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h2isometry.cpp
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h2isometry.cpp
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#include "h2isometry.h"
#include "h2point.h"
#include "sl2rmatrix.h"
H2Isometry::H2Isometry()
{
setIdentity();
}
H2Isometry::H2Isometry(uint i)
{
assert(i==1);
setIdentity();
}
H2Isometry::H2Isometry(const SL2RMatrix &A)
{
setSL2Rmatrix(A);
}
void H2Isometry::setIdentity()
{
u = 1.0;
a = 0.0;
}
void H2Isometry::getDiskCoordinates(Complex &u, Complex &a) const
{
u = this->u;
a = this->a;
}
void H2Isometry::setDiskCoordinates(const Complex &u, const Complex &a)
{
this->u=u;
this->a=a;
}
void H2Isometry::setTranslationAxisAndLength(const H2Geodesic & axis, double length)
{
H2Isometry f0;
f0.setTranslationLengthNormalized(length);
H2Isometry f;
f.setByMappingToVerticalUp(axis);
*this = f.inverse()*f0*f;
}
void H2Isometry::setTranslationLengthNormalized(double length)
{
double L = cosh(length);
a = -I*sqrt((L - 1.0)/(L + 1.0));
if (length < 0)
{
a = -a;
}
u = 1.0;
}
void H2Isometry::setVerticalTranslation(double t)
{
a = - I*t;
u = 1.0;
}
void H2Isometry::setVerticalTranslation(double t1, double t2)
{
double t = (t1 + t2)/(1 + t1 * t2);
a = - I*t;
u = 1.0;
}
void H2Isometry::setVerticalTranslation(double t1, double t2, double t3)
{
double t = (t1 + t2 + t3 + t1*t2*t3) / (1 + t1*t2 + t1*t3 + t2*t3);
a = - I*t;
u = 1.0;
}
void H2Isometry::setByMappingPointInDiskModelNormalized(const Complex &zIn, const Complex &zOut)
{
double n1 = norm(zIn), n2 = norm(zOut);
u = 1.0;
a = (zIn*(1 - n2) - zOut*(1 - n1)) / (1 - n1*n2);
}
void H2Isometry::setByMappingGeodesic(const H2Geodesic &L1, const H2Geodesic &L2)
{
H2Isometry f1,f2;
f1.setByMappingToVerticalUp(L1);
f2.setByMappingToVerticalUp(L2);
*this = f2.inverse()*f1;
}
void H2Isometry::setByMappingPointToOrigin(const H2Point &p)
{
u = 1.0;
a = p.getDiskCoordinate();
}
SL2CMatrix H2Isometry::getSU11Matrix() const
{
double s = 1.0 / sqrt(1.0 - norm(a));
Complex t = sqrt(u);
return SL2CMatrix( s*t , -a*s*t , -conj(a)*s/t , s/t );
}
SL2RMatrix H2Isometry::getSL2RMatrix() const
{
SL2CMatrix A(1.0/sqrt(2.0),-I/sqrt(2.0),-I/sqrt(2.0),1.0/sqrt(2.0));
SL2CMatrix out = A*getSU11Matrix()*A.inverse();
SL2RMatrix output;
out.getRealPart(output);
return output;
}
void H2Isometry::setSU11Matrix(const SL2CMatrix &A)
{
Complex alpha,alphaBar,beta,betaBar;
A.getCoefficients(alpha,alphaBar,beta,betaBar);
u = alpha / alphaBar;
a = -beta / alpha;
}
void H2Isometry::setSL2Rmatrix(const SL2RMatrix &A)
{
double Aa,Ab,Ac,Ad;
A.getCoefficients(Aa,Ab,Ac,Ad);
u = (Aa + Ad + I*(Ab - Ac))/(Aa + Ad - I*(Ab - Ac));
a = -(Ab + Ac + I*(Aa - Ad))/(Aa + Ad + I*(Ab - Ac));
}
bool H2Isometry::isElliptic() const
{
return (traceSquared()<4.0);
}
bool H2Isometry::isParabolic() const
{
return (traceSquared()==4.0);
}
bool H2Isometry::isHyperbolic() const
{
return (traceSquared()>4.0);
}
double H2Isometry::traceSquared() const
{
return 2.0*(1.0 + real(u))/(1.0 - norm(a));
}
double H2Isometry::translationLength() const
{
return acosh(.5*traceSquared() - 1.0);
}
double H2Isometry::error() const
{
return norm(a) + norm(u - 1.0);
}
H2Isometry H2Isometry::inverse() const
{
H2Isometry fOut;
fOut.setDiskCoordinates(conj(u),-u*a);
return fOut;
}
void H2Isometry::fixedPointsInDiskModel(CP1Point &c1, CP1Point &c2) const
{
if (a == 0.0)
{
c1 = CP1Point(0.0,1.0);
c2 = CP1Point(1.0,0.0);
return;
}
Complex D = sqrt((u - 1.0)*(u - 1.0) + 4.0*norm(a)*u);
c1 = CP1Point((1.0 - u + D),(2.0*conj(a)));
c2 = CP1Point((1.0 - u - D),(2.0*conj(a)));
}
bool H2Isometry::axis(H2Geodesic & L) const
{
if (!isHyperbolic())
{
return false;
}
else
{
Complex c1,c2;
CP1Point C1,C2;
fixedPointsInDiskModel(C1,C2);
c1 = C1.getComplexCoordinate();
c2 = C2.getComplexCoordinate();
if(norm(1.0 - conj(a)*c2) < norm(1.0 - conj(a)*c1))
{
L.setEndpointsInDiskModel(c2,c1);
return true;
}
else
{
L.setEndpointsInDiskModel(c1,c2);
return true;
}
}
}
Complex H2Isometry::hitComplexInDiskModel(const Complex &z)
{
return u*(z - a)/(1.0 - conj(a)*z);
}
H2Isometry operator *(const H2Isometry &f1, const H2Isometry &f2)
{
H2Isometry fOut;
Complex temp = 1.0 + f1.a*conj(f2.u*f2.a);
fOut.setDiskCoordinates(f1.u*f2.u*(temp*temp)/norm(temp), (f2.a + (f1.a*conj(f2.u)))/temp);
return fOut;
}
H2Point operator *(const H2Isometry & f, const H2Point & p)
{
Complex z = p.getDiskCoordinate();
H2Point pOut;
pOut.setDiskCoordinate(f.u*((z-f.a)/(1.0 - (conj(f.a)*z))));
return pOut;
}
std::vector<H2Point> operator *(const std::vector<H2Isometry> &vectorFs, const H2Point & p)
{
std::vector<H2Point> output;
output.reserve(vectorFs.size());
for (const auto & f : vectorFs)
{
output.push_back(f*p);
}
return output;
}
H2Polygon operator*(const H2Isometry &f, const H2Polygon &P)
{
H2Polygon out;
for(const auto &vertex : P.getVertices())
{
out.addVertex(f*vertex);
}
return out;
}
H2Geodesic operator*(const H2Isometry &f, const H2Geodesic &L)
{
Complex z1,z2,w1,w2;
L.getEndpointsInDiskModel(z1,z2);
w1 = f.u*((z1-f.a)/(1.0 - (conj(f.a)*z1)));
w2 = f.u*((z2-f.a)/(1.0 - (conj(f.a)*z2)));
H2Geodesic res;
res.setEndpointsInDiskModel(w1, w2);
return res;
}
H2GeodesicArc operator*(const H2Isometry &f, const H2GeodesicArc &L)
{
Complex z1,z2,w1,w2;
L.getEndpointsInDiskModel(z1,z2);
w1 = f.u*((z1-f.a)/(1.0 - (conj(f.a)*z1)));
w2 = f.u*((z2-f.a)/(1.0 - (conj(f.a)*z2)));
H2Point p1, p2;
p1.setDiskCoordinate(w1);
p2.setDiskCoordinate(w2);
return H2GeodesicArc(p1,p2);
}
void H2Isometry::setByMappingToVerticalUp(const H2Geodesic &L)
{
Complex a1,a2;
L.getEndpointsInDiskModel(a1,a2);
a = L.closestPointToOriginInDiskModel();
u = I*((conj(a)*(a1 + a2) - 2.0)/(a1 - a2));
}
void H2Isometry::setByMappingToVerticalDown(const H2Geodesic &L)
{
Complex a1,a2;
L.getEndpointsInDiskModel(a1,a2);
a = L.closestPointToOriginInDiskModel();
u = -I*((conj(a)*(a1 + a2) - 2.0)/(a1 - a2));
}
void H2Isometry::setByNormalizingPairOnLeftHandSide(const H2Isometry &f1, const H2Isometry &f1left)
{
H2Geodesic L1, L1left, L1leftNew;
f1.axis(L1);
f1left.axis(L1left);
H2Isometry fFirst, fSecond;
fFirst.setByMappingToVerticalDown(L1);
L1leftNew = fFirst*L1left;
double t = H2Isometry::geodesicNormalizer(L1leftNew);
fSecond.setVerticalTranslation(t);
*this = fSecond*fFirst;
}
void H2Isometry::setByNormalizingPairOnRightHandSide(const H2Isometry &f1, const H2Isometry &f1left)
{
H2Geodesic L1, L1left, L1leftNew;
f1.axis(L1);
f1left.axis(L1left);
H2Isometry fFirst, fSecond;
fFirst.setByMappingToVerticalUp(L1);
L1leftNew = fFirst*L1left;
double t = H2Isometry::geodesicNormalizer(L1leftNew);
fSecond.setVerticalTranslation(t);
*this = fSecond*fFirst;
}
H2Isometry H2Isometry::findConjugatorForGluing(const H2Isometry & f1, const H2Isometry & f1left,
const H2Isometry & f2, const H2Isometry &f2left, double twistNormalized)
{
double error = 0.00000001;
if (std::abs(f1.traceSquared()-f2.traceSquared()) > error)
{
throw(QString("ERROR in H2Isometry::findConjugatorForGluing: different translation lengths!"));
}
H2Isometry c1, c2, c;
H2Geodesic L1, L1left, L1leftNew;
f1.axis(L1);
f1left.axis(L1left);
c1.setByMappingToVerticalUp(L1);
L1leftNew = c1*L1left;
double t1 = H2Isometry::geodesicNormalizer(L1leftNew);
f2.axis(L1);
f2left.axis(L1left);
c2.setByMappingToVerticalDown(L1);
L1leftNew = c2*L1left;
double t2 = H2Isometry::geodesicNormalizer(L1leftNew);
c.setVerticalTranslation(-t1, twistNormalized, t2);
return c1.inverse()*c*c2;
}
bool H2Isometry::almostEqual(const H2Isometry & f1, const H2Isometry & f2)
{
double tol = 0.000000001;
return (norm(f1.u -f2.u) + norm(f1.a - f2.a)<tol);
}
std::ostream & operator<<(std::ostream & out, const H2Isometry &f)
{
if (f.isHyperbolic())
{
H2Geodesic axis;
f.axis(axis);
out << "Hyperbolic, axis = " << axis << ", translation length = " << f.translationLength();
}
if (f.isParabolic())
{
CP1Point Z1,Z2;
Complex z1;
f.fixedPointsInDiskModel(Z1,Z2);
z1 = Z1.getComplexCoordinate();
out << "Parabolic, fixed point = " << z1;
}
if (f.isElliptic())
{
CP1Point Z1,Z2;
Complex z1,z2;
f.fixedPointsInDiskModel(Z1,Z2);
z1 = Z1.getComplexCoordinate();
z2 = Z2.getComplexCoordinate();
out << "Elliptic, fixed points z1= " << z1 << " and z2= " << z2;
}
out << " {u= " << f.u << ", a= " << f.a << "}";
return out;
}
double H2Isometry::geodesicNormalizer(const H2Geodesic &L)
{
Complex z1, z2;
L.getEndpointsInDiskModel(z1, z2);
double b = -2 * (imag(z1) + imag(z2)) / norm(1.0 - z1*z2);
double delta = sqrt(b*b - 1);
double t1 = -b - delta;
double t2 = -b + delta;
return std::abs(t1) < std::abs(t2) ? -t1 : -t2;
}