#ifndef ROTATION_HH
#define ROTATION_HH
/** \file
\brief Define rotations in Euclidean spaces
*/
#include <dune/common/fvector.hh>
#include <dune/common/array.hh>
#include <dune/common/fmatrix.hh>
#include <dune/common/exceptions.hh>
#include "quaternion.hh"
template <int dim, class T>
class Rotation
{
};
/** \brief Specialization for dim==2
*/
template <class T>
class Rotation<2,T>
{
public:
/** \brief Member of the corresponding Lie algebra. This really is a skew-symmetric matrix */
typedef Dune::FieldVector<T,1> TangentVector;
/** \brief Default constructor, create the identity rotation */
Rotation()
: angle_(0)
{}
/** \brief Return the identity element */
static Rotation<2,T> identity() {
// Default constructor creates an identity
Rotation<2,T> id;
return id;
}
/** \brief The exponential map from a given point $p \in SO(3)$. */
static Rotation<2,T> exp(const Rotation<2,T>& p, const TangentVector& v) {
Rotation<2,T> result = p;
result.angle_ += v;
return result;
}
//private:
// We store the rotation as an angle
double angle_;
};
/** \brief Specialization for dim==3
Uses unit quaternion coordinates.
*/
template <class T>
class Rotation<3,T> : public Quaternion<T>
{
/** \brief Computes sin(x/2) / x without getting unstable for small x */
static T sincHalf(const T& x) {
return (x < 1e-4) ? 0.5 + (x*x/48) : std::sin(x/2)/x;
}
public:
/** \brief The type used for coordinates */
typedef T ctype;
/** \brief Member of the corresponding Lie algebra. This really is a skew-symmetric matrix */
typedef Dune::FieldVector<T,3> TangentVector;
/** \brief A tangent vector as a vector in the surrounding coordinate space */
typedef Quaternion<T> EmbeddedTangentVector;
/** \brief Default constructor creates the identity element */
Rotation()
: Quaternion<T>(0,0,0,1)
{}
Rotation<3,T>(Dune::FieldVector<T,3> axis, T angle)
: Quaternion<T>(axis, angle)
{}
/** \brief Assignment from a quaternion
\deprecated Using this is bad design.
*/
Rotation& operator=(const Quaternion<T>& other) {
(*this)[0] = other[0];
(*this)[1] = other[1];
(*this)[2] = other[2];
(*this)[3] = other[3];
return *this;
}
/** \brief Return the identity element */
static Rotation<3,T> identity() {
// Default constructor creates an identity
Rotation<3,T> id;
return id;
}
/** \brief Right multiplication */
Rotation<3,T> mult(const Rotation<3,T>& other) const {
Rotation<3,T> q;
q[0] = (*this)[3]*other[0] - (*this)[2]*other[1] + (*this)[1]*other[2] + (*this)[0]*other[3];
q[1] = (*this)[2]*other[0] + (*this)[3]*other[1] - (*this)[0]*other[2] + (*this)[1]*other[3];
q[2] = - (*this)[1]*other[0] + (*this)[0]*other[1] + (*this)[3]*other[2] + (*this)[2]*other[3];
q[3] = - (*this)[0]*other[0] - (*this)[1]*other[1] - (*this)[2]*other[2] + (*this)[3]*other[3];
return q;
}
/** \brief Right multiplication with a quaternion
\todo do we really need this?*/
Rotation<3,T> mult(const Quaternion<T>& other) const {
Rotation<3,T> q;
q[0] = (*this)[3]*other[0] - (*this)[2]*other[1] + (*this)[1]*other[2] + (*this)[0]*other[3];
q[1] = (*this)[2]*other[0] + (*this)[3]*other[1] - (*this)[0]*other[2] + (*this)[1]*other[3];
q[2] = - (*this)[1]*other[0] + (*this)[0]*other[1] + (*this)[3]*other[2] + (*this)[2]*other[3];
q[3] = - (*this)[0]*other[0] - (*this)[1]*other[1] - (*this)[2]*other[2] + (*this)[3]*other[3];
return q;
}
/** \brief The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Rotation<3,T> exp(const Dune::FieldVector<T,3>& v) {
return exp(v[0], v[1], v[2]);
}
/** \brief The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Rotation<3,T> exp(const T& v0, const T& v1, const T& v2) {
Rotation<3,T> q;
T normV = std::sqrt(v0*v0 + v1*v1 + v2*v2);
// Stabilization for small |v| due to Grassia
T sin = sincHalf(normV);
// if normV == 0 then q = (0,0,0,1)
assert(!isnan(sin));
q[0] = sin * v0;
q[1] = sin * v1;
q[2] = sin * v2;
q[3] = std::cos(normV/2);
return q;
}
/** \brief The exponential map from a given point $p \in SO(3)$. */
static Rotation<3,T> exp(const Rotation<3,T>& p, const TangentVector& v) {
Rotation<3,T> corr = exp(v);
return p.mult(corr);
}
static Dune::FieldMatrix<T,4,3> Dexp(const Dune::FieldVector<T,3>& v) {
Dune::FieldMatrix<T,4,3> result(0);
T norm = v.two_norm();
for (int i=0; i<3; i++) {
for (int m=0; m<3; m++) {
result[m][i] = (norm<1e10)
/** \todo Isn't there a better way to implement this stably? */
? 0.5 * (i==m)
: 0.5 * std::cos(norm/2) * v[i] * v[m] / (norm*norm) + sincHalf(norm) * ( (i==m) - v[i]*v[m]/(norm*norm));
}
result[3][i] = - 0.5 * sincHalf(norm) * v[i];
}
return result;
}
static void DDexp(const Dune::FieldVector<T,3>& v,
Dune::array<Dune::FieldMatrix<T,3,3>, 4>& result) {
T norm = v.two_norm();
if (norm<=1e-10) {
for (int m=0; m<4; m++)
result[m] = 0;
for (int i=0; i<3; i++)
result[3][i][i] = -0.25;
} else {
for (int i=0; i<3; i++) {
for (int j=0; j<3; j++) {
for (int m=0; m<3; m++) {
result[m][i][j] = -0.25*std::sin(norm/2)*v[i]*v[j]*v[m]/(norm*norm*norm)
+ ((i==j)*v[m] + (j==m)*v[i] + (i==m)*v[j] - 3*v[i]*v[j]*v[m]/(norm*norm))
* (0.5*std::cos(norm/2) - sincHalf(norm)) / (norm*norm);
}
result[3][i][j] = -0.5/(norm*norm)
* ( 0.5*std::cos(norm/2)*v[i]*v[j] + std::sin(norm/2) * ((i==j)*norm - v[i]*v[j]/norm));
}
}
}
}
/** \brief The inverse of the exponential map */
static Dune::FieldVector<T,3> expInv(const Rotation<3,T>& q) {
// Compute v = exp^{-1} q
// Due to numerical dirt, q[3] may be larger than 1.
// In that case, use 1 instead of q[3].
Dune::FieldVector<T,3> v;
if (q[3] > 1.0) {
v = 0;
} else {
T invSinc = 1/sincHalf(2*std::acos(q[3]));
v[0] = q[0] * invSinc;
v[1] = q[1] * invSinc;
v[2] = q[2] * invSinc;
}
return v;
}
/** \brief The derivative of the inverse of the exponential map, evaluated at q */
static Dune::FieldMatrix<T,3,4> DexpInv(const Rotation<3,T>& q) {
// Compute v = exp^{-1} q
Dune::FieldVector<T,3> v = expInv(q);
// The derivative of exp at v
Dune::FieldMatrix<T,4,3> A = Dexp(v);
// Compute the Moore-Penrose pseudo inverse A^+ = (A^T A)^{-1} A^T
Dune::FieldMatrix<T,3,3> ATA;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++) {
ATA[i][j] = 0;
for (int k=0; k<4; k++)
ATA[i][j] += A[k][i] * A[k][j];
}
ATA.invert();
Dune::FieldMatrix<T,3,4> APseudoInv;
for (int i=0; i<3; i++)
for (int j=0; j<4; j++) {
APseudoInv[i][j] = 0;
for (int k=0; k<3; k++)
APseudoInv[i][j] += ATA[i][k] * A[j][k];
}
return APseudoInv;
}
static T distance(const Rotation<3,T>& a, const Rotation<3,T>& b) {
Quaternion<T> diff = a;
diff.invert();
diff = diff.mult(b);
// Compute the geodesical distance between a and b on SO(3)
// Due to numerical dirt, diff[3] may be larger than 1.
// In that case, use 1 instead of diff[3].
return (diff[3] > 1.0)
? 0
: 2*std::acos( std::min(diff[3],1.0) );
}
/** \brief Compute the vector in T_aSO(3) that is mapped by the exponential map
to the geodesic from a to b
*/
static Dune::FieldVector<T,3> difference(const Rotation<3,T>& a, const Rotation<3,T>& b) {
Quaternion<T> diff = a;
diff.invert();
diff = diff.mult(b);
// Compute the geodesical distance between a and b on SO(3)
// Due to numerical dirt, diff[3] may be larger than 1.
// In that case, use 1 instead of diff[3].
Dune::FieldVector<T,3> v;
if (diff[3] > 1.0) {
v = 0;
} else {
T dist = 2*std::acos( std::min(diff[3],1.0) );
T invSinc = 1/sincHalf(dist);
// Compute difference on T_a SO(3)
v[0] = diff[0] * invSinc;
v[1] = diff[1] * invSinc;
v[2] = diff[2] * invSinc;
}
return v;
}
/** \brief Interpolate between two rotations */
static Rotation<3,T> interpolate(const Rotation<3,T>& a, const Rotation<3,T>& b, double omega) {
// Compute difference on T_a SO(3)
Dune::FieldVector<T,3> v = difference(a,b);
v *= omega;
return a.mult(exp(v[0], v[1], v[2]));
}
/** \brief Interpolate between two rotations
\param omega must be between 0 and 1
*/
static Quaternion<T> interpolateDerivative(const Rotation<3,T>& a, const Rotation<3,T>& b,
double omega) {
Quaternion<T> result(0);
// Compute difference on T_a SO(3)
Dune::FieldVector<double,3> xi = difference(a,b);
Dune::FieldVector<double,3> v = xi;
v *= omega;
// //////////////////////////////////////////////////////////////
// v now contains the derivative at 'a'. The derivative at
// the requested site is v pushed forward by Dexp.
// /////////////////////////////////////////////////////////////
Dune::FieldMatrix<double,4,3> diffExp = Dexp(v);
diffExp.umv(xi,result);
return a.Quaternion<T>::mult(result);
}
/** \brief Return the corresponding orthogonal matrix */
void matrix(Dune::FieldMatrix<T,3,3>& m) const {
m[0][0] = (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[0][1] = 2 * ( (*this)[0]*(*this)[1] - (*this)[2]*(*this)[3] );
m[0][2] = 2 * ( (*this)[0]*(*this)[2] + (*this)[1]*(*this)[3] );
m[1][0] = 2 * ( (*this)[0]*(*this)[1] + (*this)[2]*(*this)[3] );
m[1][1] = - (*this)[0]*(*this)[0] + (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[1][2] = 2 * ( -(*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][0] = 2 * ( (*this)[0]*(*this)[2] - (*this)[1]*(*this)[3] );
m[2][1] = 2 * ( (*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][2] = - (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] + (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
}
/** \brief Set rotation from orthogonal matrix
We tacitly assume that the matrix really is orthogonal */
void set(const Dune::FieldMatrix<T,3,3>& m) {
// Easier writing
Dune::FieldVector<T,4>& p = (*this);
// The following equations for the derivation of a unit quaternion from a rotation
// matrix comes from 'E. Salamin, Application of Quaternions to Computation with
// Rotations, Technical Report, Stanford, 1974'
p[0] = (1 + m[0][0] - m[1][1] - m[2][2]) / 4;
p[1] = (1 - m[0][0] + m[1][1] - m[2][2]) / 4;
p[2] = (1 - m[0][0] - m[1][1] + m[2][2]) / 4;
p[3] = (1 + m[0][0] + m[1][1] + m[2][2]) / 4;
// avoid rounding problems
if (p[0] >= p[1] && p[0] >= p[2] && p[0] >= p[3]) {
p[0] = std::sqrt(p[0]);
// r_x r_y = (R_12 + R_21) / 4
p[1] = (m[0][1] + m[1][0]) / 4 / p[0];
// r_x r_z = (R_13 + R_31) / 4
p[2] = (m[0][2] + m[2][0]) / 4 / p[0];
// r_0 r_x = (R_32 - R_23) / 4
p[3] = (m[2][1] - m[1][2]) / 4 / p[0];
} else if (p[1] >= p[0] && p[1] >= p[2] && p[1] >= p[3]) {
p[1] = std::sqrt(p[1]);
// r_x r_y = (R_12 + R_21) / 4
p[0] = (m[0][1] + m[1][0]) / 4 / p[1];
// r_y r_z = (R_23 + R_32) / 4
p[2] = (m[1][2] + m[2][1]) / 4 / p[1];
// r_0 r_y = (R_13 - R_31) / 4
p[3] = (m[0][2] - m[2][0]) / 4 / p[1];
} else if (p[2] >= p[0] && p[2] >= p[1] && p[2] >= p[3]) {
p[2] = std::sqrt(p[2]);
// r_x r_z = (R_13 + R_31) / 4
p[0] = (m[0][2] + m[2][0]) / 4 / p[2];
// r_y r_z = (R_23 + R_32) / 4
p[1] = (m[1][2] + m[2][1]) / 4 / p[2];
// r_0 r_z = (R_21 - R_12) / 4
p[3] = (m[1][0] - m[0][1]) / 4 / p[2];
} else {
p[3] = std::sqrt(p[3]);
// r_0 r_x = (R_32 - R_23) / 4
p[0] = (m[2][1] - m[1][2]) / 4 / p[3];
// r_0 r_y = (R_13 - R_31) / 4
p[1] = (m[0][2] - m[2][0]) / 4 / p[3];
// r_0 r_z = (R_21 - R_12) / 4
p[2] = (m[1][0] - m[0][1]) / 4 / p[3];
}
}
/** \brief Create three vectors which form an orthonormal basis of \mathbb{H} together
with this one.
This is used to compute the strain in rod problems.
See: Dichmann, Li, Maddocks, 'Hamiltonian Formulations and Symmetries in
Rod Mechanics', page 83
*/
Quaternion<T> B(int m) const {
assert(m>=0 && m<3);
Quaternion<T> r;
if (m==0) {
r[0] = (*this)[3];
r[1] = (*this)[2];
r[2] = -(*this)[1];
r[3] = -(*this)[0];
} else if (m==1) {
r[0] = -(*this)[2];
r[1] = (*this)[3];
r[2] = (*this)[0];
r[3] = -(*this)[1];
} else {
r[0] = (*this)[1];
r[1] = -(*this)[0];
r[2] = (*this)[3];
r[3] = -(*this)[2];
}
return r;
}
};
#endif