dual_quaternion.h

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#ifndef DUAL_QUATERNION_HPP
#define DUAL_QUATERNION_HPP

#include "math3d.h"

using math3d::point3d;
using math3d::matrix;
using math3d::quaternion;


template<typename T> inline int sign(T v)
{
  return (v < 0) ? -1 : 1;
}

void set_quaternion_matrix(matrix<double>&M, const quaternion<double>& q, int i = 0, int j = 0, double w = 1.0)
{
  //{{a, -b, -c, -d}, {b, a, -d, c}, {c, d, a, -b}, {d, -c, b, a}}
  M(i, j)  = w * q.w;
  M(i, j + 1)  = -w * q.i;
  M(i, j + 2)  = -w * q.j;
  M(i, j + 3)  = -w * q.k;
  M(i + 1, j) = w * q.i;
  M(i + 1, j + 1) = w * q.w;
  M(i + 1, j + 2) = -w * q.k;
  M(i + 1, j + 3) = w * q.j;
  M(i + 2, j) = w * q.j;
  M(i + 2, j + 1) = w * q.k;
  M(i + 2, j + 2) = w * q.w;
  M(i + 2, j + 3) = -w * q.i;
  M(i + 3, j) = w * q.k;
  M(i + 3, j + 1) = -w * q.j;
  M(i + 3, j + 2) = w * q.i;
  M(i + 3, j + 3) = w * q.w;
}

struct dual_quaternion
{
  quaternion<double> R, tR_2;

  dual_quaternion(double v = 1.0) : R(v), tR_2(0) {}

  static constexpr double dq_epsilon = 1e-8;

  static dual_quaternion rigid_transformation(const quaternion<double>& r, const point3d& t)
  {
    dual_quaternion result;
    result.R = r;
    result.tR_2 = (quaternion<double>::convert(t) * r) *= 0.5;
    return result;
  }
  static dual_quaternion convert(const double* p)
  {
    dual_quaternion result;
    result.R = quaternion<double>::convert(p);
    result.tR_2 = quaternion<double>::convert(p + 4);
    return result;
  }

  dual_quaternion& normalize()
  {
    double n = norm(R) * sign(R.w);
    R *= 1.0 / n;
    tR_2 *= 1.0 / n;
    double d = dot(R, tR_2);
    //tR_2 += (-d)*R;
    quaternion<double> r2 = R;
    r2 *= -d;
    tR_2 += r2;
    return *this;
  }

  point3d get_translation()
  {
    quaternion<double> t = tR_2 * ~R;
    point3d result;
    result.x = 2 * t.i;
    result.y = 2 * t.j;
    result.z = 2 * t.k;
    return result;
  }

  void to_vector(double* p)
  {
    R.to_vector(p);
    tR_2.to_vector(p + 4);
  }

  dual_quaternion& operator += (const dual_quaternion& a)
  {
    R += a.R;
    tR_2 += a.tR_2;
    return *this;
  }

  dual_quaternion& operator *= (double a)
  {
    R *= a;
    tR_2 *= a;
    return *this;
  }

  dual_quaternion& log()  //computes log map tangent at identity
  {
    //assumes qual_quaternion is unitary
    const double h0 = std::acos(R.w);
    if (h0 * h0 < dq_epsilon) //small angle approximation: sin(h0)=h0, cos(h0)=1
    {
      R.w = 0.0;
      R *= 0.5;
      tR_2.w = 0.0;
      tR_2 *= 0.5;
    }
    else
    {
      R.w = 0.0;
      const double ish0 = 1.0 / norm(R);
      //R *= ish0;
      math3d::normalize(R); //R=s0
      const double he = -tR_2.w * ish0;
      tR_2.w = 0.0;

      quaternion<double> Rp(R);
      Rp *= -dot(R, tR_2) / dot(R, R);
      tR_2 += Rp;
      tR_2 *= ish0; //tR_2=se

      tR_2 *= h0;
      Rp = R;
      Rp *= he;
      tR_2 += Rp;
      tR_2 *= 0.5;
      R *= h0 * 0.5;
    }

    return *this;
  }

  dual_quaternion& exp()  //computes exp map tangent at identity
  {
    //assumes qual_quaternion is on tangent space
    const double h0 = 2.0 * norm(R);

    if (h0 * h0 < dq_epsilon) //small angle approximation: sin(h0)=h0, cos(h0)=1
    {
      R *= 2.0;
      R.w = 1.0;
      tR_2 *= 2.0;
      //normalize();
    }
    else
    {
      const double he = 4.0 * math3d::dot(tR_2, R) / h0;
      const double sh0 = sin(h0), ch0 = cos(h0);
      quaternion<double> Rp(R);
      Rp *= -dot(R, tR_2) / dot(R, R);
      tR_2 += Rp;
      tR_2 *= 2.0 / h0; //tR_2=se


      tR_2 *= sh0;
      Rp = R;
      Rp *= he * ch0 * 2.0 / h0;
      tR_2 += Rp;
      tR_2.w = -he * sh0;

      R *= sh0 * 2.0 / h0;
      R.w = ch0;
    }
    normalize();
    return *this;
  }
};


dual_quaternion operator * (const dual_quaternion&a, const dual_quaternion& b)
{
  dual_quaternion result;
  result.R = a.R * b.R;
  result.tR_2 = a.R * b.tR_2 + a.tR_2 * b.R;
  return result;
}

dual_quaternion operator ~(const dual_quaternion& a)
{
  dual_quaternion result;
  result.R = ~a.R;
  result.tR_2 = ((~a.tR_2) *= -1);
  return result;
}

dual_quaternion operator !(const dual_quaternion& a)
{
  dual_quaternion result;
  result.R = ~a.R;
  result.tR_2 = ~a.tR_2;
  return result;
}

double dot(const dual_quaternion& a, const dual_quaternion& b)
{
  return dot(a.R, b.R) + dot(a.tR_2, b.tR_2);
}

void set_dual_quaternion_matrix(matrix<double>& M, const dual_quaternion& dq, int i = 0, int j = 0, double w = 1.0)
{
  set_quaternion_matrix(M, dq.R, i, j, w);
  M(i, j + 4) = M(i, j + 5) = M(i, j + 6) = M(i, j + 7) = 0;
  M(i + 1, j + 4) = M(i + 1, j + 5) = M(i + 1, j + 6) = M(i + 1, j + 7) = 0;
  M(i + 2, j + 4) = M(i + 2, j + 5) = M(i + 2, j + 6) = M(i + 2, j + 7) = 0;
  M(i + 3, j + 4) = M(i + 3, j + 5) = M(i + 3, j + 6) = M(i + 3, j + 7) = 0;
  set_quaternion_matrix(M, dq.tR_2, i + 4, j, w);
  set_quaternion_matrix(M, dq.R, i + 4, j + 4, w);
}

dual_quaternion log(dual_quaternion a)
{
  return a.log();
}
dual_quaternion exp(dual_quaternion a)
{
  return a.exp();
}


std::ostream& operator << (std::ostream& out, const dual_quaternion& dq)
{
  return out << "( " << dq.R.w << ", " << dq.R.i << ", " << dq.R.j << ", " << dq.R.k << ",  "
         << dq.tR_2.w << ", " << dq.tR_2.i << ", " << dq.tR_2.j << ", " << dq.tR_2.k << " )";
}

#endif