private static Circle3D HoneycombEdgeUHS(int p, int q, int r)
{
Sphere[] simplex = Mirrors(p, q, r, moveToBall: false);
Sphere s1 = simplex[0].Clone();
Sphere s2 = s1.Clone();
s2.Reflect(simplex[1]);
Circle3D intersection = s1.Intersection(s2);
return(intersection);
}
/// <summary>
/// NOTE: s must be geodesic! (orthogonal to boundary).
/// </summary>
public static Sphere UHSToBall( Sphere s )
{
Vector3D center;
double rad;
if( s.IsPlane )
{
// Planes through the origin will stay unchanged.
if( s.Offset == new Vector3D() )
return s.Clone();
// It must be vertical (because it is orthogonal).
Vector3D b1 = H3Models.UHSToBall( Infinity.InfinityVector );
Vector3D b2 = H3Models.UHSToBall( s.Offset );
H3Models.Ball.OrthogonalCircle( b1, b2, out center, out rad ); // Safer to use OrthogonalSphere?
}
else
{
Vector3D temp;
if( s.Center.IsOrigin )
{
temp = new Vector3D( s.Radius, 0, 0 );
}
else
{
temp = s.Center;
temp.Normalize();
temp *= s.Radius;
}
Vector3D centerUhs = s.Center;
Vector3D b1 = H3Models.UHSToBall( centerUhs - temp );
Vector3D b2 = H3Models.UHSToBall( centerUhs + temp );
H3Models.Ball.OrthogonalCircle( b1, b2, out center, out rad ); // Safer to use OrthogonalSphere?
// Did we project to a plane?
if( Infinity.IsInfinite( rad ) )
{
temp.RotateXY( Math.PI / 2 );
Vector3D b3 = H3Models.UHSToBall( centerUhs + temp );
center = b1.Cross( b3 );
rad = double.PositiveInfinity;
}
}
return new Sphere()
{
Center = center,
Radius = rad
};
}
/// <summary>
/// Transform a geodesic sphere in the ball model to the Klein model.
/// Output will be a plane.
/// </summary>
public static Sphere BallToKlein( Sphere s )
{
// If we are already a plane, no transformation is needed.
if( s.IsPlane )
return s.Clone();
Vector3D closest = Ball.ClosestToOrigin( s );
Vector3D p1 = HyperbolicModels.PoincareToKlein( closest );
// Ideal points are the same in the Klein/Poincare models, so grab
// two more plane points from the ideal circle.
Vector3D p2, p3, dummy;
Ball.IdealPoints( s, out p2, out dummy, out p3 );
Vector3D offset = p1;
Vector3D normal = (p2 - p1).Cross( p3 - p1 );
normal.Normalize();
if( !s.Invert )
normal *= -1;
return Sphere.Plane( offset, normal );
}
/// <summary>
/// This applies the same Mobius transform to all vertical planes through the z axis.
/// NOTE: m must therefore be a mobius transform that keeps the imaginary axis constant!
/// NOTE: s must be geodesic! (orthogonal to boundary).
/// </summary>
public static Sphere TransformInBall( Sphere s, Mobius m )
{
Vector3D center;
double rad;
if( s.IsPlane )
{
// All planes in the ball go through the origin.
if( !s.Offset.IsOrigin )
throw new System.ArgumentException();
// Vertical planes remain unchanged.
if( Tolerance.Equal( s.Normal.Z, 0 ) )
return s.Clone();
// Other planes will become spheres.
Vector3D pointOnSphere = s.Normal.Perpendicular();
pointOnSphere.Normalize();
Vector3D b1 = H3Models.TransformHelper( pointOnSphere, m );
Vector3D b2 = H3Models.TransformHelper( -pointOnSphere, m );
pointOnSphere.RotateAboutAxis( s.Normal, Math.PI / 2 );
Vector3D b3 = H3Models.TransformHelper( pointOnSphere, m );
return H3Models.Ball.OrthogonalSphere( b1, b2, b3 );
}
else
{
Vector3D s1, s2, s3;
H3Models.Ball.IdealPoints( s, out s1, out s2, out s3 );
// Transform the points.
Vector3D b1 = H3Models.TransformHelper( s1, m );
Vector3D b2 = H3Models.TransformHelper( s2, m );
Vector3D b3 = H3Models.TransformHelper( s3, m );
return H3Models.Ball.OrthogonalSphere( b1, b2, b3 );
}
return new Sphere()
{
Center = center,
Radius = rad
};
}
public static Sphere GeodesicOffset( Sphere s, double offset, bool ball = true )
{
Sphere offsetSphere;
if( ball )
{
// Geodesic offset (ball).
{ // Hyperbolic honeycomb
double mag = s.Center.Abs() - s.Radius;
mag = s.IsPlane ? DonHatch.h2eNorm( offset ) :
DonHatch.h2eNorm( DonHatch.e2hNorm( mag ) - offset );
Vector3D closestPointToOrigin = s.IsPlane ? s.Normal : s.Center;
closestPointToOrigin.Normalize();
closestPointToOrigin *= mag;
offsetSphere = H3Models.Ball.OrthogonalSphereInterior( closestPointToOrigin );
// There are multiple ultraparallel spheres.
// This experiments with picking others.
Mobius m = new Mobius();
m.Isometry( Geometry.Hyperbolic, 0, new Vector3D( 0, -0.2 ) );
//H3Models.TransformInBall2( offsetSphere, m );
}
{ // Spherical honeycomb
//offset *= -1;
double mag = -s.Center.Abs() + s.Radius;
Spherical2D.s2eNorm( Spherical2D.e2sNorm( mag ) + offset );
offsetSphere = s.Clone();
offsetSphere.Radius += offset*10;
}
}
else
{
// Geodesic offset (UHS).
// XXX - not scaled right.
offsetSphere = s.Clone();
offsetSphere.Radius += offset;
}
return offsetSphere;
}