/// <summary>
/// Move from a point p1 -> p2 along a geodesic.
/// Also somewhat from Don.
/// factor can be used to only go some fraction of the distance from p1 to p2.
/// </summary>
public void Geodesic(Geometry g, Complex p1, Complex p2, double factor = 1.0)
{
Mobius t = new Mobius();
t.Isometry(g, 0, p1 * -1);
Complex p2t = t.Apply(p2);
// Only implemented for hyperbolic so far.
if (factor != 1.0 && g == Geometry.Hyperbolic)
{
double newMag = DonHatch.h2eNorm(DonHatch.e2hNorm(p2t.Magnitude) * factor);
Vector3D temp = Vector3D.FromComplex(p2t);
temp.Normalize();
temp *= newMag;
p2t = temp.ToComplex();
}
Mobius m1 = new Mobius(), m2 = new Mobius();
m1.Isometry(g, 0, p1 * -1);
m2.Isometry(g, 0, p2t);
Mobius m3 = m1.Inverse();
this = m3 * m2 * m1;
}
/// <summary>
/// This transform will map the z points to the respective w points.
/// </summary>
public void MapPoints(Complex z1, Complex z2, Complex z3, Complex w1, Complex w2, Complex w3)
{
Mobius m1 = new Mobius(), m2 = new Mobius();
m1.MapPoints(z1, z2, z3);
m2.MapPoints(w1, w2, w3);
this = m2.Inverse() * m1;
}
/// <summary>
/// Does a circle inversion in an arbitrary, generalized circle.
/// IOW, the three points may be collinear, in which case we are talking about a reflection.
/// </summary>
private void CacheCircleInversion(Complex c1, Complex c2, Complex c3)
{
Mobius toUnitCircle = new Mobius();
toUnitCircle.MapPoints(
c1, c2, c3,
new Complex(1, 0),
new Complex(-1, 0),
new Complex(0, 1));
m_cache1 = toUnitCircle;
m_cache2 = m_cache1.Inverse();
}
public void Elliptic(Geometry g, Complex fixedPlus, double angle)
{
// To the origin.
Mobius origin = new Mobius();
origin.Isometry(g, 0, fixedPlus * -1);
// Rotate.
Mobius rotate = new Mobius();
rotate.Isometry(g, angle, new Complex());
// Conjugate.
this = origin.Inverse() * rotate * origin;
}
/// <summary>
/// Move from a point p1 -> p2 along a geodesic.
/// Also somewhat from Don.
/// </summary>
public void Geodesic(Geometry g, Complex p1, Complex p2)
{
Mobius t = new Mobius();
t.Isometry(g, 0, p1 * -1);
Complex p2t = t.Apply(p2);
Mobius m1 = new Mobius(), m2 = new Mobius();
m1.Isometry(g, 0, p1 * -1);
m2.Isometry(g, 0, p2t);
Mobius m3 = m1.Inverse();
this = m3 * m2 * m1;
}
/// <summary>
/// This transform will map the z points to the respective w points.
/// </summary>
public void MapPoints( Complex z1, Complex z2, Complex z3, Complex w1, Complex w2, Complex w3 )
{
Mobius m1 = new Mobius(), m2 = new Mobius();
m1.MapPoints( z1, z2, z3 );
m2.MapPoints( w1, w2, w3 );
this = m2.Inverse() * m1;
}
/// <summary>
/// Move from a point p1 -> p2 along a geodesic.
/// Also somewhat from Don.
/// </summary>
public void Geodesic( Geometry g, Complex p1, Complex p2 )
{
Mobius t = new Mobius();
t.Isometry( g, 0, p1 * -1 );
Complex p2t = t.Apply( p2 );
Mobius m1 = new Mobius(), m2 = new Mobius();
m1.Isometry( g, 0, p1 * -1 );
m2.Isometry( g, 0, p2t );
Mobius m3 = m1.Inverse();
this = m3 * m2 * m1;
}
public void Elliptic( Geometry g, Complex fixedPlus, double angle )
{
// To the origin.
Mobius origin = new Mobius();
origin.Isometry( g, 0, fixedPlus * -1 );
// Rotate.
Mobius rotate = new Mobius();
rotate.Isometry( g, angle, new Complex() );
// Conjugate.
this = origin.Inverse() * rotate * origin;
}
/// <summary>
/// Subdivides a segment from p1->p2 with the two endpoints not on the origin, in the respective geometry.
/// </summary>
private static Vector3D[] SubdivideSegmentInGeometry( Vector3D p1, Vector3D p2, int divisions, Geometry g )
{
// Handle this specially, so we can keep things 3D if needed.
if( g == Geometry.Euclidean )
{
Segment seg = Segment.Line( p1, p2 );
return seg.Subdivide( divisions );
}
Mobius p1ToOrigin = new Mobius();
p1ToOrigin.Isometry( g, 0, -p1 );
Mobius inverse = p1ToOrigin.Inverse();
Vector3D newP2 = p1ToOrigin.Apply( p2 );
Segment radial = Segment.Line( new Vector3D(), newP2 );
Vector3D[] temp = SubdivideRadialInGeometry( radial, divisions, g );
List<Vector3D> result = new List<Vector3D>();
foreach( Vector3D v in temp )
result.Add( inverse.Apply( v ) );
return result.ToArray();
}
/// <summary>
/// Does a circle inversion in an arbitrary, generalized circle.
/// IOW, the three points may be collinear, in which case we are talking about a reflection.
/// </summary>
private void CacheCircleInversion( Complex c1, Complex c2, Complex c3 )
{
Mobius toUnitCircle = new Mobius();
toUnitCircle.MapPoints(
c1, c2, c3,
new Complex( 1, 0 ),
new Complex( -1, 0 ),
new Complex( 0, 1 ) );
m_cache1 = toUnitCircle;
m_cache2 = m_cache1.Inverse();
}
/// <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>
/// Calculate the hyperbolic midpoint of an edge.
/// Only works for non-ideal edges at the moment.
/// </summary>
public static Vector3D Midpoint( H3.Cell.Edge edge )
{
// Special case if edge has endpoint on origin.
// XXX - Really this should be special case anytime edge goes through origin.
Vector3D e1 = edge.Start;
Vector3D e2 = edge.End;
if( e1.IsOrigin || e2.IsOrigin )
{
if( e2.IsOrigin )
Utils.Swap<Vector3D>( ref e1, ref e2 );
return HalfTo( e2 );
}
// No doubt there is a much better way, but
// work in H2 slice transformed to xy plane, with e1 on x-axis.
double angle = e1.AngleTo( e2 ); // always <= 180
e1 = new Vector3D( e1.Abs(), 0 );
e2 = new Vector3D( e2.Abs(), 0 );
e2.RotateXY( angle );
// Mobius that will move e1 to origin.
Mobius m = new Mobius();
m.Isometry( Geometry.Hyperbolic, 0, -e1 );
e2 = m.Apply( e2 );
Vector3D midOnPlane = HalfTo( e2 );
midOnPlane= m.Inverse().Apply( midOnPlane );
double midAngle = e1.AngleTo( midOnPlane );
Vector3D mid = edge.Start;
mid.RotateAboutAxis( edge.Start.Cross( edge.End ), midAngle );
mid.Normalize( midOnPlane.Abs() );
return mid;
}
private Vector3D ApplyTransformationToSphere( Vector3D v, double t )
{
v = H3Models.BallToUHS( v );
// 437 (hyperbolic)
//v *= Math.Pow( 4.259171776329806, t*10-5 );
// 36i (parabolic)
//v += new Vector3D( Math.Cos( Math.PI / 6 ), Math.Sin( Math.PI / 6 ) ) * t;
// iii (loxodromic)
//Complex c = v.ToComplex();
//double x = Math.Sqrt( 2 ) - 1;
//Mobius m = new Mobius( new Complex( x, 0 ), Complex.One, new Complex( -x, 0 ) );
// 12,12,12 loxodromic
//m = new Mobius( new Complex( 0, 1 ), Complex.One, new Complex( 0, -1 ) );
/*
c = m.Apply( c );
c *= Complex.Exp( new Complex( 2.5, 4 * Math.PI ) * t );
c = m.Inverse().Apply( c );
v = Vector3D.FromComplex( c );
*/
// Center cell head in KolorEyes.
Mobius m = new Mobius();
m.UpperHalfPlane();
v = m.Inverse().Apply( v );
return H3Models.UHSToBall( v );
}
public static Sphere[] Mirrors( int p, int q, int r, ref Vector3D cellCenter, bool moveToBall = true, double scaling = -1 )
{
Geometry g = Util.GetGeometry( p, q, r );
if( g == Geometry.Spherical )
return SimplexCalcs.MirrorsSpherical( p, q, r );
else if( g == Geometry.Euclidean )
return SimplexCalcs.MirrorsEuclidean();
// This is a rotation we'll apply to the mirrors at the end.
// This is to try to make our image outputs have vertical bi-lateral symmetry and the most consistent in all cases.
// NOTE: + is CW, not CCW. (Because the way I did things, our images have been reflected vertically, and I'm too lazy to go change this.)
double rotation = Math.PI / 2;
// Some construction points we need.
Vector3D p1, p2, p3;
Segment seg = null;
TilePoints( p, q, out p1, out p2, out p3, out seg );
//
// Construct in UHS
//
Geometry cellGeometry = Geometry2D.GetGeometry( p, q );
Vector3D center = new Vector3D();
double radius = 0;
if( cellGeometry == Geometry.Spherical )
{
// Finite or Infinite r
// Spherical trig
double halfSide = Geometry2D.GetTrianglePSide( q, p );
double mag = Math.Sin( halfSide ) / Math.Cos( Util.PiOverNSafe( r ) );
mag = Math.Asin( mag );
// e.g. 43j
//mag *= 0.95;
// Move mag to p1.
mag = Spherical2D.s2eNorm( mag );
H3Models.Ball.DupinCyclideSphere( p1, mag, Geometry.Spherical, out center, out radius );
}
else if( cellGeometry == Geometry.Euclidean )
{
center = p1;
radius = p1.Dist( p2 ) / Math.Cos( Util.PiOverNSafe( r ) );
}
else if( cellGeometry == Geometry.Hyperbolic )
{
if( Infinite( p ) && Infinite( q ) && FiniteOrInfinite( r ) )
{
//double iiiCellRadius = 2 - Math.Sqrt( 2 );
//Circle3D iiiCircle = new Circle3D() { Center = new Vector3D( 1 - iiiCellRadius, 0, 0 ), Radius = iiiCellRadius };
//radius = iiiCellRadius; // infinite r
//center = new Vector3D( 1 - radius, 0, 0 );
// For finite r, it was easier to calculate cell facet in a more symmetric position,
// then move into position with the other mirrors via a Mobius transformation.
double rTemp = 1 / ( Math.Cos( Util.PiOverNSafe( r ) ) + 1 );
Mobius m = new Mobius();
m.Isometry( Geometry.Hyperbolic, -Math.PI / 4, new Vector3D( 0, Math.Sqrt( 2 ) - 1 ) );
Vector3D c1 = m.Apply( new Vector3D( 1 - 2 * rTemp, 0, 0 ) );
Vector3D c2 = c1;
c2.Y *= -1;
Vector3D c3 = new Vector3D( 1, 0 );
Circle3D c = new Circle3D( c1, c2, c3 );
radius = c.Radius;
center = c.Center;
}
else if( Infinite( p ) && Finite( q ) && FiniteOrInfinite( r ) )
{
// http://www.wolframalpha.com/input/?i=r%2Bx+%3D+1%2C+sin%28pi%2Fp%29+%3D+r%2Fx%2C+solve+for+r
// radius = 2 * Math.Sqrt( 3 ) - 3; // Appolonian gasket wiki page
//radius = Math.Sin( Math.PI / q ) / ( Math.Sin( Math.PI / q ) + 1 );
//center = new Vector3D( 1 - radius, 0, 0 );
// For finite r, it was easier to calculate cell facet in a more symmetric position,
// then move into position with the other mirrors via a Mobius transformation.
double rTemp = 1 / ( Math.Cos( Util.PiOverNSafe( r ) ) + 1 );
Mobius m = new Mobius();
m.Isometry( Geometry.Hyperbolic, 0, p2 );
Vector3D findingAngle = m.Inverse().Apply( new Vector3D( 1, 0 ) );
double angle = Math.Atan2( findingAngle.Y, findingAngle.X );
m.Isometry( Geometry.Hyperbolic, angle, p2 );
Vector3D c1 = m.Apply( new Vector3D( 1 - 2 * rTemp, 0, 0 ) );
Vector3D c2 = c1;
c2.Y *= -1;
Vector3D c3 = new Vector3D( 1, 0 );
Circle3D c = new Circle3D( c1, c2, c3 );
radius = c.Radius;
center = c.Center;
}
else if( Finite( p ) && Infinite( q ) && FiniteOrInfinite( r ) )
{
radius = p2.Abs(); // infinite r
radius = DonHatch.asinh( Math.Sinh( DonHatch.e2hNorm( p2.Abs() ) ) / Math.Cos( Util.PiOverNSafe( r ) ) ); // hyperbolic trig
// 4j3
//m_jOffset = radius * 0.02;
//radius += m_jOffset ;
radius = DonHatch.h2eNorm( radius );
center = new Vector3D();
rotation *= -1;
}
else if( /*Finite( p ) &&*/ Finite( q ) )
{
// Infinite r
//double mag = Geometry2D.GetTrianglePSide( q, p );
// Finite or Infinite r
double halfSide = Geometry2D.GetTrianglePSide( q, p );
double mag = DonHatch.asinh( Math.Sinh( halfSide ) / Math.Cos( Util.PiOverNSafe( r ) ) ); // hyperbolic trig
H3Models.Ball.DupinCyclideSphere( p1, DonHatch.h2eNorm( mag ), out center, out radius );
}
else
throw new System.NotImplementedException();
}
Sphere cellBoundary = new Sphere()
{
Center = center,
Radius = radius
};
Sphere[] interior = InteriorMirrors( p, q );
Sphere[] surfaces = new Sphere[] { cellBoundary, interior[0], interior[1], interior[2] };
// Apply rotations.
bool applyRotations = true;
if( applyRotations )
{
foreach( Sphere s in surfaces )
RotateSphere( s, rotation );
p1.RotateXY( rotation );
}
// Apply scaling
bool applyScaling = scaling != -1;
if( applyScaling )
{
//double scale = 1.0/0.34390660467269524;
//scale = 0.58643550768408892;
foreach( Sphere s in surfaces )
Sphere.ScaleSphere( s, scaling );
}
bool facetCentered = false;
if( facetCentered )
{
PrepForFacetCentering( p, q, surfaces, ref cellCenter );
}
// Move to ball if needed.
Sphere[] result = surfaces.Select( s => moveToBall ? H3Models.UHSToBall( s ) : s ).ToArray();
cellCenter = moveToBall ? H3Models.UHSToBall( cellCenter ) : cellCenter;
return result;
}
