| public ReflectPoint ( Vector3D p ) : Vector3D | ||
| p | Vector3D | |
| return | Vector3D | |
private static void Transform(Mesh mesh)
{
Sphere sphere = new Sphere();
sphere.Radius = 0.1;
for (int i = 0; i < mesh.Triangles.Count; i++)
{
mesh.Triangles[i] = new Mesh.Triangle(
sphere.ReflectPoint(mesh.Triangles[i].a),
sphere.ReflectPoint(mesh.Triangles[i].b),
sphere.ReflectPoint(mesh.Triangles[i].c));
}
}
public void Reflect(Sphere sphere)
{
for (int i = 0; i < 4; i++)
{
Verts[i] = sphere.ReflectPoint(Verts[i]);
}
CalcFaces();
}
/// <summary>
/// Reflect ourselves about another sphere.
/// </summary>
public void Reflect(Sphere sphere)
{
// An interior point used to calculate whether we get inverted.
Vector3D interiorPoint;
if (IsPlane)
{
Debug.Assert(!this.Normal.IsOrigin);
interiorPoint = this.Offset - this.Normal;
}
else
{
// We don't want it to be the center, because that will reflect to infinity.
interiorPoint = (this.Center + new Vector3D(this.Radius / 2, 0));
}
if (Invert)
{
interiorPoint = ReflectPoint(interiorPoint);
}
Debug.Assert(IsPointInside(interiorPoint));
interiorPoint = sphere.ReflectPoint(interiorPoint);
Debug.Assert(!interiorPoint.DNE);
if (this.Equals(sphere))
{
if (IsPlane)
{
//this.Center = -this.Center; // Same as inverting, but we need to do it this way because of Pov-Ray
this.Invert = !this.Invert;
}
else
{
this.Invert = !this.Invert;
}
Debug.Assert(this.IsPointInside(interiorPoint));
return;
}
// Both planes?
if (IsPlane && sphere.IsPlane)
{
// XXX - not general, but I know the planes I'll be dealing with go through the origin.
//if( !sphere.Offset.IsOrigin )
// throw new System.NotImplementedException();
/*Vector3D p1 = this.Normal.Cross( sphere.Normal );
* if( !p1.Normalize() )
* {
* this.Center *= -1;
* return;
* }
*
* Vector3D p2 = p1.Cross( this.Normal );
* p2.Normalize();
* p1 = sphere.ReflectPoint( p1 );
* p2 = sphere.ReflectPoint( p2 );
* Vector3D newNormal = p2.Cross( p1 );
* if( !newNormal.Normalize() )
* throw new System.Exception( "Reflection impl" );
* this.Center = newNormal;*/
// Reflect the normal relative to the plane (conjugate with sphere.Offset).
Vector3D newNormal = this.Normal + sphere.Offset;
newNormal = sphere.ReflectPoint(newNormal);
newNormal -= sphere.Offset;
newNormal.Normalize();
this.Center = newNormal;
// Calc the new offset (so far we have considered planes through origin).
this.Offset = sphere.ReflectPoint(this.Offset);
//Debug.Assert( Offset.IsOrigin ); // XXX - should handle more generality.
Debug.Assert(this.IsPointInside(interiorPoint));
return;
}
// We are a plane and reflecting in a sphere.
if (IsPlane)
{
// Think of 2D case here (circle and line)...
Vector3D projected = Euclidean3D.ProjectOntoPlane(this.Normal, this.Offset, sphere.Center);
Vector3D p = sphere.ReflectPoint(projected);
if (Infinity.IsInfinite(p))
{
// This can happen if we go through sphere.Center.
// This reflection does not change our orientation (does not invert us).
return;
}
Center = sphere.Center + (p - sphere.Center) / 2;
Radius = Center.Dist(sphere.Center);
// Did this invert us?
if (!this.IsPointInside(interiorPoint))
{
Invert = !Invert;
}
return;
}
// Is mirror a plane?
if (sphere.IsPlane)
{
Vector3D projected = Euclidean3D.ProjectOntoPlane(sphere.Normal, sphere.Offset, Center);
Vector3D diff = Center - projected;
Center -= 2 * diff;
// Radius remains unchanged.
// NOTE: This does not invert us.
Debug.Assert(this.IsPointInside(interiorPoint));
return;
}
//
// Now sphere reflecting in a sphere.
//
// Reflecting to a plane?
if (IsPointOn(sphere.Center))
{
// Concentric spheres?
if (Center == sphere.Center)
{
throw new System.Exception();
}
// Center
Vector3D center = Center - sphere.Center;
// Offset
Vector3D direction = center;
direction.Normalize();
Vector3D offset = direction * Radius * 2;
offset = sphere.ReflectPoint(offset);
// We are a line now.
Center = center;
//Offset = offset; // Not working?? Caused issues in old generation code for 435.
Radius = double.PositiveInfinity;
// Did this invert us?
if (!this.IsPointInside(interiorPoint))
{
this.Invert = !this.Invert;
}
Debug.Assert(this.IsPointInside(interiorPoint));
return;
}
// XXX - Could try to share code below with Circle class.
// NOTE: We can't just reflect the center.
// See http://mathworld.wolfram.com/Inversion.html
double a = Radius;
double k = sphere.Radius;
Vector3D v = Center - sphere.Center;
double s = k * k / (v.MagSquared() - a * a);
Center = sphere.Center + v * s;
Radius = Math.Abs(s) * a;
// Did this invert us?
if (!this.IsPointInside(interiorPoint))
{
Invert = !Invert;
}
}
private static void ReflectEdgesRecursive(Sphere[] simplex, Edge[] edges, Settings settings,
HashSet <Edge> completedEdges)
{
if (0 == edges.Length)
{
return;
}
HashSet <Edge> newEdges = new HashSet <Edge>(new H3.Cell.EdgeEqualityComparer());
foreach (Edge edge in edges)
{
//foreach( Sphere mirror in simplex )
for (int m = 0; m < simplex.Length; m++)
{
Sphere mirror = simplex[m];
if (completedEdges.Count > settings.MaxEdges)
{
throw new System.Exception("Maxing out edges - will result in uneven filling.");
}
Vector3D r1 = mirror.ReflectPoint(edge.Start);
Vector3D r2 = mirror.ReflectPoint(edge.End);
Edge newEdge = new Edge(r1, r2);
newEdge.CopyDepthsFrom(edge);
if (!EdgeOk(newEdge, settings))
{
continue;
}
// This tracks reflections across the cell facets.
newEdge.Depths[m]++;
// Edge color.
// This also controls resolution of the edges, and can have a big effect on file size.
// Make the threshold cutoff black, or the background color.
double percentWhite = 1;
if (settings.ThreshType == EdgeThreshType.Length)
{
percentWhite = (r1.Dist(r2) - settings.Threshold) / 0.015;
}
else
{
double closestToOrigin = Math.Min(r1.Abs(), r2.Abs()); // Mainly ranges from 0 to 1
if (closestToOrigin < 0.9)
{
percentWhite = 1.0;
}
else
{
percentWhite = 1.0 - Math.Pow(closestToOrigin - 0.9, 1.3) / 0.1;
}
}
if (percentWhite < 0)
{
percentWhite = 0;
}
if (percentWhite > 1)
{
percentWhite = 1;
}
//newEdge.Color = new Vector3D( percentWhite, percentWhite, percentWhite );
newEdge.Color = m_background;
newEdge.Color.Z = 0.1 + 0.9 * percentWhite;
if (completedEdges.Add(newEdge))
{
// Haven't seen this edge yet, so
// we'll need to recurse on it.
newEdges.Add(newEdge);
}
}
}
ReflectEdgesRecursive(simplex, newEdges.ToArray(), settings, completedEdges);
}
/// <summary>
/// Reflect ourselves about another sphere.
/// </summary>
public void Reflect( Sphere sphere )
{
// An interior point used to calculate whether we get inverted.
Vector3D interiorPoint;
if( IsPlane )
{
Debug.Assert( !this.Normal.IsOrigin );
interiorPoint = -this.Normal;
}
else
{
// We don't want it to be the center, because that will reflect to infinity.
interiorPoint = ( this.Center + new Vector3D( this.Radius / 2, 0 ) );
}
if( Invert )
interiorPoint = ReflectPoint( interiorPoint );
Debug.Assert( IsPointInside( interiorPoint ) );
interiorPoint = sphere.ReflectPoint( interiorPoint );
Debug.Assert( !interiorPoint.DNE );
if( this.Equals( sphere ) )
{
if( IsPlane )
{
//this.Center = -this.Center; // Same as inverting, but we need to do it this way because of Pov-Ray
this.Invert = !this.Invert;
}
else
this.Invert = !this.Invert;
Debug.Assert( this.IsPointInside( interiorPoint ) );
return;
}
// Both planes?
if( IsPlane && sphere.IsPlane )
{
// XXX - not general, but I know the planes I'll be dealing with go through the origin.
//if( !sphere.Offset.IsOrigin )
// throw new System.NotImplementedException();
/*Vector3D p1 = this.Normal.Cross( sphere.Normal );
if( !p1.Normalize() )
{
this.Center *= -1;
return;
}
Vector3D p2 = p1.Cross( this.Normal );
p2.Normalize();
p1 = sphere.ReflectPoint( p1 );
p2 = sphere.ReflectPoint( p2 );
Vector3D newNormal = p2.Cross( p1 );
if( !newNormal.Normalize() )
throw new System.Exception( "Reflection impl" );
this.Center = newNormal;*/
// Reflect the normal relative to the plane (conjugate with sphere.Offset).
Vector3D newNormal = this.Normal + sphere.Offset;
newNormal = sphere.ReflectPoint( newNormal );
newNormal -= sphere.Offset;
newNormal.Normalize();
this.Center = newNormal;
// Calc the new offset (so far we have considered planes through origin).
this.Offset = sphere.ReflectPoint( this.Offset );
//Debug.Assert( Offset.IsOrigin ); // XXX - should handle more generality.
Debug.Assert( this.IsPointInside( interiorPoint ) );
return;
}
// We are a plane and reflecting in a sphere.
if( IsPlane )
{
// Think of 2D case here (circle and line)...
Vector3D projected = Euclidean3D.ProjectOntoPlane( this.Normal, this.Offset, sphere.Center );
Vector3D p = sphere.ReflectPoint( projected );
if( Infinity.IsInfinite( p ) )
{
// This can happen if we go through sphere.Center.
// This reflection does not change our orientation (does not invert us).
return;
}
Center = sphere.Center + ( p - sphere.Center ) / 2;
Radius = Center.Dist( sphere.Center );
// Did this invert us?
if( !this.IsPointInside( interiorPoint ) )
Invert = !Invert;
return;
}
// Is mirror a plane?
if( sphere.IsPlane )
{
Vector3D projected = Euclidean3D.ProjectOntoPlane( sphere.Normal, sphere.Offset, Center );
Vector3D diff = Center - projected;
Center -= 2 * diff;
// Radius remains unchanged.
// NOTE: This does not invert us.
Debug.Assert( this.IsPointInside( interiorPoint ) );
return;
}
//
// Now sphere reflecting in a sphere.
//
// Reflecting to a plane?
if( IsPointOn( sphere.Center ) )
{
// Concentric spheres?
if( Center == sphere.Center )
throw new System.Exception();
// Center
Vector3D center = Center - sphere.Center;
// Offset
Vector3D direction = center;
direction.Normalize();
Vector3D offset = direction * Radius * 2;
offset = sphere.ReflectPoint( offset );
// We are a line now.
Center = center;
//Offset = offset; // Not working?? Caused issues in old generation code for 435.
Radius = double.PositiveInfinity;
// Did this invert us?
if( !this.IsPointInside( interiorPoint ) )
this.Invert = !this.Invert;
Debug.Assert( this.IsPointInside( interiorPoint ) );
return;
}
// XXX - Could try to share code below with Circle class.
// NOTE: We can't just reflect the center.
// See http://mathworld.wolfram.com/Inversion.html
double a = Radius;
double k = sphere.Radius;
Vector3D v = Center - sphere.Center;
double s = k * k / ( v.MagSquared() - a * a );
Center = sphere.Center + v * s;
Radius = Math.Abs( s ) * a;
// Did this invert us?
if( !this.IsPointInside( interiorPoint ) )
Invert = !Invert;
}
/// <summary>
/// Given 2 points in the interior of the ball, calculate the center and radius of the orthogonal circle.
/// One point may optionally be on the boundary, but one shoudl be in the interior.
/// If both points are on the boundary, we'll fall back on our other method.
/// </summary>
public static void OrthogonalCircleInterior( Vector3D v1, Vector3D v2, out Circle3D circle )
{
if( Tolerance.Equal( v1.Abs(), 1 ) &&
Tolerance.Equal( v2.Abs(), 1 ) )
{
circle = OrthogonalCircle( v1, v2 );
return;
}
// http://www.math.washington.edu/~king/coursedir/m445w06/ortho/01-07-ortho-to3.html
// http://www.youtube.com/watch?v=Bkvo09KE1zo
Vector3D interior = Tolerance.Equal( v1.Abs(), 1 ) ? v2 : v1;
Sphere ball = new Sphere();
Vector3D reflected = ball.ReflectPoint( interior );
circle = new Circle3D( reflected, v1, v2 );
}
/// <summary>
/// Using this to move the view around in interesting ways.
/// </summary>
private Vector3D ApplyTransformation( Vector3D v, double t = 0.0 )
{
//v.RotateXY( Math.PI / 4 + 0.01 );
bool applyNone = true;
if( applyNone )
return v;
Mobius m0 = new Mobius(), m1 = new Mobius(), m2 = new Mobius(), m3 = new Mobius();
Sphere unitSphere = new Sphere();
// self-similar scale for 437
//v*= 4.259171776329806;
double s = 6.5;
v *= s;
v += new Vector3D( s/3, -s/3 );
v = unitSphere.ReflectPoint( v );
v.RotateXY( Math.PI/6 );
//v /= 3;
//v.RotateXY( Math.PI );
//v.RotateXY( Math.PI/2 );
return v;
//v.Y = v.Y / Math.Cos( Math.PI / 6 ); // 637 repeatable
//return v;
// 12,12,12
m0.Isometry( Geometry.Hyperbolic, 0, new Complex( .0, .0 ) );
m1 = Mobius.Identity();
m2 = Mobius.Identity();
m3 = Mobius.Identity();
v = (m0 * m1 * m2 * m3).Apply( v );
return v;
// i64
m0.Isometry( Geometry.Hyperbolic, 0, new Complex( .5, .5 ) );
m1.UpperHalfPlane();
m2 = Mobius.Scale( 1.333333 );
m3.Isometry( Geometry.Euclidean, 0, new Vector3D( 0, -1.1 ) );
v = (m1 * m2 * m3).Apply( v );
return v;
// 464
// NOTE: Also, don't apply rotations during simplex generation.
m1.UpperHalfPlane();
m2 = Mobius.Scale( 1.3 );
m3.Isometry( Geometry.Euclidean, 0, new Vector3D( 1.55, -1.1 ) );
v = ( m1 * m2 * m3 ).Apply( v );
return v;
// iii
m1.Isometry( Geometry.Hyperbolic, 0, new Complex( 0, Math.Sqrt( 2 ) - 1 ) );
m2.Isometry( Geometry.Euclidean, -Math.PI / 4, 0 );
m3 = Mobius.Scale( 5 );
//v = ( m1 * m2 * m3 ).Apply( v );
// Vertical Line
/*v = unitSphere.ReflectPoint( v );
m1.MapPoints( new Vector3D(-1,0), new Vector3D(), new Vector3D( 1, 0 ) );
m2 = Mobius.Scale( .5 );
v = (m1*m2).Apply( v ); */
/*
m1 = Mobius.Scale( 0.175 );
v = unitSphere.ReflectPoint( v );
v = m1.Apply( v );
* */
// Inversion
//v = unitSphere.ReflectPoint( v );
//return v;
/*Mobius m1 = new Mobius(), m2 = new Mobius(), m3 = new Mobius();
m1.Isometry( Geometry.Spherical, 0, new Complex( 0, 1 ) );
m2.Isometry( Geometry.Euclidean, 0, new Complex( 0, -1 ) );
m3 = Mobius.Scale( 0.5 );
v = (m1 * m3 * m2).Apply( v );*/
//Mobius m = new Mobius();
//m.Isometry( Geometry.Hyperbolic, 0, new Complex( -0.88, 0 ) );
//m.Isometry( Geometry.Hyperbolic, 0, new Complex( 0, Math.Sqrt(2) - 1 ) );
//m = Mobius.Scale( 0.17 );
//m.Isometry( Geometry.Spherical, 0, new Complex( 0, 3.0 ) );
//v = m.Apply( v );
// 63i, 73i
m1 = Mobius.Scale( 6.0 ); // Scale {3,i} to unit disk.
m1 = Mobius.Scale( 1.0 / 0.14062592996431983 ); // 73i (constant is abs of midpoint of {3,7} tiling, if we want to calc later for other tilings).
m2.MapPoints( Infinity.InfinityVector, new Vector3D( 1, 0 ), new Vector3D() ); // swap interior/exterior
m3.UpperHalfPlane();
v *= 2.9;
// iii
/*m1.MapPoints( new Vector3D(), new Vector3D(1,0), new Vector3D( Math.Sqrt( 2 ) - 1, 0 ) );
m2.Isometry( Geometry.Euclidean, -Math.PI / 4, 0 );
m3 = Mobius.Scale( 0.75 );*/
Mobius m = m3 * m2 * m1;
v = m.Inverse().Apply( v ); // Strange that we have to do inverse here.
return v;
}