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HSO3.hpp
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/********************************************************************************************/
/* */
/* HSO3.hpp header file */
/* */
/* This file is not currently part of the Boost library. It is simply an example of the use */
/* quaternions can be put to. Hopefully it will be useful too. */
/* */
/* This file provides tools to convert between quaternions and R^3 rotation matrices. */
/* */
/********************************************************************************************/
// (C) Copyright Hubert Holin 2001.
// Distributed under the Boost Software License, Version 1.0. (See
// accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
#ifndef TEST_HSO3_HPP
#define TEST_HSO3_HPP
#include <algorithm>
#if defined(__GNUC__) && (__GNUC__ < 3)
#include <boost/limits.hpp>
#else
#include <limits>
#endif
#include <stdexcept>
#include <string>
#include <boost/math/quaternion.hpp>
#if defined(__GNUC__) && (__GNUC__ < 3)
// gcc 2.x ignores function scope using declarations, put them here instead:
using namespace ::std;
using namespace ::boost::math;
#endif
template<typename TYPE_FLOAT>
struct R3_matrix
{
TYPE_FLOAT a11, a12, a13;
TYPE_FLOAT a21, a22, a23;
TYPE_FLOAT a31, a32, a33;
};
// Note: the input quaternion need not be of norm 1 for the following function
template<typename TYPE_FLOAT>
R3_matrix<TYPE_FLOAT> quaternion_to_R3_rotation(::boost::math::quaternion<TYPE_FLOAT> const & q)
{
using ::std::numeric_limits;
TYPE_FLOAT a = q.R_component_1();
TYPE_FLOAT b = q.R_component_2();
TYPE_FLOAT c = q.R_component_3();
TYPE_FLOAT d = q.R_component_4();
TYPE_FLOAT aa = a*a;
TYPE_FLOAT ab = a*b;
TYPE_FLOAT ac = a*c;
TYPE_FLOAT ad = a*d;
TYPE_FLOAT bb = b*b;
TYPE_FLOAT bc = b*c;
TYPE_FLOAT bd = b*d;
TYPE_FLOAT cc = c*c;
TYPE_FLOAT cd = c*d;
TYPE_FLOAT dd = d*d;
TYPE_FLOAT norme_carre = aa+bb+cc+dd;
if (norme_carre <= numeric_limits<TYPE_FLOAT>::epsilon())
{
::std::string error_reporting("Argument to quaternion_to_R3_rotation is too small!");
::std::underflow_error bad_argument(error_reporting);
throw(bad_argument);
}
R3_matrix<TYPE_FLOAT> out_matrix;
out_matrix.a11 = (aa+bb-cc-dd)/norme_carre;
out_matrix.a12 = 2*(-ad+bc)/norme_carre;
out_matrix.a13 = 2*(ac+bd)/norme_carre;
out_matrix.a21 = 2*(ad+bc)/norme_carre;
out_matrix.a22 = (aa-bb+cc-dd)/norme_carre;
out_matrix.a23 = 2*(-ab+cd)/norme_carre;
out_matrix.a31 = 2*(-ac+bd)/norme_carre;
out_matrix.a32 = 2*(ab+cd)/norme_carre;
out_matrix.a33 = (aa-bb-cc+dd)/norme_carre;
return(out_matrix);
}
template<typename TYPE_FLOAT>
void find_invariant_vector( R3_matrix<TYPE_FLOAT> const & rot,
TYPE_FLOAT & x,
TYPE_FLOAT & y,
TYPE_FLOAT & z)
{
using ::std::sqrt;
using ::std::numeric_limits;
TYPE_FLOAT b11 = rot.a11 - static_cast<TYPE_FLOAT>(1);
TYPE_FLOAT b12 = rot.a12;
TYPE_FLOAT b13 = rot.a13;
TYPE_FLOAT b21 = rot.a21;
TYPE_FLOAT b22 = rot.a22 - static_cast<TYPE_FLOAT>(1);
TYPE_FLOAT b23 = rot.a23;
TYPE_FLOAT b31 = rot.a31;
TYPE_FLOAT b32 = rot.a32;
TYPE_FLOAT b33 = rot.a33 - static_cast<TYPE_FLOAT>(1);
TYPE_FLOAT minors[9] =
{
b11*b22-b12*b21,
b11*b23-b13*b21,
b12*b23-b13*b22,
b11*b32-b12*b31,
b11*b33-b13*b31,
b12*b33-b13*b32,
b21*b32-b22*b31,
b21*b33-b23*b31,
b22*b33-b23*b32
};
TYPE_FLOAT * where = ::std::max_element(minors, minors+9);
TYPE_FLOAT det = *where;
if (det <= numeric_limits<TYPE_FLOAT>::epsilon())
{
::std::string error_reporting("Underflow error in find_invariant_vector!");
::std::underflow_error processing_error(error_reporting);
throw(processing_error);
}
switch (where-minors)
{
case 0:
z = static_cast<TYPE_FLOAT>(1);
x = (-b13*b22+b12*b23)/det;
y = (-b11*b23+b13*b21)/det;
break;
case 1:
y = static_cast<TYPE_FLOAT>(1);
x = (-b12*b23+b13*b22)/det;
z = (-b11*b22+b12*b21)/det;
break;
case 2:
x = static_cast<TYPE_FLOAT>(1);
y = (-b11*b23+b13*b21)/det;
z = (-b12*b21+b11*b22)/det;
break;
case 3:
z = static_cast<TYPE_FLOAT>(1);
x = (-b13*b32+b12*b33)/det;
y = (-b11*b33+b13*b31)/det;
break;
case 4:
y = static_cast<TYPE_FLOAT>(1);
x = (-b12*b33+b13*b32)/det;
z = (-b11*b32+b12*b31)/det;
break;
case 5:
x = static_cast<TYPE_FLOAT>(1);
y = (-b11*b33+b13*b31)/det;
z = (-b12*b31+b11*b32)/det;
break;
case 6:
z = static_cast<TYPE_FLOAT>(1);
x = (-b23*b32+b22*b33)/det;
y = (-b21*b33+b23*b31)/det;
break;
case 7:
y = static_cast<TYPE_FLOAT>(1);
x = (-b22*b33+b23*b32)/det;
z = (-b21*b32+b22*b31)/det;
break;
case 8:
x = static_cast<TYPE_FLOAT>(1);
y = (-b21*b33+b23*b31)/det;
z = (-b22*b31+b21*b32)/det;
break;
default:
::std::string error_reporting("Impossible condition in find_invariant_vector");
::std::logic_error processing_error(error_reporting);
throw(processing_error);
break;
}
TYPE_FLOAT vecnorm = sqrt(x*x+y*y+z*z);
if (vecnorm <= numeric_limits<TYPE_FLOAT>::epsilon())
{
::std::string error_reporting("Overflow error in find_invariant_vector!");
::std::overflow_error processing_error(error_reporting);
throw(processing_error);
}
x /= vecnorm;
y /= vecnorm;
z /= vecnorm;
}
template<typename TYPE_FLOAT>
void find_orthogonal_vector( TYPE_FLOAT x,
TYPE_FLOAT y,
TYPE_FLOAT z,
TYPE_FLOAT & u,
TYPE_FLOAT & v,
TYPE_FLOAT & w)
{
using ::std::abs;
using ::std::sqrt;
using ::std::numeric_limits;
TYPE_FLOAT vecnormsqr = x*x+y*y+z*z;
if (vecnormsqr <= numeric_limits<TYPE_FLOAT>::epsilon())
{
::std::string error_reporting("Underflow error in find_orthogonal_vector!");
::std::underflow_error processing_error(error_reporting);
throw(processing_error);
}
TYPE_FLOAT lambda;
TYPE_FLOAT components[3] =
{
abs(x),
abs(y),
abs(z)
};
TYPE_FLOAT * where = ::std::min_element(components, components+3);
switch (where-components)
{
case 0:
if (*where <= numeric_limits<TYPE_FLOAT>::epsilon())
{
v =
w = static_cast<TYPE_FLOAT>(0);
u = static_cast<TYPE_FLOAT>(1);
}
else
{
lambda = -x/vecnormsqr;
u = static_cast<TYPE_FLOAT>(1) + lambda*x;
v = lambda*y;
w = lambda*z;
}
break;
case 1:
if (*where <= numeric_limits<TYPE_FLOAT>::epsilon())
{
u =
w = static_cast<TYPE_FLOAT>(0);
v = static_cast<TYPE_FLOAT>(1);
}
else
{
lambda = -y/vecnormsqr;
u = lambda*x;
v = static_cast<TYPE_FLOAT>(1) + lambda*y;
w = lambda*z;
}
break;
case 2:
if (*where <= numeric_limits<TYPE_FLOAT>::epsilon())
{
u =
v = static_cast<TYPE_FLOAT>(0);
w = static_cast<TYPE_FLOAT>(1);
}
else
{
lambda = -z/vecnormsqr;
u = lambda*x;
v = lambda*y;
w = static_cast<TYPE_FLOAT>(1) + lambda*z;
}
break;
default:
::std::string error_reporting("Impossible condition in find_invariant_vector");
::std::logic_error processing_error(error_reporting);
throw(processing_error);
break;
}
TYPE_FLOAT vecnorm = sqrt(u*u+v*v+w*w);
if (vecnorm <= numeric_limits<TYPE_FLOAT>::epsilon())
{
::std::string error_reporting("Underflow error in find_orthogonal_vector!");
::std::underflow_error processing_error(error_reporting);
throw(processing_error);
}
u /= vecnorm;
v /= vecnorm;
w /= vecnorm;
}
// Note: we want [[v, v, w], [r, s, t], [x, y, z]] to be a direct orthogonal basis
// of R^3. It might not be orthonormal, however, and we do not check if the
// two input vectors are colinear or not.
template<typename TYPE_FLOAT>
void find_vector_for_BOD(TYPE_FLOAT x,
TYPE_FLOAT y,
TYPE_FLOAT z,
TYPE_FLOAT u,
TYPE_FLOAT v,
TYPE_FLOAT w,
TYPE_FLOAT & r,
TYPE_FLOAT & s,
TYPE_FLOAT & t)
{
r = +y*w-z*v;
s = -x*w+z*u;
t = +x*v-y*u;
}
template<typename TYPE_FLOAT>
inline bool is_R3_rotation_matrix(R3_matrix<TYPE_FLOAT> const & mat)
{
using ::std::abs;
using ::std::numeric_limits;
return (
!(
(abs(mat.a11*mat.a11+mat.a21*mat.a21+mat.a31*mat.a31 - static_cast<TYPE_FLOAT>(1)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
(abs(mat.a11*mat.a12+mat.a21*mat.a22+mat.a31*mat.a32 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
(abs(mat.a11*mat.a13+mat.a21*mat.a23+mat.a31*mat.a33 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
//(abs(mat.a11*mat.a12+mat.a21*mat.a22+mat.a31*mat.a32 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
(abs(mat.a12*mat.a12+mat.a22*mat.a22+mat.a32*mat.a32 - static_cast<TYPE_FLOAT>(1)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
(abs(mat.a12*mat.a13+mat.a22*mat.a23+mat.a32*mat.a33 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
//(abs(mat.a11*mat.a13+mat.a21*mat.a23+mat.a31*mat.a33 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
//(abs(mat.a12*mat.a13+mat.a22*mat.a23+mat.a32*mat.a33 - static_cast<TYPE_FLOAT>(0)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())||
(abs(mat.a13*mat.a13+mat.a23*mat.a23+mat.a33*mat.a33 - static_cast<TYPE_FLOAT>(1)) > static_cast<TYPE_FLOAT>(10)*numeric_limits<TYPE_FLOAT>::epsilon())
)
);
}
template<typename TYPE_FLOAT>
::boost::math::quaternion<TYPE_FLOAT> R3_rotation_to_quaternion( R3_matrix<TYPE_FLOAT> const & rot,
::boost::math::quaternion<TYPE_FLOAT> const * hint = 0)
{
using ::boost::math::abs;
using ::std::abs;
using ::std::sqrt;
using ::std::numeric_limits;
if (!is_R3_rotation_matrix(rot))
{
::std::string error_reporting("Argument to R3_rotation_to_quaternion is not an R^3 rotation matrix!");
::std::range_error bad_argument(error_reporting);
throw(bad_argument);
}
::boost::math::quaternion<TYPE_FLOAT> q;
if (
(abs(rot.a11 - static_cast<TYPE_FLOAT>(1)) <= numeric_limits<TYPE_FLOAT>::epsilon())&&
(abs(rot.a22 - static_cast<TYPE_FLOAT>(1)) <= numeric_limits<TYPE_FLOAT>::epsilon())&&
(abs(rot.a33 - static_cast<TYPE_FLOAT>(1)) <= numeric_limits<TYPE_FLOAT>::epsilon())
)
{
q = ::boost::math::quaternion<TYPE_FLOAT>(1);
}
else
{
TYPE_FLOAT cos_theta = (rot.a11+rot.a22+rot.a33-static_cast<TYPE_FLOAT>(1))/static_cast<TYPE_FLOAT>(2);
TYPE_FLOAT stuff = (cos_theta+static_cast<TYPE_FLOAT>(1))/static_cast<TYPE_FLOAT>(2);
TYPE_FLOAT cos_theta_sur_2 = sqrt(stuff);
TYPE_FLOAT sin_theta_sur_2 = sqrt(1-stuff);
TYPE_FLOAT x;
TYPE_FLOAT y;
TYPE_FLOAT z;
find_invariant_vector(rot, x, y, z);
TYPE_FLOAT u;
TYPE_FLOAT v;
TYPE_FLOAT w;
find_orthogonal_vector(x, y, z, u, v, w);
TYPE_FLOAT r;
TYPE_FLOAT s;
TYPE_FLOAT t;
find_vector_for_BOD(x, y, z, u, v, w, r, s, t);
TYPE_FLOAT ru = rot.a11*u+rot.a12*v+rot.a13*w;
TYPE_FLOAT rv = rot.a21*u+rot.a22*v+rot.a23*w;
TYPE_FLOAT rw = rot.a31*u+rot.a32*v+rot.a33*w;
TYPE_FLOAT angle_sign_determinator = r*ru+s*rv+t*rw;
if (angle_sign_determinator > +numeric_limits<TYPE_FLOAT>::epsilon())
{
q = ::boost::math::quaternion<TYPE_FLOAT>(cos_theta_sur_2, +x*sin_theta_sur_2, +y*sin_theta_sur_2, +z*sin_theta_sur_2);
}
else if (angle_sign_determinator < -numeric_limits<TYPE_FLOAT>::epsilon())
{
q = ::boost::math::quaternion<TYPE_FLOAT>(cos_theta_sur_2, -x*sin_theta_sur_2, -y*sin_theta_sur_2, -z*sin_theta_sur_2);
}
else
{
TYPE_FLOAT desambiguator = u*ru+v*rv+w*rw;
if (desambiguator >= static_cast<TYPE_FLOAT>(1))
{
q = ::boost::math::quaternion<TYPE_FLOAT>(0, +x, +y, +z);
}
else
{
q = ::boost::math::quaternion<TYPE_FLOAT>(0, -x, -y, -z);
}
}
}
if ((hint != 0) && (abs(*hint+q) < abs(*hint-q)))
{
return(-q);
}
return(q);
}
#endif /* TEST_HSO3_HPP */