/// <summary>
/// Reflect a point in us.
/// </summary>
public Vector3D ReflectPoint(Vector3D p)
{
if (IsPlane)
{
// We used to call ProjectOntoPlane, but optimized it away.
// This is faster because we already know our normal is normalized,
// and it avoids some extra Vector3D operations.
double dist = Euclidean3D.DistancePointPlane(this.Normal, this.Offset, p);
Vector3D offset = this.Normal * dist * -2;
return(p + offset);
}
else
{
if (p == Center)
{
return(Infinity.InfinityVector);
}
if (Infinity.IsInfinite(p))
{
return(Center);
}
Vector3D v = p - Center;
double d = v.Abs();
v.Normalize();
return(Center + v * (Radius * Radius / d));
}
}
public static double GetNormalizedCircumRadius(int p, double q)
{
double hypot = GetTriangleHypotenuse(p, q);
switch (Geometry2D.GetGeometry(p, q))
{
case Geometry.Spherical:
return(Spherical2D.s2eNorm(hypot) * DiskRadius);
case Geometry.Euclidean:
return(EuclideanHypotenuse);
case Geometry.Hyperbolic:
{
if (Infinity.IsInfinite(hypot))
{
return(DiskRadius);
}
return(DonHatch.h2eNorm(hypot) * DiskRadius);
}
}
Debug.Assert(false);
return(1);
}
/// <summary>
/// Reflect a point in us.
/// </summary>
public Vector3D ReflectPoint(Vector3D p)
{
if (IsPlane)
{
//Debug.Assert( !Infinity.IsInfinite( p ) );
Vector3D v = Euclidean3D.ProjectOntoPlane(this.Normal, this.Offset, p);
v = p + (v - p) * 2;
return(v);
}
else
{
if (p == Center)
{
return(Infinity.InfinityVector);
}
if (Infinity.IsInfinite(p))
{
return(Center);
}
Vector3D v = p - Center;
double d = v.Abs();
v.Normalize();
return(Center + v * (Radius * Radius / d));
}
}
/// <summary>
/// Construct a circle from 3 points
/// </summary>
/// <returns>false if the construction failed (if we are a line).</returns>
public bool From3Points(Vector3D p1, Vector3D p2, Vector3D p3)
{
Reset();
// Check for any infinite points, in which case we are a line.
// I'm not sure these checks are smart, since our IsInfinite check is so inclusive,
// but Big Chop puzzle doesn't work if we don't do this.
// ZZZ - Still, I need to think on this more.
if (Infinity.IsInfinite(p1))
{
this.From2Points(p2, p3);
return(false);
}
else if (Infinity.IsInfinite(p2))
{
this.From2Points(p1, p3);
return(false);
}
else if (Infinity.IsInfinite(p3))
{
this.From2Points(p1, p2);
return(false);
}
/* Some links
* http://mathforum.org/library/drmath/view/54323.html
* http://delphiforfun.org/Programs/Math_Topics/circle_from_3_points.htm
* There is lots of info out there about solving via equations,
* but as with other code in this project, I wanted to use geometrical constructions. */
// Midpoints.
Vector3D m1 = (p1 + p2) / 2;
Vector3D m2 = (p1 + p3) / 2;
// Perpendicular bisectors.
Vector3D b1 = (p2 - p1) / 2;
Vector3D b2 = (p3 - p1) / 2;
b1.Normalize();
b2.Normalize();
b1.Rotate90();
b2.Rotate90();
Vector3D newCenter;
int found = Euclidean2D.IntersectionLineLine(m1, m1 + b1, m2, m2 + b2, out newCenter);
Center = newCenter;
if (0 == found)
{
// The points are collinear, so we are a line.
From2Points(p1, p2);
return(false);
}
Radius = (p1 - Center).Abs();
Debug.Assert(Tolerance.Equal(Radius, (p2 - Center).Abs()));
Debug.Assert(Tolerance.Equal(Radius, (p3 - Center).Abs()));
return(true);
}
public static Vector3D InfinitySafe(Vector3D input)
{
if (Infinity.IsInfinite(input))
{
return(Infinity.LargeFiniteVector);
}
return(input);
}
public static Vector3D PlaneToSphereSafe(Vector3D planePoint)
{
if (Infinity.IsInfinite(planePoint))
{
return(new Vector3D(0, 0, 1));
}
return(PlaneToSphere(planePoint));
}
/// <summary>
/// https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection
/// </summary>
private static double StereoToEqualArea(double dist)
{
if (Infinity.IsInfinite(dist))
{
return(1);
}
double dot = dist * dist; // X^2 + Y^2 + Z^2
double w = (dot - 1) / (dot + 1);
double r = Math.Sqrt(2 * (1 + w));
return(r / 2);
}
private static double StereoToEquidistant(double dist)
{
if (Infinity.IsInfinite(dist))
{
return(1);
}
double dot = dist * dist; // X^2 + Y^2 + Z^2
double w = (dot - 1) / (dot + 1);
double x = Math.Sqrt(1 - w * w);
double r = Euclidean2D.AngleToCounterClock(new Vector3D(0, -1), new Vector3D(x, w));
return(r / Math.PI);
}
private bool NewTetAfterReflect(Cell cell, Segment s, HashSet <Vector3D> completed)
{
foreach (Tile tile in cell.Tiles)
{
CircleNE newVertexCircle = tile.VertexCircle.Clone();
newVertexCircle.Reflect(s);
if (completed.Contains(Infinity.InfinitySafe(newVertexCircle.CenterNE)))
{
return(false);
}
}
return(true);
}
public void Reflect(Segment seg)
{
foreach (Tile tile in this.Tiles)
{
tile.Reflect(seg);
if (Infinity.IsInfinite(tile.Boundary.Center))
{
tile.Boundary.Center = Infinity.LargeFiniteVector;
}
if (Infinity.IsInfinite(tile.Drawn.Center))
{
tile.Drawn.Center = Infinity.LargeFiniteVector;
}
}
}
/// <summary>
/// Apply a transform to us.
/// </summary>
private void TransformInternal <T>(T transform) where T : ITransform
{
// NOTES:
// Arcs can go to lines, and lines to arcs.
// Rotations may reverse arc directions as well.
// Arc centers can't be transformed directly.
// NOTE: We must calc this before altering the endpoints.
Vector3D mid = Midpoint;
if (Infinity.IsInfinite(mid))
{
mid = Infinity.IsInfinite(P1) ? P2 * Infinity.FiniteScale : P1 * Infinity.FiniteScale;
}
P1 = transform.Apply(P1);
P2 = transform.Apply(P2);
mid = transform.Apply(mid);
// Can we make a circle out of the transformed points?
Circle temp = new Circle();
if (!Infinity.IsInfinite(P1) && !Infinity.IsInfinite(P2) && !Infinity.IsInfinite(mid) &&
temp.From3Points(P1, mid, P2))
{
Type = SegmentType.Arc;
Center = temp.Center;
// Work out the orientation of the arc.
Vector3D t1 = P1 - Center;
Vector3D t2 = mid - Center;
Vector3D t3 = P2 - Center;
double a1 = Euclidean2D.AngleToCounterClock(t2, t1);
double a2 = Euclidean2D.AngleToCounterClock(t3, t1);
Clockwise = a2 > a1;
}
else
{
// The circle construction fails if the points
// are colinear (if the arc has been transformed into a line).
Type = SegmentType.Line;
// XXX - need to do something about this.
// Turn into 2 segments?
//if( isInfinite( mid ) )
// Actually the check should just be whether mid is between p1 and p2.
}
}
public static Vector3D R3toS3(Vector3D p)
{
if (Infinity.IsInfinite(p))
{
return(new Vector3D(0, 0, 0, 1));
}
p.W = 0;
double dot = p.Dot(p); // X^2 + Y^2 + Z^2
return(new Vector3D(
2 * p.X / (dot + 1),
2 * p.Y / (dot + 1),
2 * p.Z / (dot + 1),
(dot - 1) / (dot + 1)));
}
private static double StereoToEqualVolume(double dist)
{
if (Infinity.IsInfinite(dist))
{
return(1);
}
double dot = dist * dist; // X^2 + Y^2 + Z^2
double w = (dot - 1) / (dot + 1);
w = -w; // Because I derived formula from north pole.
double t = Math.PI / 2 - w * Math.Sqrt(1 - w * w) - Math.Asin(w);
double r = Math.Pow(t * 3 / 2, 1.0 / 3);
return(r);
}
private static double Pi_hpq(int p, int q)
{
double pi = Math.PI;
double pip = PiOverNSafe(p);
double piq = PiOverNSafe(q);
double temp = Math.Pow(Math.Cos(pip), 2) + Math.Pow(Math.Cos(piq), 2);
double hab = pi / Math.Acos(Math.Sqrt(temp));
// Infinity safe.
double pi_hpq = pi / hab;
if (Infinity.IsInfinite(hab))
{
pi_hpq = 0;
}
return(pi_hpq);
}
public Vector3D[] CalcEdgePoints(double arcResolution, int minSegs, bool checkForInfinities)
{
List <Vector3D> points = new List <Vector3D>();
for (int i = 0; i < NumSides; i++)
{
Segment s = Segments[i];
// First point.
// ZZZ - getting lazy
//Debug.Assert( ! (isInfinite( s.m_p1 ) && isInfinite( s.m_p2 )) );
Vector3D p1 = checkForInfinities && Infinity.IsInfinite(s.P1) ?
s.P2 * Infinity.FiniteScale :
s.P1;
points.Add(p1);
// For arcs, add in a bunch of extra points.
if (SegmentType.Arc == s.Type)
{
double maxAngle = s.Angle;
Vector3D vs = s.P1 - s.Center;
int numSegments = (int)(maxAngle / (arcResolution));
if (numSegments < minSegs)
{
numSegments = minSegs;
}
double angle = maxAngle / numSegments;
for (int j = 1; j < numSegments; j++)
{
vs.RotateXY(s.Clockwise ? -angle : angle);
points.Add(vs + s.Center);
}
}
// Last point.
Vector3D p2 = checkForInfinities && Infinity.IsInfinite(s.P2) ?
s.P1 * Infinity.FiniteScale :
s.P2;
points.Add(p2);
}
return(points.ToArray());
}
private void ReflectRecursive(int level, List <Cell> cells, HashSet <Vector3D> completed)
{
level++;
// Breadth first recursion.
if (0 == cells.Count)
{
return;
}
List <Cell> reflected = new List <Cell>();
foreach (Cell cell in cells)
{
foreach (Segment seg in cell.Segments)
{
// Are we done?
if (m_cells.Count >= m_cellCount)
{
return;
}
if (!NewTetAfterReflect(cell, seg, completed))
{
continue;
}
Cell newBase = cell.Clone();
newBase.Level = level;
newBase.Reflect(seg);
m_cells.Add(newBase);
reflected.Add(newBase);
foreach (Tile tile in newBase.Tiles)
{
completed.Add(Infinity.InfinitySafe(tile.VertexCircle.CenterNE));
}
}
}
ReflectRecursive(level, reflected, completed);
}
// ZZZ - I wonder if we want to do normalization of lines before comparing.
public bool Equals(CircleNE c1, CircleNE c2)
{
bool radiusEqual =
Tolerance.Equal(c1.Radius, c2.Radius) ||
(Infinity.IsInfinite(c1.Radius) && Infinity.IsInfinite(c2.Radius));
if (c1.IsLine)
{
return(c1.P1 == c2.P1 &&
c1.P2 == c2.P2 &&
radiusEqual);
}
else
{
return
(c1.Center == c2.Center &&
c1.CenterNE == c2.CenterNE &&
radiusEqual);
}
}
public Circle3D(Vector3D t1, Vector3D t2, Vector3D t3)
{
Vector3D center;
double radius;
From3Points(t1, t2, t3, out center, out radius);
Center = center;
Radius = radius;
Vector3D normal = (t2 - t1).Cross(t3 - t1);
if (Infinity.IsInfinite(Radius))
{
Center = Vector3D.DneVector();
normal = !t1.IsOrigin ? t1 : !t2.IsOrigin ? t2 : t3; // Hacky rep of line.
}
normal.Normalize();
Normal = normal;
}
/// <summary>
/// Checks to see if a point is inside us, in a non-Euclidean sense.
/// This works if we are inverted, and even if we are a line!
/// (if we are a line, half of the plane is still "inside").
/// </summary>
public bool IsPointInsideNE(Vector3D testPoint)
{
if (this.IsLine)
{
// We are inside if the test point is on the same side
// as the non-Euclidean center.
return(Euclidean2D.SameSideOfLine(P1, P2, testPoint, CenterNE));
}
else
{
// Whether we are inside in the Euclidean sense.
bool pointInside = false;
if (!Infinity.IsInfinite(testPoint))
{
pointInside = this.IsPointInside(testPoint);
}
// And in the Non-Euclidean sense.
bool inverted = this.Inverted;
return((!inverted && pointInside) ||
(inverted && !pointInside));
}
}
/// <summary>
/// Transform a mesh in the Poincare model to Dini's surface.
/// </summary>
public static Mesh Dini(Mesh mesh)
{
System.Func <Vector3D, Vector3D> transform = v =>
{
//v = DiskToUpper( v );
//v.Y = Math.Log( v.Y );
//if( v.Y < 1 || v.Y > 10 )
// return Infinity.InfinityVector;
//if( v.X < -Math.PI || v.X > Math.PI )
//if( v.X < -3*Math.PI || v.X > 3*Math.PI )
// return Infinity.InfinityVector;
//v.Y = Math.Log( v.Y );
//return v;
return(Dini(v));
};
Mesh result = new Mesh();
for (int i = 0; i < mesh.Triangles.Count; i++)
{
Vector3D a = transform(mesh.Triangles[i].a);
Vector3D b = transform(mesh.Triangles[i].b);
Vector3D c = transform(mesh.Triangles[i].c);
if (Infinity.IsInfinite(a) ||
Infinity.IsInfinite(b) ||
Infinity.IsInfinite(c))
{
continue;
}
result.Triangles.Add(new Mesh.Triangle(a, b, c));
}
return(result);
}
/// <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;
}
}