double Distance( const Conic& c1, const Conic& c2, double circle_radius )
{
const Matrix3d Q = c1.Dual * c2.C;
// const double dsq = 3 - trace(Q)* pow(determinant(Q),-1.0/3.0);
const double dsq = 3 - Q.trace()* 1.0 / cbrt(Q.determinant());
return sqrt(dsq) * circle_radius;
}
double angle_mismatch(const Matrix3d& Q, const Matrix3d& R)
{
//std::cout << "Q\n" << Q << "\nR\n" << R << std::endl;
Matrix3d S = Q.transpose()*R;
//std::cout << "S\n: " << S << std::endl;
double thecos = (S.trace()-1.0)/2.0;
thecos = std::max( std::min ( thecos, 1.0), -1.0);
//std::cout << "thecos: " << thecos << " leads to angle: " << acos(thecos) << " * " << ( S(1,2) < 0.0 ? -1.0 : 1.0) << std::endl;
return (acos(thecos) * ( S(1,2) < 0.0 ? -1.0 : 1.0));
}
Matrix3d skewlog(Matrix3d M){
Matrix3d skew;
double val = (M.trace() - 1.f)/2.f;
if(val > 1.f)
val = 1.f;
else if (val < -1.f)
val = -1.f;
double theta = acos(val);
if(theta == 0.f)
skew << 0,0,0,0,0,0,0,0,0;
else
skew << (M-M.transpose())/(2.f*sin(theta))*theta;
return skew;
}
Vector3d logarithm_map_so3(Matrix3d R){
Matrix3d Id3 = Matrix3d::Identity();
Vector3d w;
Matrix3d V = Matrix3d::Identity(), w_hat = Matrix3d::Zero();
w << 0.f, 0.f, 0.f;
double cosine = (R.trace() - 1.f)/2.f;
if(cosine > 1.f)
cosine = 1.f;
else if (cosine < -1.f)
cosine = -1.f;
double sine = sqrt(1.0-cosine*cosine);
if(sine > 1.f)
sine = 1.f;
else if (sine < -1.f)
sine = -1.f;
double theta = acos(cosine);
if( theta > 0.000001 ){
w_hat << theta*(R-R.transpose()) / (2.f*sine);
w = skewcoords(w_hat);
}
return w;
}
/*
* deviator of a tensor
*/
static Matrix3d Deviator(Matrix3d M) {
Matrix3d eye;
eye.setIdentity();
eye *= M.trace() / 3.0;
return M - eye;
}
void DecomposeEssentialUsingHorn90(double _E[9], double _R1[9], double _R2[9], double _t1[3], double _t2[3]) {
//from : http://people.csail.mit.edu/bkph/articles/Essential.pdf
#ifdef USE_EIGEN
using namespace Eigen;
Matrix3d E = Map<Matrix<double,3,3,RowMajor> >(_E);
Matrix3d EEt = E * E.transpose();
Vector3d e0e1 = E.col(0).cross(E.col(1)),e1e2 = E.col(1).cross(E.col(2)),e2e0 = E.col(2).cross(E.col(0));
Vector3d b1,b2;
#if 1
//Method 1
Matrix3d bbt = 0.5 * EEt.trace() * Matrix3d::Identity() - EEt; //Horn90 (12)
Vector3d bbt_diag = bbt.diagonal();
if (bbt_diag(0) > bbt_diag(1) && bbt_diag(0) > bbt_diag(2)) {
b1 = bbt.row(0) / sqrt(bbt_diag(0));
b2 = -b1;
} else if (bbt_diag(1) > bbt_diag(0) && bbt_diag(1) > bbt_diag(2)) {
b1 = bbt.row(1) / sqrt(bbt_diag(1));
b2 = -b1;
} else {
b1 = bbt.row(2) / sqrt(bbt_diag(2));
b2 = -b1;
}
#else
//Method 2
if (e0e1.norm() > e1e2.norm() && e0e1.norm() > e2e0.norm()) {
b1 = e0e1.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18)
b2 = -b1;
} else if (e1e2.norm() > e0e1.norm() && e1e2.norm() > e2e0.norm()) {
b1 = e1e2.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18)
b2 = -b1;
} else {
b1 = e2e0.normalized() * sqrt(0.5 * EEt.trace()); //Horn90 (18)
b2 = -b1;
}
#endif
//Horn90 (19)
Matrix3d cofactors; cofactors.col(0) = e1e2; cofactors.col(1) = e2e0; cofactors.col(2) = e0e1;
cofactors.transposeInPlace();
//B = [b]_x , see Horn90 (6) and http://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication
Matrix3d B1; B1 << 0,-b1(2),b1(1),
b1(2),0,-b1(0),
-b1(1),b1(0),0;
Matrix3d B2; B2 << 0,-b2(2),b2(1),
b2(2),0,-b2(0),
-b2(1),b2(0),0;
Map<Matrix<double,3,3,RowMajor> > R1(_R1),R2(_R2);
//Horn90 (24)
R1 = (cofactors.transpose() - B1*E) / b1.dot(b1);
R2 = (cofactors.transpose() - B2*E) / b2.dot(b2);
Map<Vector3d> t1(_t1),t2(_t2);
t1 = b1; t2 = b2;
cout << "Horn90 provided " << endl << R1 << endl << "and" << endl << R2 << endl;
#endif
}
double BODY::Angle2BD(BODY* bd2)
{
Matrix3d ar = bd2->A0.transpose()*A0;
double Ctheta = 0.5*(ar.trace() - 1);
return acos(Ctheta);
}