
// need to get these from CalcBallArc
Vector3D center = Vector3D.DneVector();
double radius = double.NaN;
Vector3D normal = Vector3D.DneVector();
double angleTot = double.NaN;
double length1 = Scale( 2.1 ); // 12-end piece
//double length1 = Scale( 1.6 ); // 6-end piece
//double length1 = Scale( 1.4 ); // 4-end piece
double length2 = Scale( 0.5 );
double outerRadStart = Scale( 0.0625 / 2 );
double outerRadEnd = Scale( 0.25 / 2 );
System.Func<Vector3D, double> outerSizeFunc = v =>
{
double angle = (v1 - center).AngleTo( v - center );
double len = radius * angle;
return outerRadStart + (outerRadEnd - outerRadStart) * (len / length1);
};
System.Func<Vector3D, double> outerSizeFunc2 = v =>
{
double angle = (v2 - center).AngleTo( v - center );
double len = radius * angle;
return outerRadStart + (outerRadEnd - outerRadStart) * (len / length1);
};
System.Func<Vector3D, double> innerSizeFunc = v =>
{
// Very slightly bigger than 1/8 inch OD.
return Scale( 0.13 / 2 );
};
Vector3D[] outerPoints = Shapeways.CalcArcPoints( center, radius, v1, normal, length1 / radius );
Vector3D[] innerPoints = Shapeways.CalcArcPoints( center, radius, outerPoints[outerPoints.Length - 1], normal * -1, length2 / radius );
mesh.AddCornucopia( outerPoints, outerSizeFunc, innerPoints, innerSizeFunc );
outerPoints = Shapeways.CalcArcPoints( center, radius, v2, normal * -1, length1 / radius );
innerPoints = Shapeways.CalcArcPoints( center, radius, outerPoints[outerPoints.Length - 1], normal, length2 / radius );
mesh.AddCornucopia( outerPoints, outerSizeFunc2, innerPoints, innerSizeFunc );
m_inventory.AddRod( Rod.Create( radius, angleTot ) );
}
/// <summary>
/// Helper to project points from S3 -> S2, then add an associated curve.
/// </summary>
private static void ProjectAndAddS3Points(Shapeways mesh, Vector3D[] pointsS3, bool shrink)
{
// Project to S3, then to R3.
List <Vector3D> projected = new List <Vector3D>();
foreach (Vector3D v in pointsS3)
{
v.Normalize();
Vector3D c = v.ProjectTo3DSafe(1.0);
// Pull R3 into a smaller open disk.
if (shrink)
{
double mag = Math.Atan(c.Abs());
c.Normalize();
c *= mag;
}
projected.Add(c);
}
System.Func <Vector3D, double> sizeFunc = v =>
{
// Constant thickness.
// return 0.08;
double sphericalThickness = 0.002;
double abs = v.Abs();
if (shrink)
{
abs = Math.Tan(abs); // The unshrunk abs.
}
// The thickness at this vector location.
double result = Spherical2D.s2eNorm(Spherical2D.e2sNorm(abs) + sphericalThickness) - abs;
if (shrink)
{
result *= Math.Atan(abs) / abs; // shrink it back down.
}
return(result);
};
mesh.AddCurve(projected.ToArray(), sizeFunc);
}
private static void AddEuclideanEdge(Shapeways mesh, HashSet <H3.Cell.Edge> completed, Vector3D start, Vector3D end)
{
double cutoff = 1.75;
if (Math.Abs(start.X) > cutoff || Math.Abs(start.Y) > cutoff || Math.Abs(start.Z) > cutoff ||
Math.Abs(end.X) > cutoff || Math.Abs(end.Y) > cutoff || Math.Abs(end.Z) > cutoff)
{
return;
}
if (mesh != null)
{
AddEuclideanEdgeToMesh(mesh, completed, start, end);
return;
}
completed.Add(new H3.Cell.Edge(start, end));
}
public static void GenEuclidean()
{
Shapeways mesh = new Shapeways();
HashSet <H3.Cell.Edge> completed = new HashSet <H3.Cell.Edge>(new H3.Cell.EdgeEqualityComparer());
int count = 2;
for (int i = -count; i < count; i++)
{
for (int j = -count; j < count; j++)
{
for (int k = -count; k < count; k++)
{
// Offset
double io = i + 0.5;
double jo = j + 0.5;
double ko = k + 0.5;
Vector3D p = new Vector3D(io, jo, ko);
// Add a sphere for this point.
Sphere s = new Sphere()
{
Center = p, Radius = 0.05
};
mesh.AddSphere(s.Center, s.Radius);
// Do every edge emanating from this point.
AddEuclideanEdge(mesh, completed, p, new Vector3D(io + 1, jo, ko));
AddEuclideanEdge(mesh, completed, p, new Vector3D(io - 1, jo, ko));
AddEuclideanEdge(mesh, completed, p, new Vector3D(io, jo + 1, ko));
AddEuclideanEdge(mesh, completed, p, new Vector3D(io, jo - 1, ko));
AddEuclideanEdge(mesh, completed, p, new Vector3D(io, jo, ko + 1));
AddEuclideanEdge(mesh, completed, p, new Vector3D(io, jo, ko - 1));
}
}
}
STL.SaveMeshToSTL(mesh.Mesh, "434.stl");
//PovRay.WriteEdges( new PovRay.Parameters() { AngularThickness = .05 }, Geometry.Euclidean, completed.ToArray(), "434.pov", append: false );
}
private static void AddEuclideanEdge(Shapeways mesh, HashSet <H3.Cell.Edge> completed, Vector3D start, Vector3D end)
{
H3.Cell.Edge edge = new H3.Cell.Edge(start, end);
if (completed.Contains(edge))
{
return;
}
Shapeways tempMesh = new Shapeways();
Segment seg = Segment.Line(start, end);
int div = 20 - (int)(start.Abs() * 4);
if (div < 1)
{
div = 1;
}
tempMesh.AddCurve(seg.Subdivide(div), .05);
Transform(tempMesh.Mesh);
mesh.Mesh.Triangles.AddRange(tempMesh.Mesh.Triangles);
completed.Add(edge);
}
private static void ShapewaysPolytopes()
{
VEF loader = new VEF();
loader.Load( @"C:\Users\roice\Documents\projects\vZome\VefProjector\data\24cell-cellFirst.vef" );
int divisions = 25;
Shapeways mesh = new Shapeways();
//int count = 0;
foreach( Edge edge in loader.Edges )
{
Segment seg = Segment.Line(
loader.Vertices[edge.V1].ConvertToReal(),
loader.Vertices[edge.V2].ConvertToReal() );
Vector3D[] points = seg.Subdivide( divisions );
bool shrink = true;
ProjectAndAddS3Points( mesh, points, shrink );
//if( count++ > 10 )
// break;
}
STL.SaveMeshToSTL( mesh.Mesh, @"D:\p4\R3\sample\out1.stl" );
}
/// <summary>
/// Helper to project points from S3 -> S2, then add an associated curve.
/// XXX - Not completely correct.
/// </summary>
private static void ProjectAndAddS3Points( Shapeways mesh, Vector3D[] pointsS3, bool shrink )
{
List<Vector3D> projected = new List<Vector3D>();
foreach( Vector3D v in pointsS3 )
{
v.Normalize();
Vector3D c = v.ProjectTo3DSafe( 1.0 );
// Pull R3 into a smaller open disk.
if( shrink )
{
double mag = Math.Atan( c.Abs() );
c.Normalize();
c *= mag;
}
projected.Add( c );
}
System.Func<Vector3D, double> sizeFunc = v =>
{
// Constant thickness.
// return 0.08;
double sphericalThickness = 0.05;
double abs = v.Abs();
if( shrink )
abs = Math.Tan( abs ); // The unshrunk abs.
// The thickness at this vector location.
double result = Spherical2D.s2eNorm( Spherical2D.e2sNorm( abs ) + sphericalThickness ) - abs;
if( shrink )
result *= Math.Atan( abs ) / abs; // shrink it back down.
return result;
};
mesh.AddCurve( projected.ToArray(), sizeFunc );
}
private static void ProjectAndAddS3Points( Shapeways mesh, Vector3D[] pointsS3 )
{
double r = 0.02;
List<Vector3D> projected = new List<Vector3D>();
List<double> radii = new List<double>();
foreach( Vector3D v in pointsS3 )
{
v.Normalize();
Vector3D c = v.ProjectTo3DSafe( 1.0 );
Vector3D p;
double d;
H3Models.Ball.DupinCyclideSphere( c, r, Geometry.Spherical, out p, out d );
projected.Add( p );
radii.Add( d );
}
mesh.AddCurve( projected.ToArray(), radii.ToArray() );
}
/// <summary>
/// Our approach will be:
/// (1) Generate a portion of one cell.
/// (2) Reflect all facets in the central facet, to get things filled-in inside the central facet. (Trim small edges here?)
/// (3) Copy this region around the plane, and go back to step (2) if density is not high enough.
/// (4) Map to Ball, trimming edges that become too small.
/// NOTE: All verts are on the boundary, so we can reflect around
// in circles on the plane at infinity, rather than spheres.
/// </summary>
public static void GenerateExotic( EHoneycomb honeycomb, H3.Settings settings )
{
settings.AngularThickness = 0.17;
Tiling tiling;
Tile baseTile;
GetAssociatedTiling( honeycomb, out tiling, out baseTile );
List<H3.Cell.Edge> edges = new List<H3.Cell.Edge>();
foreach( Segment seg in baseTile.Boundary.Segments )
edges.Add( new H3.Cell.Edge( seg.P1, seg.P2 ) );
settings.Position = Polytope.Projection.FaceCentered;
double scale = 1;
Vector3D offset = new Vector3D();
if( settings.Position == Polytope.Projection.FaceCentered )
{
scale = FaceCenteredScale( baseTile.VertexCircle );
offset = new Vector3D();
}
else if( settings.Position == Polytope.Projection.EdgeCentered )
{
scale = EdgeCenteredScale( baseTile );
offset = baseTile.Boundary.Segments[0].Midpoint;
}
int iterations = m_params.Iterations;
for( int i=0; i<iterations; i++ )
edges = DoOneStep( edges, tiling, baseTile.VertexCircle );
edges = CopyAndProject( edges, tiling, scale, offset );
if( m_params.RemoveDangling )
{
Dictionary<H3.Cell.Edge,int> edgeDict = edges.ToDictionary( e => e, e => 1 );
H3.RemoveDanglingEdgesRecursive( edgeDict );
edges = edgeDict.Keys.ToList();
}
string outputFileName = H3.m_baseDir + Honeycomb.String( honeycomb, false );
System.IO.File.Delete( outputFileName );
if( m_params.Output == H3.Output.STL )
{
outputFileName += ".stl";
// Now mesh the edges.
Shapeways mesh = new Shapeways();
foreach( H3.Cell.Edge edge in edges )
{
// Append to the file vs. writing out all at once because I was running out of memory otherwise.
mesh = new Shapeways();
int div;
H3Models.Ball.LODThin( edge.Start, edge.End, out div );
mesh.Div = div;
H3.Util.AddToMeshInternal( mesh, edge.Start, edge.End );
mesh.Mesh.Scale( settings.Scale );
STL.AppendMeshToSTL( mesh.Mesh, outputFileName );
}
}
else
{
outputFileName += ".pov";
PovRay.WriteH3Edges( new PovRay.Parameters()
{
AngularThickness = settings.AngularThickness,
Halfspace = settings.Halfspace,
ThinEdges = settings.ThinEdges,
},
edges.ToArray(), outputFileName );
}
}
private static void HopfFibration( Tiling tiling )
{
int segDivisions = 10;
Shapeways mesh = new Shapeways();
HashSet<Vector3D> done = new HashSet<Vector3D>();
foreach( Tile tile in tiling.Tiles )
foreach( Segment seg in tile.Boundary.Segments )
{
if( done.Contains( seg.Midpoint ) )
continue;
// Subdivide the segment, and project points to S2.
Vector3D[] points = seg.Subdivide( segDivisions ).Select( v => Spherical2D.PlaneToSphere( v ) ).ToArray();
foreach( Vector3D point in points )
{
Vector3D[] circlePoints = OneHopfCircle( point );
ProjectAndAddS3Points( mesh, circlePoints, shrink: false );
}
done.Add( seg.Midpoint );
}
STL.SaveMeshToSTL( mesh.Mesh, @"D:\p4\R3\sample\out1.stl" );
}
public static void Generate(EHoneycomb honeycomb, H3.Settings settings)
{
// XXX - Block the same as in H3. Share code better.
H3.Cell template = null;
{
int p, q, r;
Honeycomb.PQR(honeycomb, out p, out q, out r);
// Get data we need to generate the honeycomb.
Polytope.Projection projection = Polytope.Projection.FaceCentered;
double phi, chi, psi;
H3.HoneycombData(honeycomb, out phi, out chi, out psi);
H3.SetupCentering(honeycomb, settings, phi, chi, psi, ref projection);
Tiling tiling = new Tiling();
TilingConfig config = new TilingConfig(p, q);
tiling.GenerateInternal(config, projection);
H3.Cell first = new H3.Cell(p, H3.GenFacets(tiling));
first.ToSphere(); // Work in ball model.
first.ScaleToCircumSphere(1.0);
first.ApplyMobius(settings.Mobius);
template = first;
}
// Center
Vector3D center = template.Center;
// Face
H3.Cell.Facet facet = template.Facets[0];
Sphere s = H3Models.Ball.OrthogonalSphereInterior(facet.Verts[0], facet.Verts[1], facet.Verts[2]);
Vector3D face = s.Center;
face.Normalize();
face *= DistOriginToOrthogonalSphere(s.Radius);
// Edge
Circle3D c;
H3Models.Ball.OrthogonalCircleInterior(facet.Verts[0], facet.Verts[1], out c);
Vector3D edge = c.Center;
edge.Normalize();
edge *= DistOriginToOrthogonalSphere(c.Radius);
// Vertex
Vector3D vertex = facet.Verts[0];
Tet fundamental = new Tet(center, face, edge, vertex);
// Recurse.
int level = 1;
Dictionary <Tet, int> completedTets = new Dictionary <Tet, int>(new TetEqualityComparer());
completedTets.Add(fundamental, level);
List <Tet> tets = new List <Tet>();
tets.Add(fundamental);
ReflectRecursive(level, tets, completedTets, settings);
Shapeways mesh = new Shapeways();
foreach (KeyValuePair <Tet, int> kvp in completedTets)
{
if (Utils.Odd(kvp.Value))
{
continue;
}
Tet tet = kvp.Key;
// XXX - really want sphere surfaces here.
mesh.Mesh.Triangles.Add(new Mesh.Triangle(tet.Verts[0], tet.Verts[1], tet.Verts[2]));
mesh.Mesh.Triangles.Add(new Mesh.Triangle(tet.Verts[0], tet.Verts[3], tet.Verts[1]));
mesh.Mesh.Triangles.Add(new Mesh.Triangle(tet.Verts[0], tet.Verts[2], tet.Verts[3]));
mesh.Mesh.Triangles.Add(new Mesh.Triangle(tet.Verts[1], tet.Verts[3], tet.Verts[2]));
}
mesh.Mesh.Scale(settings.Scale);
STL.SaveMeshToSTL(mesh.Mesh, H3.m_baseDir + "fundamental" + ".stl");
}
/// <summary>
/// Add an ideal banana to our mesh. Passed in edge should be in Ball model.
/// </summary>
public static void AddIdealBanana( Shapeways mesh, Vector3D e1, Vector3D e2, H3.Settings settings )
{
Vector3D z1 = H3Models.BallToUHS( e1 );
Vector3D z2 = H3Models.BallToUHS( e2 );
// Mobius taking z1,z2 to origin,inf
Complex dummy = new Complex( Math.E, Math.PI );
Mobius m = new Mobius( z1, dummy, z2 );
// Make our truncated cone. We need to deal with the two ideal endpoints specially.
List<Vector3D> points = new List<Vector3D>();
double logHeight = 2; // XXX - magic number, and going to cause problems for infinity checks if too big.
int div1, div2;
H3Models.Ball.LOD_Ideal( e1, e2, out div1, out div2, settings );
double increment = logHeight / div1;
for( int i=-div1; i<=div1; i+=2 )
points.Add( new Vector3D( 0, 0, Math.Exp( increment * i ) ) );
Shapeways tempMesh = new Shapeways();
tempMesh.Div = div2;
System.Func<Vector3D, double> sizeFunc = v => H3Models.UHS.SizeFunc( v, settings.AngularThickness );
//Mesh.OpenCylinder... pass in two ideal endpoints?
tempMesh.AddCurve( points.ToArray(), sizeFunc, new Vector3D(), Infinity.InfinityVector );
// Unwind the transforms.
TakePointsBack( tempMesh.Mesh, m.Inverse(), settings );
mesh.Mesh.Triangles.AddRange( tempMesh.Mesh.Triangles );
}
/// <summary>
/// A helper for adding a sphere. center should be passed in the ball model.
/// The approach is similar to how we do the bananas below.
/// </summary>
public static void AddSphere( Shapeways mesh, Vector3D center, H3.Settings settings )
{
Vector3D centerUHS = H3Models.BallToUHS( center );
// Find the Mobius we need.
// We'll do this in two steps.
// (1) Find a mobius taking center to (0,0,h).
// (2) Deal with scaling to a height of 1.
Vector3D flattened = centerUHS;
flattened.Z = 0;
Mobius m1 = new Mobius( flattened, Complex.One, Infinity.InfinityVector );
Vector3D centerUHS_transformed = m1.ApplyToQuaternion( centerUHS );
double scale = 1.0 / centerUHS_transformed.Z;
Mobius m2 = new Mobius( scale, Complex.Zero, Complex.Zero, Complex.One );
Mobius m = m2 * m1; // Compose them (multiply in reverse order).
// Add the sphere at the Ball origin.
// It will *always* be generated with the same radius.
Shapeways tempMesh = new Shapeways();
tempMesh.AddSphere( new Vector3D(), H3Models.Ball.SizeFunc( new Vector3D(), settings.AngularThickness ) );
// Unwind the transforms.
for( int i=0; i<tempMesh.Mesh.Triangles.Count; i++ )
{
tempMesh.Mesh.Triangles[i] = new Mesh.Triangle(
H3Models.BallToUHS( tempMesh.Mesh.Triangles[i].a ),
H3Models.BallToUHS( tempMesh.Mesh.Triangles[i].b ),
H3Models.BallToUHS( tempMesh.Mesh.Triangles[i].c ) );
}
Banana.TakePointsBack( tempMesh.Mesh, m.Inverse(), settings );
mesh.Mesh.Triangles.AddRange( tempMesh.Mesh.Triangles );
}