rotation.hh

#ifndef ROTATION_HH
#define ROTATION_HH

/** \file
    \brief Define rotations in Euclidean spaces
*/

#include <array>

#include <dune/common/fvector.hh>
#include <dune/common/fmatrix.hh>
#include <dune/common/exceptions.hh>
#include <dune/common/math.hh>

#include "quaternion.hh"
#include <dune/gfe/tensor3.hh>
#include <dune/gfe/unitvector.hh>
#include <dune/gfe/skewmatrix.hh>
#include <dune/gfe/symmetricmatrix.hh>

template <class T, int dim>
class Rotation
{

};

/** \brief Specialization for dim==2
    \tparam T The type used for coordinates
*/
template <class T>
class Rotation<T,2>
{
public:
    /** \brief The type used for coordinates */
    typedef T ctype;

    /** \brief Dimension of the manifold formed by the 2d rotations */
    static const int dim = 1;

    /** \brief Coordinates are embedded in the euclidean space */
    static const int embeddedDim = 1;

    /** \brief Member of the corresponding Lie algebra.  This really is a skew-symmetric matrix */
    typedef Dune::FieldVector<T,1> TangentVector;

    /** \brief Member of the corresponding Lie algebra.  This really is a skew-symmetric matrix

    This vector is not really embedded in anything.  I have to make my notation more consistent! */
    typedef Dune::FieldVector<T,1> EmbeddedTangentVector;

    /** \brief The global convexity radius of the rotation group */
    static constexpr double convexityRadius = 0.5 * M_PI;

    /** \brief Default constructor, create the identity rotation */
    Rotation()
        : angle_(0)
    {}

    Rotation(const T& angle)
        : angle_(angle)
    {}

    /** \brief Return the identity element */
    static Rotation<T,2> identity() {
        // Default constructor creates an identity
        Rotation<T,2> id;
        return id;
    }

    static T distance(const Rotation<T,2>& a, const Rotation<T,2>& b) {
        T dist = a.angle_ - b.angle_;
        while (dist < 0)
            dist += 2*M_PI;
        while (dist > 2*M_PI)
            dist -= 2*M_PI;

        return (dist <= M_PI) ? dist : 2*M_PI - dist;
    }

    /** \brief The exponential map from a given point $p \in SO(3)$. */
    static Rotation<T,2> exp(const Rotation<T,2>& p, const TangentVector& v) {
        Rotation<T,2> result = p;
        result.angle_ += v;
        return result;
    }

    /** \brief The exponential map from \f$ \mathfrak{so}(2) \f$ to \f$ SO(2) \f$
     */
    static Rotation<T,2> exp(const Dune::FieldVector<T,1>& v) {
        Rotation<T,2> result;
        result.angle_ = v[0];
        return result;
    }

    static TangentVector derivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,2>& a,
                                                                      const Rotation<T,2>& b) {
        // This assertion is here to remind me of the following laziness:
        // The difference has to be computed modulo 2\pi
        assert( std::abs(a.angle_ - b.angle_) <= M_PI );
        return -2 * (a.angle_ - b.angle_);
    }

    static Dune::FieldMatrix<T,1,1> secondDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,2>& a,
                                                                                            const Rotation<T,2>& b) {
        return 2;
    }

    /** \brief Right multiplication */
    Rotation<T,2> mult(const Rotation<T,2>& other) const {
        Rotation<T,2> q = *this;
        q.angle_ += other.angle_;
        return q;
    }

    /** \brief Compute an orthonormal basis of the tangent space of SO(3).

    This basis is of course not globally continuous.
    */
    Dune::FieldMatrix<T,1,1> orthonormalFrame() const {
        return Dune::FieldMatrix<T,1,1>(1);
    }

    //private:

    // We store the rotation as an angle
    T angle_;
};

//! Send configuration to output stream
template <class T>
std::ostream& operator<< (std::ostream& s, const Rotation<T,2>& c)
  {
      return s << "[" << c.angle_ << "  (" << std::sin(c.angle_) << " " << std::cos(c.angle_) << ") ]";
  }


/** \brief Specialization for dim==3

Uses unit quaternion coordinates.
\todo Reimplement the method inverse() such that it returns a Rotation instead of a Quaternion.
Then remove the cast in the method setRotation, file averageinterface.hh
*/
template <class T>
class Rotation<T,3> : public Quaternion<T>
{

    /** \brief Computes sin(x/2) / x without getting unstable for small x */
    static T sincHalf(const T& x) {
        using std::sin;
        return (x < 1e-1) ? 0.5 - (x*x/48)  + Dune::power(x,4)/3840 - Dune::power(x,6)/645120  : sin(x/2)/x;
    }

    /** \brief Computes sin(sqrt(x)/2) / sqrt(x) without getting unstable for small x
     *
     * Using this somewhat exotic series allows to avoid a few calls to std::sqrt near 0,
     * which ADOL-C doesn't like.
     */
    static T sincHalfOfSquare(const T& x) {
        using std::sin;
        using std::sqrt;
        return (x < 1e-15) ? 0.5 - (x/48)  + (x*x)/(120*32) : sin(sqrt(x)/2)/sqrt(x);
    }

    /** \brief Computes sin(sqrt(x)) / sqrt(x) without getting unstable for small x
     *
     * Using this somewhat exotic series allows to avoid a few calls to std::sqrt near 0,
     * which ADOL-C doesn't like.
     */
    static T sincOfSquare(const T& x) {
        using std::sin;
        using std::sqrt;
        // we need here lots of terms to be sure that the numerical derivatives are also within maschine precision
        return (x < 1e-2) ?
        1-x/6
        +x*x/120
        -Dune::power(x,3)/5040
        +Dune::power(x,4)/362880
        -Dune::power(x,5)/39916800
        +Dune::power(x,6)/6227020800
        -Dune::power(x,7)/1307674368000: sin(sqrt(x))/sqrt(x);
    }

public:

    /** \brief The type used for coordinates */
    typedef T ctype;

    /** \brief The type used for global coordinates */
    typedef Dune::FieldVector<T,4> CoordinateType;

    /** \brief Dimension of the manifold formed by the 3d rotations */
    static const int dim = 3;

    /** \brief Coordinates are embedded into a four-dimension Euclidean space */
    static const int embeddedDim = 4;

    /** \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 The global convexity radius of the rotation group */
    static constexpr double convexityRadius = 0.5 * M_PI;

    /** \brief Default constructor creates the identity element */
    Rotation()
        : Quaternion<T>(0,0,0,1)
    {}

    explicit Rotation(const std::array<T,4>& c)
    {
        for (int i=0; i<4; i++)
            (*this)[i] = c[i];

        *this /= this->two_norm();
    }

    explicit Rotation(const Dune::FieldVector<T,4>& c)
        : Quaternion<T>(c)
    {
        *this /= this->two_norm();
    }

    Rotation(Dune::FieldVector<T,3> axis, T angle)
    {
        axis /= axis.two_norm();
        axis *= std::sin(angle/2);
        (*this)[0] = axis[0];
        (*this)[1] = axis[1];
        (*this)[2] = axis[2];
        (*this)[3] = std::cos(angle/2);
    }

    /** \brief Rebind the Rotation to another coordinate type */
    template<class U>
    struct rebind
    {
      typedef Rotation<U,3> other;
    };

    Rotation& operator= (const Dune::FieldVector<T,4>& other)
    {
      for (int i=0; i<4; i++)
        (*this)[i] = other[i];
      *this /= this->two_norm();
      return *this;
    }

    /** \brief Assignment from Rotation with different type -- used for automatic differentiation with ADOL-C */
    template <class T2>
    Rotation& operator <<= (const Rotation<T2,3>& other) {
        for (int i=0; i<4; i++)
            (*this)[i] <<= other[i];
        return *this;
    }

    /** \brief Return the identity element */
    static Rotation<T,3> identity() {
        // Default constructor creates an identity
        Rotation<T,3> id;
        return id;
    }

    /** \brief Right multiplication */
    Rotation<T,3> mult(const Rotation<T,3>& other) const {
        Rotation<T,3> 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<T,3> exp(const SkewMatrix<T,3>& v) {
        using std::cos;
        using std::sqrt;

        Rotation<T,3> q;

        Dune::FieldVector<T,3> vAxial = v.axial();
        T normV2 = vAxial.two_norm2();

        // Stabilization for small |v|
        T sin = sincHalfOfSquare(normV2);

        assert(!std::isnan(sin));

        q[0] = sin * vAxial[0];
        q[1] = sin * vAxial[1];
        q[2] = sin * vAxial[2];

        // The series expansion of cos(x) at x=0 is
        // 1 - x*x/2 + x*x*x*x/24 - ...
        // hence the series of cos(x/2) is
        // 1 - x*x/8 + x*x*x*x/384 - ...
        q[3] = (normV2 < 1e-4)
          ? 1 - normV2/8 + normV2*normV2 / 384-Dune::power(normV2,3)/46080 + Dune::power(normV2,4)/10321920
          : cos(sqrt(normV2)/2);

        return q;
    }


    /** \brief The exponential map from a given point $p \in SO(3)$.

     * \param v A tangent vector *at the identity*!
     */
    static Rotation<T,3> exp(const Rotation<T,3>& p, const SkewMatrix<T,3>& v) {
        Rotation<T,3> corr = exp(v);
        return p.mult(corr);
    }

    /** \brief The exponential map from a given point $p \in SO(3)$.

        There may be a more direct way to implement this

        \param v A tangent vector in quaternion coordinates
     */
    static Rotation<T,3> exp(const Rotation<T,3>& p, const Dune::FieldVector<T,4>& v) {

        using std::abs;
        using std::cos;
        using std::sqrt;
        assert( abs(p*v) < 1e-8 );

        const T norm2 = v.two_norm2();

        Rotation<T,3> result = p;

        // The series expansion of cos(x) at x=0 is
        // 1 - x*x/2 + x*x*x*x/24 - ...
        T cosValue = (norm2 < 1e-5)
          ? 1 - norm2/2 + norm2*norm2 / 24 - Dune::power(norm2,3)/720+Dune::power(norm2,4)/40320
          : cos(sqrt(norm2));
        result *= cosValue;

        result.axpy(sincOfSquare(norm2), v);
        return result;
    }


     /** \brief The exponential map from a given point $p \in SO(3)$.

        \param v A tangent vector.
     */
    static Rotation<T,3> exp(const Rotation<T,3>& p, const TangentVector& v) {

        // embedded tangent vector
        Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();
        Quaternion<T> embeddedTangent;
        basis.mtv(v, embeddedTangent);

        return exp(p,embeddedTangent);

    }

    /** \brief Compute tangent vector from given basepoint and skew symmetric matrix. */
    static TangentVector skewToTangentVector(const Rotation<T,3>& p, const SkewMatrix<T,3>& v ) {

        // embedded tangent vector at identity
        Quaternion<T> vAtIdentity(0);
        vAtIdentity[0] = 0.5*v.axial()[0];
        vAtIdentity[1] = 0.5*v.axial()[1];
        vAtIdentity[2] = 0.5*v.axial()[2];

        // multiply with base point to get real embedded tangent vector
        Quaternion<T> vQuat = ((Quaternion<T>) p).mult(vAtIdentity);

        //get basis of the tangent space
        Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();

        // transform coordinates
        TangentVector tang;
        basis.mv(vQuat,tang);

        return tang;
    }

    /** \brief Compute skew matrix from given basepoint and tangent vector. */
    static SkewMatrix<T,3> tangentToSkew(const Rotation<T,3>& p, const TangentVector& tangent) {

        // embedded tangent vector
        Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();
        Quaternion<T> embeddedTangent;
        basis.mtv(tangent, embeddedTangent);

        return tangentToSkew(p,embeddedTangent);
    }

    /** \brief Compute skew matrix from given basepoint and an embedded tangent vector. */
    static SkewMatrix<T,3> tangentToSkew(const Rotation<T,3>& p, const EmbeddedTangentVector& q) {

        // left multiplication by the inverse base point yields a tangent vector at the identity
        Quaternion<T> vAtIdentity = p.inverse().mult(q);
        assert( std::abs(vAtIdentity[3]) < 1e-8 );

        SkewMatrix<T,3> skew;
        skew.axial()[0] = 2*vAtIdentity[0];
        skew.axial()[1] = 2*vAtIdentity[1];
        skew.axial()[2] = 2*vAtIdentity[2];

        return skew;
    }

    /** \brief Derivative of the exponential map at the identity
     *
     * The exponential map at the identity is a map from a neighborhood of zero to the neighborhood of the identity rotation.
     *
     * \param v Where to evaluate the derivative of the (exponential map at the identity)
     */
    static Dune::FieldMatrix<T,4,3> Dexp(const SkewMatrix<T,3>& v) {

        using std::cos;

        Dune::FieldMatrix<T,4,3> result(0);
        Dune::FieldVector<T,3> vAxial = v.axial();
        T norm = vAxial.two_norm();

        for (int i=0; i<3; i++) {

            for (int m=0; m<3; m++) {

                result[m][i] = (norm<1e-10)
                    /** \todo Isn't there a better way to implement this stably? */
                    ? T(0.5) * (i==m)
                    : 0.5 * cos(norm/2) * vAxial[i] * vAxial[m] / (norm*norm) + sincHalf(norm) * ( (i==m) - vAxial[i]*vAxial[m]/(norm*norm));

            }

            result[3][i] = - 0.5 * sincHalf(norm) * vAxial[i];

        }
        return result;
    }

    static void DDexp(const Dune::FieldVector<T,3>& v,
                      std::array<Dune::FieldMatrix<T,3,3>, 4>& result) {

        using std::cos;
        using std::sin;

        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*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*cos(norm/2) - sincHalf(norm)) / (norm*norm);


                    }

                    result[3][i][j] = -0.5/(norm*norm)
                        * ( 0.5*cos(norm/2)*v[i]*v[j] + 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<T,3>& 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 {

            using std::acos;
            T invSinc = 1/sincHalf(2*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<T,3>& 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(SkewMatrix<T,3>(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;
    }

    /** \brief The cayley mapping from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$.
     *
     *  The formula is taken from 'J.C.Simo, N.Tarnom, M.Doblare - Non-linear dynamics of
     *  three-dimensional rods:Exact energy and momentum conserving algorithms'
     *  (but the corrected version with 0.25 instead of 0.5 in the denominator)
     */
    static Rotation<T,3> cayley(const SkewMatrix<T,3>& s) {
        Rotation<T,3> q;

        Dune::FieldVector<T,3> vAxial = s.axial();
        T norm = 0.25*vAxial.two_norm2() + 1;

        Dune::FieldMatrix<T,3,3> mat = s.toMatrix();
        mat *= 0.5;
        Dune::FieldMatrix<T,3,3> skewSquare = mat;
        skewSquare.rightmultiply(mat);
        mat += skewSquare;
        mat *= 2/norm;

        for (int i=0;i<3;i++)
            mat[i][i] += 1;

        q.set(mat);
        return q;
    }

    /** \brief The inverse of the Cayley mapping.
     *
     *  The formula is taken from J.M.Selig - Cayley Maps for SE(3).
     */
    static SkewMatrix<T,3>  cayleyInv(const Rotation<T,3> q) {

        Dune::FieldMatrix<T,3,3> mat;

        // compute the trace of the rotation matrix
        T trace = -q[0]*q[0] -q[1]*q[1] -q[2]*q[2]+3*q[3]*q[3];

        if ( (trace+1)>1e-6 || (trace+1)<-1e-6) { // if this term doesn't vanish we can use a direct formula

            q.matrix(mat);
            Dune::FieldMatrix<T,3,3> matT;
            Rotation<T,3>(q.inverse()).matrix(matT);
            mat -= matT;
            mat *= 2/(1+trace);
        }
        else { // use the formula that involves the computation of an inverse
            Dune::FieldMatrix<T,3,3> inv;
            q.matrix(inv);
            Dune::FieldMatrix<T,3,3> notInv = inv;

            for (int i=0;i<3;i++) {
                inv[i][i] +=1;
                notInv[i][i] -=1;
            }
            inv.invert();
            mat = notInv.leftmultiply(inv);
            mat *= 2;
        }

        // result is a skew symmetric matrix
        SkewMatrix<T,3> res;
        res.axial()[0] = mat[2][1];
        res.axial()[1] = mat[0][2];
        res.axial()[2] = mat[1][0];

        return res;

    }

    static T distance(const Rotation<T,3>& a, const Rotation<T,3>& b) {

        // Distance in the unit quaternions: 2*arccos( ((a^{-1) b)_3 )
        // But note that (a^{-1}b)_3 is actually <a,b> (in R^4)
        T sp = a.globalCoordinates() * b.globalCoordinates();

        // Scalar product may be larger than 1.0, due to numerical dirt
        using std::acos;
        using std::min;
        T dist = 2*acos( min(sp,T(1.0)) );

        // Make sure we do the right thing if a and b are not in the same sheet
        // of the double covering of the unit quaternions over SO(3)
        return (dist>=M_PI) ? (2*M_PI - dist) : dist;
    }

    /** \brief Compute the vector in T_aSO(3) that is mapped by the exponential map
        to the geodesic from a to b
    */
    static EmbeddedTangentVector log(const Rotation<T,3>& a, const Rotation<T,3>& b) {

        // embedded tangent vector at identity
        Quaternion<T> v;

        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].
        if (diff[3] > 1.0) {

            v = 0;

        } else {

            // TODO: ADOL-C does not like this part of the code,
            // because arccos is not differentiable at -1 and 1.
            // (Even though the overall 'log' function is differentiable.)
            using std::acos;
            T dist = 2*acos( diff[3] );

            // Make sure we do the right thing if a and b are not in the same sheet
            // of the double covering of the unit quaternions over SO(3)
            if (dist>=M_PI) {
                dist = 2*M_PI - dist;
                diff *= -1;
            }

            T invSinc = 1/sincHalf(dist);

            // Compute log on T_a SO(3)
            v[0] = 0.5 * diff[0] * invSinc;
            v[1] = 0.5 * diff[1] * invSinc;
            v[2] = 0.5 * diff[2] * invSinc;
            v[3] = 0;
        }

        // multiply with base point to get real embedded tangent vector
        return ((Quaternion<T>) a).mult(v);
    }

    /** \brief Compute the derivatives of the director vectors with respect to the quaternion coordinates
     *
     * Let \f$ d_k(q) = (d_{k,1}, d_{k,2}, d_{k,3})\f$ be the k-th director vector at \f$ q \f$.
     * Then the return value of this method is
     * \f[ A_{ijk} = \frac{\partial d_{i,j}}{\partial q_k} \f]
     */
    void getFirstDerivativesOfDirectors(Tensor3<T,3, 3, 4>& dd_dq) const
    {
        const Quaternion<T>& q = (*this);

        dd_dq[0][0] = { 2*q[0], -2*q[1], -2*q[2],  2*q[3]};
        dd_dq[0][1] = { 2*q[1],  2*q[0],  2*q[3],  2*q[2]};
        dd_dq[0][2] = { 2*q[2], -2*q[3],  2*q[0], -2*q[1]};

        dd_dq[1][0] = { 2*q[1],  2*q[0], -2*q[3], -2*q[2]};
        dd_dq[1][1] = {-2*q[0],  2*q[1], -2*q[2],  2*q[3]};
        dd_dq[1][2] = { 2*q[3],  2*q[2],  2*q[1],  2*q[0]};

        dd_dq[2][0] = { 2*q[2],  2*q[3],  2*q[0],  2*q[1]};
        dd_dq[2][1] = {-2*q[3],  2*q[2],  2*q[1], -2*q[0]};
        dd_dq[2][2] = {-2*q[0], -2*q[1],  2*q[2],  2*q[3]};

    }

    /** \brief Compute the second derivatives of the director vectors with respect to the quaternion coordinates
     *
     * Let \f$ d_k(q) = (d_{k,1}, d_{k,2}, d_{k,3})\f$ be the k-th director vector at \f$ q \f$.
     * Then the return value of this method is
     * \f[ A_{ijkl} = \frac{\partial^2 d_{i,j}}{\partial q_k \partial q_l} \f]
     */
    static void getSecondDerivativesOfDirectors(std::array<Tensor3<T,3, 4, 4>, 3>& dd_dq_dq)
    {
        for (int i=0; i<3; i++)
            dd_dq_dq[i] = T(0);

        dd_dq_dq[0][0][0][0] =  2;  dd_dq_dq[0][0][1][1] = -2;  dd_dq_dq[0][0][2][2] = -2;  dd_dq_dq[0][0][3][3] =  2;
        dd_dq_dq[0][1][0][1] =  2;  dd_dq_dq[0][1][1][0] =  2;  dd_dq_dq[0][1][2][3] =  2;  dd_dq_dq[0][1][3][2] =  2;
        dd_dq_dq[0][2][0][2] =  2;  dd_dq_dq[0][2][1][3] = -2;  dd_dq_dq[0][2][2][0] =  2;  dd_dq_dq[0][2][3][1] = -2;

        dd_dq_dq[1][0][0][1] =  2;  dd_dq_dq[1][0][1][0] =  2;  dd_dq_dq[1][0][2][3] = -2;  dd_dq_dq[1][0][3][2] = -2;
        dd_dq_dq[1][1][0][0] = -2;  dd_dq_dq[1][1][1][1] =  2;  dd_dq_dq[1][1][2][2] = -2;  dd_dq_dq[1][1][3][3] =  2;
        dd_dq_dq[1][2][0][3] =  2;  dd_dq_dq[1][2][1][2] =  2;  dd_dq_dq[1][2][2][1] =  2;  dd_dq_dq[1][2][3][0] =  2;

        dd_dq_dq[2][0][0][2] =  2;  dd_dq_dq[2][0][1][3] =  2;  dd_dq_dq[2][0][2][0] =  2;  dd_dq_dq[2][0][3][1] =  2;
        dd_dq_dq[2][1][0][3] = -2;  dd_dq_dq[2][1][1][2] =  2;  dd_dq_dq[2][1][2][1] =  2;  dd_dq_dq[2][1][3][0] = -2;
        dd_dq_dq[2][2][0][0] = -2;  dd_dq_dq[2][2][1][1] = -2;  dd_dq_dq[2][2][2][2] =  2;  dd_dq_dq[2][2][3][3] =  2;

    }

    /** \brief Transform tangent vectors from quaternion representation to matrix representation
     *
     * This class represents rotations as unit quaternions, and therefore variations of rotations
     * (i.e., tangent vector) are represented in quaternion space, too.  However, some applications
     * require the tangent vectors as matrices. To obtain matrix coordinates we use the
     * chain rule, which means that we have to multiply the given derivative with
     * the derivative of the embedding of the unit quaternion into the space of 3x3 matrices.
     * This second derivative is almost given by the method getFirstDerivativesOfDirectors.
     * However, since the directors of a given unit quaternion are the _columns_ of the
     * corresponding orthogonal matrix, we need to invert the i and j indices
     *
     * As a typical GFE assembler will require this for several tangent vectors at once,
     * the implementation here is a vector one: It allows to treat several tangent vectors
     * together, which have to come as the columns of the `derivative` parameter.
     *
     * So, if I am not mistaken, result[i][j][k] contains \partial R_ij / \partial k
     *
     * \param derivative Tangent vector in quaternion coordinates
     * \returns DR Tangent vector in matrix coordinates
     */
    template <int blocksize>
    Tensor3<T,3,3,blocksize> quaternionTangentToMatrixTangent(const Dune::FieldMatrix<T,4,blocksize>& derivative) const
    {
        Tensor3<T,3,3,blocksize> result = T(0);

        Tensor3<T,3 , 3, 4> dd_dq;
        getFirstDerivativesOfDirectors(dd_dq);

        for (int i=0; i<3; i++)
            for (int j=0; j<3; j++)
                for (int k=0; k<blocksize; k++)
                    for (int l=0; l<4; l++)
                        result[i][j][k] += dd_dq[j][i][l] * derivative[l][k];

        return result;
    }

    /** \brief Compute the derivative of the squared distance function with respect to the second argument
     *
     * The squared distance function is
     * \f[ 4 \arccos^2 (|<p,q>|) \f]
     * Deriving this with respect to the second coordinate gives
     * \f[ 4 (\arccos^2)'(x)|_{x = |<p,q>|} * P_qp \f]
     * The whole thing has to be multiplied by -1 if \f$ <p,q> \f$ is negative
     */
    static EmbeddedTangentVector derivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p,
                                                                      const Rotation<T,3>& q) {
        T sp = p.globalCoordinates() * q.globalCoordinates();

        // Compute the projection of p onto the tangent space of q
        EmbeddedTangentVector result = p;
        result.axpy(-1*(q*p), q);

        // The ternary operator comes from the derivative of the absolute value function
        using std::abs;
        result *= 4 * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp)) * ( (sp<0) ? -1 : 1 );

        return result;
    }

    /** \brief Compute the Hessian of the squared distance function keeping the first argument fixed */
    static Dune::SymmetricMatrix<T,4> secondDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {

        T sp = p.globalCoordinates() * q.globalCoordinates();

        EmbeddedTangentVector pProjected = q.projectOntoTangentSpace(p.globalCoordinates());

        Dune::SymmetricMatrix<T,4> A;
        for (int i=0; i<4; i++)
            for (int j=0; j<=i; j++)
                A(i,j) = pProjected[i]*pProjected[j];

        using std::abs;
        A *= 4*UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp));

        // Compute matrix B (see notes)
        Dune::SymmetricMatrix<T,4> Pq;
        for (int i=0; i<4; i++)
            for (int j=0; j<=i; j++)
                Pq(i,j) = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];

        // Bring it all together
        A.axpy(-4* ((sp<0) ? -1 : 1) * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp))*sp, Pq);

        return A;
    }

    /** \brief Compute the mixed second derivate \partial d^2 / \partial dp dq */
    static Dune::FieldMatrix<T,4,4> secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {

        T sp = p.globalCoordinates() * q.globalCoordinates();

        // Compute vector A (see notes)
        Dune::FieldMatrix<T,1,4> row;
        row[0] = q.projectOntoTangentSpace(p.globalCoordinates());

        EmbeddedTangentVector tmp = p.projectOntoTangentSpace(q.globalCoordinates());
        Dune::FieldMatrix<T,4,1> column;
        for (int i=0; i<4; i++)  // turn row vector into column vector
            column[i] = tmp[i];

        Dune::FieldMatrix<T,4,4> A;
        // A = row * column
        Dune::FMatrixHelp::multMatrix(column,row,A);
        using std::abs;
        A *= 4 * UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp));

        // Compute matrix B (see notes)
        Dune::FieldMatrix<T,4,4> Pp, Pq;
        for (int i=0; i<4; i++)
            for (int j=0; j<4; j++) {
                Pp[i][j] = (i==j) - p.globalCoordinates()[i]*p.globalCoordinates()[j];
                Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
            }

        Dune::FieldMatrix<T,4,4> B;
        Dune::FMatrixHelp::multMatrix(Pp,Pq,B);

        // Bring it all together
        A.axpy(4 * ( (sp<0) ? -1 : 1) * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp)), B);

        return A;
    }

    /** \brief Compute the third derivative \partial d^3 / \partial dq^3

    Unlike the distance itself the squared distance is differentiable at zero
     */
    static Tensor3<T,4,4,4> thirdDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {

        Tensor3<T,4,4,4> result;

        T sp = p.globalCoordinates() * q.globalCoordinates();

        // The projection matrix onto the tangent space at p and q
        Dune::FieldMatrix<T,4,4> Pq;
        for (int i=0; i<4; i++)
            for (int j=0; j<4; j++)
                Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];

        EmbeddedTangentVector pProjected = q.projectOntoTangentSpace(p.globalCoordinates());

        double plusMinus = (sp < 0) ? -1 : 1;

        using std::abs;
        for (int i=0; i<4; i++)
            for (int j=0; j<4; j++)
                for (int k=0; k<4; k++) {

                    result[i][j][k] = plusMinus * UnitVector<T,4>::thirdDerivativeOfArcCosSquared(abs(sp)) * pProjected[i] * pProjected[j] * pProjected[k]
                                    - UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp)) * ((i==j)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[j])*pProjected[k]
                                    - UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp)) * ((i==k)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[k])*pProjected[j]
                                    - UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp)) * pProjected[i] * Pq[j][k] * sp
                                    + plusMinus * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp)) * ((i==j)*q.globalCoordinates()[k] + (i==k)*q.globalCoordinates()[j]) * sp
                                    - plusMinus * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp)) * p.globalCoordinates()[i] * Pq[j][k];
                }

        Pq *= 4;
        result = Pq * result;

        return result;
    }

    /** \brief Compute the mixed third derivative \partial d^3 / \partial da db^2

    Unlike the distance itself the squared distance is differentiable at zero
     */
    static Tensor3<T,4,4,4> thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(const Rotation<T,3>& p, const Rotation<T,3>& q) {

        Tensor3<T,4,4,4> result;

        T sp = p.globalCoordinates() * q.globalCoordinates();

        // The projection matrix onto the tangent space at p and q
        Dune::FieldMatrix<T,4,4> Pp, Pq;
        for (int i=0; i<4; i++)
            for (int j=0; j<4; j++) {
                Pp[i][j] = (i==j) - p.globalCoordinates()[i]*p.globalCoordinates()[j];
                Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
            }

        EmbeddedTangentVector pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
        EmbeddedTangentVector qProjected = p.projectOntoTangentSpace(q.globalCoordinates());

        Tensor3<T,4,4,4> derivativeOfPqOTimesPq;
        for (int i=0; i<4; i++)
            for (int j=0; j<4; j++)
                for (int k=0; k<4; k++) {
                    derivativeOfPqOTimesPq[i][j][k] = 0;
                    for (int l=0; l<4; l++)
                        derivativeOfPqOTimesPq[i][j][k] += Pp[i][l] * (Pq[j][l]*pProjected[k] + pProjected[j]*Pq[k][l]);
                }

        double plusMinus = (sp < 0) ? -1 : 1;

        using std::abs;
        result = plusMinus * UnitVector<T,4>::thirdDerivativeOfArcCosSquared(abs(sp))         * Tensor3<T,4,4,4>::product(qProjected,pProjected,pProjected)
                 + UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp))      * derivativeOfPqOTimesPq
                 - UnitVector<T,4>::secondDerivativeOfArcCosSquared(abs(sp)) * sp * Tensor3<T,4,4,4>::product(qProjected,Pq)
                 - plusMinus * UnitVector<T,4>::derivativeOfArcCosSquared(abs(sp))            * Tensor3<T,4,4,4>::product(qProjected,Pq);

        result *= 4;

        return result;

    }




    /** \brief Interpolate between two rotations */
    static Rotation<T,3> interpolate(const Rotation<T,3>& a, const Rotation<T,3>& b, T omega) {

        // Compute log on T_a SO(3)
        EmbeddedTangentVector v = log(a,b);
        return exp(a, omega*v);
    }

    /** \brief Interpolate between two rotations
        \param omega must be between 0 and 1
    */
    static Quaternion<T> interpolateDerivative(const Rotation<T,3>& a, const Rotation<T,3>& b,
                                               T omega) {
        Quaternion<T> result(0);

        // Compute log on T_a SO(3)
        SkewMatrix<T,3> xi = log(a,b);

        SkewMatrix<T,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<T,4,3> diffExp = Dexp(v);

        diffExp.umv(xi.axial(),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 Derivative of the map from unit quaternions to orthogonal matrices
     */
    static Tensor3<T,3,3,4> derivativeOfQuaternionToMatrix(const Dune::FieldVector<T,4>& p)
    {
      Tensor3<T,3,3,4> result;

      result[0][0] = { 2*p[0], -2*p[1], -2*p[2],  2*p[3]};
      result[0][1] = { 2*p[1],  2*p[0], -2*p[3], -2*p[2]};
      result[0][2] = { 2*p[2],  2*p[3],  2*p[0],  2*p[1]};

      result[1][0] = { 2*p[1],  2*p[0],  2*p[3],  2*p[2]};
      result[1][1] = {-2*p[0],  2*p[1], -2*p[2],  2*p[3]};
      result[1][2] = {-2*p[3],  2*p[2],  2*p[1], -2*p[0]};

      result[2][0] = { 2*p[2], -2*p[3],  2*p[0], -2*p[1]};
      result[2][1] = { 2*p[3],  2*p[2],  2*p[1],  2*p[0]};
      result[2][2] = {-2*p[0], -2*p[1],  2*p[2],  2*p[3]};

      return result;
    }

    /** \brief Set rotation from orthogonal matrix