colvarmodule::quaternion Class Reference

1-dimensional vector of real numbers with four components and a quaternion algebra More...

#include <colvartypes.h>

Public Member Functions
	quaternion (cvm::real x, cvm::real y, cvm::real z)
	Constructor from a 3-d vector.
	quaternion (cvm::real const qv[4])
	Constructor component by component.
	quaternion (cvm::real q0i, cvm::real q1i, cvm::real q2i, cvm::real q3i)
	Constructor component by component.
	quaternion (cvm::vector1d< cvm::real > const &v)
void	set_from_euler_angles (cvm::real phi_in, cvm::real theta_in, cvm::real psi_in)
	quaternion ()
	Default constructor.
void	set (cvm::real value)
	Set all components to a scalar.
void	reset ()
	Set all components to zero (null quaternion)
void	reset_rotation ()
	Set the q0 component to 1 and the others to 0 (quaternion representing no rotation)
std::string	to_simple_string () const
int	from_simple_string (std::string const &s)
cvm::real &	operator[] (int i)
	Access the quaternion as a 4-d array (return a reference)
cvm::real	operator[] (int i) const
	Access the quaternion as a 4-d array (return a value)
cvm::vector1d< cvm::real > const	as_vector () const
cvm::real	norm2 () const
	Square norm of the quaternion.
cvm::real	norm () const
	Norm of the quaternion.
cvm::quaternion	conjugate () const
	Return the conjugate quaternion.
void	operator*= (cvm::real a)
void	operator/= (cvm::real a)
void	set_positive ()
void	operator+= (cvm::quaternion const &h)
void	operator-= (cvm::quaternion const &h)
cvm::rvector	get_vector () const
	Return the vector component.
cvm::rvector	rotate (cvm::rvector const &v) const
	Rotate v through this quaternion (put it in the rotated reference frame)
cvm::quaternion	rotate (cvm::quaternion const &Q2) const
	Rotate Q2 through this quaternion (put it in the rotated reference frame)
cvm::rmatrix	rotation_matrix () const
	Return the 3x3 matrix associated to this quaternion.
cvm::quaternion	position_derivative_inner (cvm::rvector const &pos, cvm::rvector const &vec) const
	Multiply the given vector by the derivative of the given (rotated) position with respect to the quaternion. More...
cvm::real	cosine (cvm::quaternion const &q) const
	Return the cosine between the orientation frame associated to this quaternion and another.
cvm::real	dist2 (cvm::quaternion const &Q2) const
	Square distance from another quaternion on the 4-dimensional unit sphere: returns the square of the angle along the shorter of the two geodesics.
cvm::quaternion	dist2_grad (cvm::quaternion const &Q2) const
void	match (cvm::quaternion &Q2) const
	Choose the closest between Q2 and -Q2 and save it back. Not required for dist2() and dist2_grad()
cvm::real	inner (cvm::quaternion const &Q2) const
	Inner product (as a 4-d vector) with Q2; requires match() if the largest overlap is looked for.

Static Public Member Functions
static size_t	output_width (size_t real_width)
	Tell the number of characters required to print a quaternion, given that of a real number.

Public Attributes
cvm::real	q0
cvm::real	q1
cvm::real	q2
cvm::real	q3

Friends
std::ostream &	operator<< (std::ostream &os, cvm::quaternion const &q)
	Formatted output operator.
std::istream &	operator>> (std::istream &is, cvm::quaternion &q)
	Formatted input operator.
cvm::quaternion	operator+ (cvm::quaternion const &h, cvm::quaternion const &q)
cvm::quaternion	operator- (cvm::quaternion const &h, cvm::quaternion const &q)
cvm::quaternion	operator* (cvm::quaternion const &h, cvm::quaternion const &q)
	Provides the quaternion product. NOTE: for the inner product use: h.inner (q);
cvm::quaternion	operator* (cvm::real c, cvm::quaternion const &q)
cvm::quaternion	operator* (cvm::quaternion const &q, cvm::real c)
cvm::quaternion	operator/ (cvm::quaternion const &q, cvm::real c)

Detailed Description

1-dimensional vector of real numbers with four components and a quaternion algebra

Member Function Documentation

dist2_grad()

cvm::quaternion colvarmodule::quaternion::dist2_grad ( cvm::quaternion const &  Q2 ) const

inline

Gradient of the square distance: returns a 4-vector equivalent to that provided by slerp

position_derivative_inner()

cvm::quaternion colvarmodule::quaternion::position_derivative_inner ( cvm::rvector const &  pos, cvm::rvector const &  vec  ) const

inline

Multiply the given vector by the derivative of the given (rotated) position with respect to the quaternion.

Parameters

pos	The position \(\mathbf{x}\).
vec	The vector \(\mathbf{v}\).

Returns

A quaternion (see the detailed documentation below).

This function is mainly used for projecting the gradients or forces on the rotated atoms to the forces on quaternion. Assume this rotation can be represented as \(R(\mathbf{q})\), where \(\mathbf{q} := (q_0, q_1, q_2, q_3)\) is the current quaternion, the function returns the following new quaternion:

\[ \left(\mathbf{v}^\mathrm{T}\frac{\partial R(\mathbf{q})}{\partial q_0}\mathbf{x}, \mathbf{v}^\mathrm{T}\frac{\partial R(\mathbf{q})}{\partial q_1}\mathbf{x}, \mathbf{v}^\mathrm{T}\frac{\partial R(\mathbf{q})}{\partial q_2}\mathbf{x}, \mathbf{v}^\mathrm{T}\frac{\partial R(\mathbf{q})}{\partial q_3}\mathbf{x}\right) \]

where \(\mathbf{v}\) is usually the gradient of \(\xi\) with respect to the rotated frame \(\tilde{\mathbf{X}}\), \(\partial \xi / \partial \tilde{\mathbf{X}}\), or the force acting on it ( \(\mathbf{F}_{\tilde{\mathbf{X}}}\)). By using the following loop in pseudo C++ code, either \(\partial \xi / \partial \tilde{\mathbf{X}}\) or \(\mathbf{F}_{\tilde{\mathbf{X}}}\), can be projected to \(\partial \xi / \partial \mathbf{q}\) or \(\mathbf{F}_q\) into sum_dxdq:

cvm::real sum_dxdq[4] = {0, 0, 0, 0};
for (size_t i = 0; i < main_group_size(); ++i) {
  const cvm::rvector v = grad_or_force_on_rotated_main_group(i);
  const cvm::rvector x = unrotated_main_group_positions(i);
  cvm::quaternion const dxdq = position_derivative_inner(x, v);
  sum_dxdq[0] += dxdq[0];
  sum_dxdq[1] += dxdq[1];
  sum_dxdq[2] += dxdq[2];
  sum_dxdq[3] += dxdq[3];
}

set_from_euler_angles()

void colvarmodule::quaternion::set_from_euler_angles ( cvm::real  phi_in, cvm::real  theta_in, cvm::real  psi_in  )

inline

"Constructor" after Euler angles (in radians)

http://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles

---

The documentation for this class was generated from the following files:

• colvartypes.h
• colvartypes.cpp