// these sources don't seem to agree, and what I ended up using is still different. WHY?
// http://en.wikipedia.org/wiki/Generalised_circle
// http://www.math.ubc.ca/~cass/research/pdf/Geometry.pdf
// http://www.math.okstate.edu/~wrightd/INDRA/MobiusonCircles.mpl
public static CircLine operator *(Mobius m, CircLine circLine)
{
#if true
Mobius inverse = m.Inverse;
Mobius b = new Mobius(new Complex(circLine.a, 0), circLine.b.Conjugate, circLine.b, new Complex(circLine.c, 0));
Mobius hermitian = inverse.Transpose *
new Mobius(new Complex(circLine.a, 0), circLine.b.Conjugate, circLine.b, new Complex(circLine.c, 0)) *
inverse.Conjugate;
return CircLine.Create(hermitian.A.Re, hermitian.C, hermitian.D.Re);
#else // do it by decomposing the mobius -- slower
Complex a = m.A;
Complex b = m.B;
Complex c = m.C;
Complex d = m.D;
CircLine toInvert, inverted, scaled;
if (c == Complex.Zero) {
scaled = circLine.Scale(a / d);
return scaled.Translate(b / d);
}
toInvert = circLine.Translate(d / c);
inverted = toInvert.Inverse;
scaled = inverted.Scale(-(a * d - b * c) / (c * c));
return scaled.Translate(a / c);
#endif
}
// these sources don't seem to agree, and what I ended up using is still different. WHY?
// http://en.wikipedia.org/wiki/Generalised_circle
// http://www.math.ubc.ca/~cass/research/pdf/Geometry.pdf
// http://www.math.okstate.edu/~wrightd/INDRA/MobiusonCircles.mpl
public static CircLine operator *(Mobius m, CircLine circLine)
{
#if true
Mobius inverse = m.Inverse;
Mobius b = new Mobius(new Complex(circLine.a, 0), circLine.b.Conjugate, circLine.b, new Complex(circLine.c, 0));
Mobius hermitian = inverse.Transpose *
new Mobius(new Complex(circLine.a, 0), circLine.b.Conjugate, circLine.b, new Complex(circLine.c, 0)) *
inverse.Conjugate;
return(CircLine.Create(hermitian.A.Re, hermitian.C, hermitian.D.Re));
#else // do it by decomposing the mobius -- slower
Complex a = m.A;
Complex b = m.B;
Complex c = m.C;
Complex d = m.D;
CircLine toInvert, inverted, scaled;
if (c == Complex.Zero)
{
scaled = circLine.Scale(a / d);
return(scaled.Translate(b / d));
}
toInvert = circLine.Translate(d / c);
inverted = toInvert.Inverse;
scaled = inverted.Scale(-(a * d - b * c) / (c * c));
return(scaled.Translate(a / c));
#endif
}
public static Mobius CreateDiscTranslation(Complex a, Complex b)
{
return
(Mobius.CreateDiscAutomorphism(b, 0) *
Mobius.CreateDiscAutomorphism(a, 0).Inverse);
}
// Visual Complex Analysis p320
public static Mobius CreateDiscAutomorphism(Complex a, double phi)
{
return
(Mobius.CreateRotation(phi) *
new Mobius(Complex.One, -a, a.Conjugate, -Complex.One));
}