cv::DualQuat< _Tp > Class Template Reference

#include <opencv2/core/dualquaternion.hpp>

Collaboration diagram for cv::DualQuat< _Tp >:

Public Member Functions
	DualQuat ()
	DualQuat (const _Tp w, const _Tp x, const _Tp y, const _Tp z, const _Tp w_, const _Tp x_, const _Tp y_, const _Tp z_)
	create from eight same type numbers.
	DualQuat (const Vec< _Tp, 8 > &q)
	create from a double or float vector.
DualQuat< _Tp >	conjugate () const
	return the conjugate of a dual quaternion.
_Tp	dot (DualQuat< _Tp > p) const
	return the dot product of two dual quaternion.
DualQuat< _Tp >	exp () const
	return the value of exponential function value
Quat< _Tp >	getDualPart () const
	return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat().
Quat< _Tp >	getRealPart () const
	return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat().
Quat< _Tp >	getRotation (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the rotation in quaternion form.
Vec< _Tp, 3 >	getTranslation (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the translation vector. The rotation \(r\) in this dual quaternion \(\sigma\) is applied before translation \(t\). The dual quaternion \(\sigma\) is defined as
DualQuat< _Tp >	inv (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is
DualQuat< _Tp >	log (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of logarithm function value
DualQuat< _Tp >	norm () const
	return the norm \(||\sigma||\) of dual quaternion \(\sigma = p + \epsilon q\).
DualQuat< _Tp >	normalize () const
	return a normalized dual quaternion. A dual quaternion can be expressed as
DualQuat< _Tp >	operator* (const DualQuat< _Tp > &) const
	Multiplication operator of two dual quaternions q and p. Multiplies values on either side of the operator.
DualQuat< _Tp >	operator*= (const _Tp s)
	Multiplication assignment operator of a quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.
DualQuat< _Tp > &	operator*= (const DualQuat< _Tp > &)
	Multiplication assignment operator of two quaternions. It multiplies right operand with the left operand and assign the result to left operand.
DualQuat< _Tp >	operator+ (const DualQuat< _Tp > &) const
	Addition operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(q_i\).
DualQuat< _Tp > &	operator+= (const DualQuat< _Tp > &)
	Addition assignment operator of two dual quaternions p and q. It adds right operand to the left operand and assign the result to left operand.
DualQuat< _Tp >	operator- () const
	Return opposite dual quaternion \(-p\) which satisfies \(p + (-p) = 0.\).
DualQuat< _Tp >	operator- (const DualQuat< _Tp > &) const
	Subtraction operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(-q_i\).
DualQuat< _Tp > &	operator-= (const DualQuat< _Tp > &)
	Subtraction assignment operator of two dual quaternions p and q. It subtracts right operand from the left operand and assign the result to left operand.
DualQuat< _Tp >	operator/ (const _Tp s) const
	Division operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.
DualQuat< _Tp >	operator/ (const DualQuat< _Tp > &) const
	Division operator of two dual quaternions p and q. Divides left hand operand by right hand operand.
Quat< _Tp > &	operator/= (const _Tp s)
	Division assignment operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.
DualQuat< _Tp > &	operator/= (const DualQuat< _Tp > &)
	Division assignment operator of two dual quaternions p and q; It divides left operand with the right operand and assign the result to left operand.
bool	operator== (const DualQuat< _Tp > &) const
	return true if two dual quaternions p and q are nearly equal, i.e. when the absolute value of each \(p_i\) and \(q_i\) is less than CV_DUAL_QUAT_EPS.
DualQuat< _Tp >	power (const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:
DualQuat< _Tp >	power (const DualQuat< _Tp > &q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:
Affine3< _Tp >	toAffine3 (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	Transform this dual quaternion to a instance of Affine3.
Matx< _Tp, 4, 4 >	toMat (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	Transform this dual quaternion to a affine transformation matrix the form of matrix, see createFromMat().
Vec< _Tp, 8 >	toVec () const
	Transform this dual quaternion to a vector.

Static Public Member Functions
static DualQuat< _Tp >	createFromAffine3 (const Affine3< _Tp > &R)
	create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat()
static DualQuat< _Tp >	createFromAngleAxisTrans (const _Tp angle, const Vec< _Tp, 3 > &axis, const Vec< _Tp, 3 > &translation)
	create a dual quaternion from a rotation angle \(\theta\), a rotation axis \(\boldsymbol{u}\) and a translation \(\boldsymbol{t}\). It generates a dual quaternion \(\sigma\) in the form of
static DualQuat< _Tp >	createFromMat (InputArray _R)
	Transform this dual quaternion to an affine transformation matrix \(M\). Dual quaternion consists of a rotation \(r=[a,b,c,d]\) and a translation \(t=[\Delta x,\Delta y,\Delta z]\). The affine transformation matrix \(M\) has the form.
static DualQuat< _Tp >	createFromPitch (const _Tp angle, const _Tp d, const Vec< _Tp, 3 > &axis, const Vec< _Tp, 3 > &moment)
	A dual quaternion is a vector in form of.
static DualQuat< _Tp >	createFromQuat (const Quat< _Tp > &realPart, const Quat< _Tp > &dualPart)
	create Dual Quaternion from two same type quaternions p and q. A Dual Quaternion \(\sigma\) has the form:
static DualQuat< _Tp >	dqblend (const DualQuat< _Tp > &q1, const DualQuat< _Tp > &q2, const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
	The method of Dual Quaternion linear Blending(DQB) is to compute a transformation between dual quaternion \(q_1\) and \(q_2\) and can be defined as:
template<int cn>
static DualQuat< _Tp >	gdqblend (const Vec< DualQuat< _Tp >, cn > &dualquat, InputArray weights, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
	The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.
static DualQuat< _Tp >	gdqblend (InputArray dualquat, InputArray weights, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
	The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.
static DualQuat< _Tp >	sclerp (const DualQuat< _Tp > &q1, const DualQuat< _Tp > &q2, const _Tp t, bool directChange=true, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
	The screw linear interpolation(ScLERP) is an extension of spherical linear interpolation of dual quaternion. If \(\sigma_1\) and \(\sigma_2\) are two dual quaternions representing the initial and final pose. The interpolation of ScLERP function can be defined as:

Public Attributes
_Tp	w
_Tp	w_
_Tp	x
_Tp	x_
_Tp	y
_Tp	y_
_Tp	z
_Tp	z_

Static Public Attributes
static constexpr _Tp	CV_DUAL_QUAT_EPS = (_Tp)1.e-6

Friends
template<typename T >
DualQuat< T >	conjugate (const DualQuat< T > &dq)
	return the conjugate of a dual quaternion.
template<typename T >
DualQuat< T >	cv::operator* (const DualQuat< T > &, const T s)
	Multiplication operator of a dual quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.
template<typename T >
DualQuat< T >	cv::operator* (const T s, const DualQuat< T > &)
	Multiplication operator of a scalar and a dual quaternions. It multiplies right operand with the left operand and assign the result to left operand.
template<typename T >
DualQuat< T >	cv::operator+ (const DualQuat< T > &, const T s)
	Addition operator of a dual quaternions and a scalar. Adds right hand operand from left hand operand.
template<typename T >
DualQuat< T >	cv::operator+ (const T s, const DualQuat< T > &)
	Addition operator of a scalar and a dual quaternions. Adds right hand operand from left hand operand.
template<typename T >
DualQuat< T >	cv::operator- (const DualQuat< T > &, const T s)
	Subtraction operator of a dual quaternion and a scalar. Subtracts right hand operand from left hand operand.
template<typename T >
DualQuat< T >	cv::operator- (const T s, const DualQuat< T > &)
	Subtraction operator of a scalar and a dual quaternions. Subtracts right hand operand from left hand operand.
template<typename S >
std::ostream &	cv::operator<< (std::ostream &, const DualQuat< S > &)
template<typename T >
DualQuat< T >	exp (const DualQuat< T > &dq)
	return the value of exponential function value
template<typename T >
DualQuat< T >	inv (const DualQuat< T > &dq, QuatAssumeType assumeUnit)
	if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is
template<typename T >
DualQuat< T >	log (const DualQuat< T > &dq, QuatAssumeType assumeUnit)
	return the value of logarithm function value
template<typename T >
DualQuat< T >	power (const DualQuat< T > &dq, const T t, QuatAssumeType assumeUnit)
	return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:
template<typename T >
DualQuat< T >	power (const DualQuat< T > &p, const DualQuat< T > &q, QuatAssumeType assumeUnit)
	return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:

Detailed Description

template<typename _Tp>
class cv::DualQuat< _Tp >

Dual quaternions were introduced to describe rotation together with translation while ordinary quaternions can only describe rotation. It can be used for shortest path pose interpolation, local pose optimization or volumetric deformation. More details can be found

• https://en.wikipedia.org/wiki/Dual_quaternion
• "A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies", Ben Kenwright, 2012
• "Dual Quaternions", Yan-Bin Jia, 2013
• "Geometric Skinning with Approximate Dual Quaternion Blending", Kavan, 2008
• http://rodolphe-vaillant.fr/?e=29

A unit dual quaternion can be classically represented as:

\[ \begin{equation} \begin{split} \sigma &= \left(r+\frac{\epsilon}{2}tr\right)\\ &= [w, x, y, z, w\_, x\_, y\_, z\_] \end{split} \end{equation} \]

where \(r, t\) represents the rotation (ordinary unit quaternion) and translation (pure ordinary quaternion) respectively.

A general dual quaternions which consist of two quaternions is usually represented in form of:

\[ \sigma = p + \epsilon q \]

where the introduced dual unit \(\epsilon\) satisfies \(\epsilon^2 = \epsilon^3 =...=0\), and \(p, q\) are quaternions.

Alternatively, dual quaternions can also be interpreted as four components which are all dual numbers:

\[ \sigma = \hat{q}_w + \hat{q}_xi + \hat{q}_yj + \hat{q}_zk \]

If we set \(\hat{q}_x, \hat{q}_y\) and \(\hat{q}_z\) equal to 0, a dual quaternion is transformed to a dual number. see normalize().

If you want to create a dual quaternion, you can use:

using namespace cv;
double angle = CV_PI;
 
// create from eight number
DualQuatd dq1(1, 2, 3, 4, 5, 6, 7, 8); //p = [1,2,3,4]. q=[5,6,7,8]
 
// create from Vec
Vec<double, 8> v{1,2,3,4,5,6,7,8};
DualQuatd dq_v{v};
 
// create from two quaternion
Quatd p(1, 2, 3, 4);
Quatd q(5, 6, 7, 8);
DualQuatd dq2 = DualQuatd::createFromQuat(p, q);
 
// create from an angle, an axis and a translation
Vec3d axis{0, 0, 1};
Vec3d trans{3, 4, 5};
DualQuatd dq3 = DualQuatd::createFromAngleAxisTrans(angle, axis, trans);
 
// If you already have an instance of class Affine3, then you can use
Affine3d R = dq3.toAffine3();
DualQuatd dq4 = DualQuatd::createFromAffine3(R);
 
// or create directly by affine transformation matrix Rt
// see createFromMat() in detail for the form of Rt
Matx44d Rt = dq3.toMat();
DualQuatd dq5 = DualQuatd::createFromMat(Rt);
 
// Any rotation + translation movement can
// be expressed as a rotation + translation around the same line in space (expressed by Plucker
// coords), and here's a way to represent it this way.
Vec3d axis{1, 1, 1}; // axis will be normalized in createFromPitch
Vec3d trans{3, 4 ,5};
axis = axis / std::sqrt(axis.dot(axis));// The formula for computing moment that I use below requires a normalized axis
Vec3d moment = 1.0 / 2 * (trans.cross(axis) + axis.cross(trans.cross(axis)) *
                           std::cos(rotation_angle / 2) / std::sin(rotation_angle / 2));
double d = trans.dot(qaxis);
DualQuatd dq6 = DualQuatd::createFromPitch(angle, d, axis, moment);

A point \(v=(x, y, z)\) in form of dual quaternion is \([1+\epsilon v]=[1,0,0,0,0,x,y,z]\). The transformation of a point \(v_1\) to another point \(v_2\) under the dual quaternion \(\sigma\) is

\[ 1 + \epsilon v_2 = \sigma * (1 + \epsilon v_1) * \sigma^{\star} \]

where \(\sigma^{\star}=p^*-\epsilon q^*.\)

A line in the \(Pl\ddot{u}cker\) coordinates \((\hat{l}, m)\) defined by the dual quaternion \(l=\hat{l}+\epsilon m\). To transform a line,

\[l_2 = \sigma * l_1 * \sigma^*,\]

where \(\sigma=r+\frac{\epsilon}{2}rt\) and \(\sigma^*=p^*+\epsilon q^*\).

To extract the Vec<double, 8> or Vec<float, 8>, see toVec();

To extract the affine transformation matrix, see toMat();

To extract the instance of Affine3, see toAffine3();

If two quaternions \(q_0, q_1\) are needed to be interpolated, you can use sclerp()

DualQuatd::sclerp(q0, q1, t)

or dqblend().

DualQuatd::dqblend(q0, q1, t)

With more than two dual quaternions to be blended, you can use generalize linear dual quaternion blending with the corresponding weights, i.e. gdqblend().

Constructor & Destructor Documentation

DualQuat() [1/3]

template<typename _Tp >

cv::DualQuat< _Tp >::DualQuat

(

)

DualQuat() [2/3]

template<typename _Tp >

cv::DualQuat< _Tp >::DualQuat	(	const _Tp	w,
		const _Tp	x,
		const _Tp	y,
		const _Tp	z,
		const _Tp	w_,
		const _Tp	x_,
		const _Tp	y_,
		const _Tp	z_
	)

create from eight same type numbers.

DualQuat() [3/3]

template<typename _Tp >

cv::DualQuat< _Tp >::DualQuat

(

const Vec< _Tp, 8 > &

q

)

create from a double or float vector.

Member Function Documentation

conjugate()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::conjugate

(

)

const

return the conjugate of a dual quaternion.

\[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \]

createFromAffine3()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::createFromAffine3 ( const Affine3< _Tp > &  R )

static

create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat()

createFromAngleAxisTrans()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::createFromAngleAxisTrans ( const _Tp  angle, const Vec< _Tp, 3 > &  axis, const Vec< _Tp, 3 > &  translation  )

static

create a dual quaternion from a rotation angle \(\theta\), a rotation axis \(\boldsymbol{u}\) and a translation \(\boldsymbol{t}\). It generates a dual quaternion \(\sigma\) in the form of

\[\begin{equation} \begin{split} \sigma &= r + \frac{\epsilon}{2}\boldsymbol{t}r \\ &= [\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})] + \frac{\epsilon}{2}[0, \boldsymbol{t}][[\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})]]\\ &= \cos(\frac{\theta}{2}) + \boldsymbol{u}\sin(\frac{\theta}{2}) + \frac{\epsilon}{2}(-(\boldsymbol{t} \cdot \boldsymbol{u})\sin(\frac{\theta}{2}) + \boldsymbol{t}\cos(\frac{\theta}{2}) + \boldsymbol{u} \times \boldsymbol{t} \sin(\frac{\theta}{2})). \end{split} \end{equation}\]

Parameters

angle	rotation angle.
axis	rotation axis.
translation	a vector of length 3.

Note

Axis will be normalized in this function. And translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

See also

Quat

createFromMat()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::createFromMat ( InputArray  _R )

static

Transform this dual quaternion to an affine transformation matrix \(M\). Dual quaternion consists of a rotation \(r=[a,b,c,d]\) and a translation \(t=[\Delta x,\Delta y,\Delta z]\). The affine transformation matrix \(M\) has the form.

\[ \begin{bmatrix} 1-2(e_2^2 +e_3^2) &2(e_1e_2-e_0e_3) &2(e_0e_2+e_1e_3) &\Delta x\\ 2(e_0e_3+e_1e_2) &1-2(e_1^2+e_3^2) &2(e_2e_3-e_0e_1) &\Delta y\\ 2(e_1e_3-e_0e_2) &2(e_0e_1+e_2e_3) &1-2(e_1^2-e_2^2) &\Delta z\\ 0&0&0&1 \end{bmatrix} \]

if A is a matrix consisting of n points to be transformed, this could be achieved by

\[ new\_A = M * A \]

where A has the form

\[ \begin{bmatrix} x_0& x_1& x_2&...&x_n\\ y_0& y_1& y_2&...&y_n\\ z_0& z_1& z_2&...&z_n\\ 1&1&1&...&1 \end{bmatrix} \]

where the same subscript represent the same point. The size of A should be \([4,n]\). and the same size for matrix new_A.

Parameters

_R	4x4 matrix that represents rotations and translation.

Note

Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

createFromPitch()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::createFromPitch ( const _Tp  angle, const _Tp  d, const Vec< _Tp, 3 > &  axis, const Vec< _Tp, 3 > &  moment  )

static

A dual quaternion is a vector in form of.

\[ \begin{equation} \begin{split} \sigma &=\boldsymbol{p} + \epsilon \boldsymbol{q}\\ &= \cos\hat{\frac{\theta}{2}}+\overline{\hat{l}}\sin\frac{\hat{\theta}}{2} \end{split} \end{equation} \]

where \(\hat{\theta}\) is dual angle and \(\overline{\hat{l}}\) is dual axis:

\[ \hat{\theta}=\theta + \epsilon d,\\ \overline{\hat{l}}= \hat{l} +\epsilon m. \]

In this representation, \(\theta\) is rotation angle and \((\hat{l},m)\) is the screw axis, d is the translation distance along the axis.

Parameters

angle	rotation angle.
d	translation along the rotation axis.
axis	rotation axis represented by quaternion with w = 0.
moment	the moment of line, and it should be orthogonal to axis.

Note

Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

createFromQuat()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::createFromQuat ( const Quat< _Tp > &  realPart, const Quat< _Tp > &  dualPart  )

static

create Dual Quaternion from two same type quaternions p and q. A Dual Quaternion \(\sigma\) has the form:

\[\sigma = p + \epsilon q\]

where p and q are defined as follows:

\[\begin{equation} \begin{split} p &= w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}\\ q &= w\_ + x\_\boldsymbol{i} + y\_\boldsymbol{j} + z\_\boldsymbol{k}. \end{split} \end{equation} \]

The p and q are the real part and dual part respectively.

Parameters

realPart	a quaternion, real part of dual quaternion.
dualPart	a quaternion, dual part of dual quaternion.

See also

Quat

dot()

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::dot

(

DualQuat< _Tp >

p

)

const

return the dot product of two dual quaternion.

Parameters

p	other dual quaternion.

dqblend()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::dqblend ( const DualQuat< _Tp > &  q1, const DualQuat< _Tp > &  q2, const _Tp  t, QuatAssumeType  assumeUnit = QUAT_ASSUME_NOT_UNIT  )

static

The method of Dual Quaternion linear Blending(DQB) is to compute a transformation between dual quaternion \(q_1\) and \(q_2\) and can be defined as:

\[ DQB(t;{\boldsymbol{q}}_1,{\boldsymbol{q}}_2)= \frac{(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2}{||(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2||}. \]

where \(q_1\) and \(q_2\) are unit dual quaternions representing the input transformations. If you want to use DQB that works for more than two rigid transformations, see gdqblend

Parameters

q1	a unit dual quaternion representing the input transformations.
q2	a unit dual quaternion representing the input transformations.
t	parameter \(t\in[0,1]\).
assumeUnit	if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

See also

gdqblend

exp()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::exp

(

)

const

return the value of exponential function value

gdqblend() [1/2]

template<typename _Tp >

template<int cn>

static DualQuat< _Tp > cv::DualQuat< _Tp >::gdqblend ( const Vec< DualQuat< _Tp >, cn > &  dualquat, InputArray  weights, QuatAssumeType  assumeUnit = QUAT_ASSUME_NOT_UNIT  )

static

The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.

\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]

Parameters

dualquat	vector of dual quaternions
weights	vector of weights, the size of weights should be the same as dualquat, and the weights should satisfy \(\sum_0^n w_{i} = 1\) and \(w_i>0\).
assumeUnit	if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

Note

the type of weights' element should be the same as the date type of dual quaternion inside the dualquat.

gdqblend() [2/2]

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::gdqblend ( InputArray  dualquat, InputArray  weights, QuatAssumeType  assumeUnit = QUAT_ASSUME_NOT_UNIT  )

static

The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.

\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]

Parameters

dualquat	The dual quaternions which have 8 channels and 1 row or 1 col.
weights	vector of weights, the size of weights should be the same as dualquat, and the weights should satisfy \(\sum_0^n w_{i} = 1\) and \(w_i>0\).
assumeUnit	if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

Note

the type of weights' element should be the same as the date type of dual quaternion inside the dualquat.

getDualPart()

template<typename _Tp >

Quat< _Tp > cv::DualQuat< _Tp >::getDualPart

(

)

const

return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat().

See also

createFromQuat, getRealPart

getRealPart()

template<typename _Tp >

Quat< _Tp > cv::DualQuat< _Tp >::getRealPart

(

)

const

return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat().

See also

createFromQuat, getDualPart

getRotation()

template<typename _Tp >

Quat< _Tp > cv::DualQuat< _Tp >::getRotation

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

return the rotation in quaternion form.

getTranslation()

template<typename _Tp >

Vec< _Tp, 3 > cv::DualQuat< _Tp >::getTranslation

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

return the translation vector. The rotation \(r\) in this dual quaternion \(\sigma\) is applied before translation \(t\). The dual quaternion \(\sigma\) is defined as

\[\begin{equation} \begin{split} \sigma &= p + \epsilon q \\ &= r + \frac{\epsilon}{2}{t}r. \end{split} \end{equation}\]

Thus, the translation can be obtained as follows

\[t = 2qp^*.\]

Parameters

assumeUnit

if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

Note

This dual quaternion's translation is applied after the rotation.

inv()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::inv

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is

\[\sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \]

or equivalentlly,

\[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\]

Parameters

assumeUnit

if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

log()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::log

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

return the value of logarithm function value

Parameters

assumeUnit

if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

norm()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::norm

(

)

const

return the norm \(||\sigma||\) of dual quaternion \(\sigma = p + \epsilon q\).

\[ \begin{equation} \begin{split} ||\sigma|| &= \sqrt{\sigma * \sigma^*} \\ &= ||p|| + \epsilon \frac{p \cdot q}{||p||}. \end{split} \end{equation} \]

Generally speaking, the norm of a not unit dual quaternion is a dual number. For convenience, we return it in the form of a dual quaternion , i.e.

\[ ||\sigma|| = [||p||, 0, 0, 0, \frac{p \cdot q}{||p||}, 0, 0, 0].\]

Note

The data type of dual number is dual quaternion.

normalize()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::normalize

(

)

const

return a normalized dual quaternion. A dual quaternion can be expressed as

\[ \begin{equation} \begin{split} \sigma &= p + \epsilon q\\ &=||\sigma||\left(r+\frac{1}{2}tr\right) \end{split} \end{equation} \]

where \(r, t\) represents the rotation (ordinary quaternion) and translation (pure ordinary quaternion) respectively, and \(||\sigma||\) is the norm of dual quaternion(a dual number). A dual quaternion is unit if and only if

\[ ||p||=1, p \cdot q=0 \]

where \(\cdot\) means dot product. The process of normalization is

\[ \sigma_{u}=\frac{\sigma}{||\sigma||} \]

Next, we simply proof \(\sigma_u\) is a unit dual quaternion:

\[ \renewcommand{\Im}{\operatorname{Im}} \begin{equation} \begin{split} \sigma_{u}=\frac{\sigma}{||\sigma||}&=\frac{p + \epsilon q}{||p||+\epsilon\frac{p\cdot q}{||p||}}\\ &=\frac{p}{||p||}+\epsilon\left(\frac{q}{||p||}-p\frac{p\cdot q}{||p||^3}\right)\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\left(qp^{*}-p\cdot q\right)\frac{p}{||p||}\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\Im(qp^*)\frac{p}{||p||}.\\ \end{split} \end{equation} \]

As expected, the real part is a rotation and dual part is a pure quaternion.

operator*()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator*

(

const DualQuat< _Tp > &

)

const

Multiplication operator of two dual quaternions q and p. Multiplies values on either side of the operator.

Rule of dual quaternion multiplication: The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus

\[ \begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p * q << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]

operator*=() [1/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator*=

(

const _Tp

s

)

Multiplication assignment operator of a quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication with a scalar:

\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
p *= s;
std::cout << p << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]

Note

the type of scalar should be equal to the dual quaternion.

operator*=() [2/2]

template<typename _Tp >

DualQuat< _Tp > & cv::DualQuat< _Tp >::operator*=

(

const DualQuat< _Tp > &

)

Multiplication assignment operator of two quaternions. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication: The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus

\[ \begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p *= q;
std::cout << p << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]

operator+()

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator+

(

const DualQuat< _Tp > &

)

const

Addition operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(q_i\).

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p + q << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]

operator+=()

template<typename _Tp >

DualQuat< _Tp > & cv::DualQuat< _Tp >::operator+=

(

const DualQuat< _Tp > &

)

Addition assignment operator of two dual quaternions p and q. It adds right operand to the left operand and assign the result to left operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p += q; // equivalent to p = p + q
std::cout << p << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]

operator-() [1/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator-

(

)

const

Return opposite dual quaternion \(-p\) which satisfies \(p + (-p) = 0.\).

For example

DualQuatd q{1, 2, 3, 4, 5, 6, 7, 8};
std::cout << -q << std::endl; // [-1, -2, -3, -4, -5, -6, -7, -8]

operator-() [2/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator-

(

const DualQuat< _Tp > &

)

const

Subtraction operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(-q_i\).

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p - q << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]

operator-=()

template<typename _Tp >

DualQuat< _Tp > & cv::DualQuat< _Tp >::operator-=

(

const DualQuat< _Tp > &

)

Subtraction assignment operator of two dual quaternions p and q. It subtracts right operand from the left operand and assign the result to left operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p -= q; // equivalent to p = p - q
std::cout << p << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]

operator/() [1/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator/

(

const _Tp

s

)

const

Division operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a scalar:

\[ \begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_, z\_] / s\\ &=[w/s, x/s, y/s, z/s, w\_/s, x\_/s, y\_/s, z\_/s]. \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4]

Note

the type of scalar should be equal to this dual quaternion.

operator/() [2/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::operator/

(

const DualQuat< _Tp > &

)

const

Division operator of two dual quaternions p and q. Divides left hand operand by right hand operand.

Rule of dual quaternion division with a dual quaternion:

\[ \begin{equation} \begin{split} p / q &= p * q.inv()\\ \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p / q << std::endl; // equivalent to p * q.inv()

operator/=() [1/2]

template<typename _Tp >

Quat< _Tp > & cv::DualQuat< _Tp >::operator/=

(

const _Tp

s

)

Division assignment operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a scalar:

\[ \begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_ ,z\_] / s\\ &=[w / s, x / s, y / s, z / s, w\_ / \space s, x\_ / \space s, y\_ / \space s, z\_ / \space s]. \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0]

Note

the type of scalar should be equal to the dual quaternion.

operator/=() [2/2]

template<typename _Tp >

DualQuat< _Tp > & cv::DualQuat< _Tp >::operator/=

(

const DualQuat< _Tp > &

)

Division assignment operator of two dual quaternions p and q; It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a quaternion:

\[ \begin{equation} \begin{split} p / q&= p * q.inv()\\ \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p /= q; // equivalent to p = p * q.inv()
std::cout << p << std::endl;

operator==()

template<typename _Tp >

bool cv::DualQuat< _Tp >::operator==

(

const DualQuat< _Tp > &

)

const

return true if two dual quaternions p and q are nearly equal, i.e. when the absolute value of each \(p_i\) and \(q_i\) is less than CV_DUAL_QUAT_EPS.

power() [1/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::power	(	const _Tp	t,
		QuatAssumeType	assumeUnit = QUAT_ASSUME_NOT_UNIT
	)		const

return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:

\[ p^t = \exp(t\ln p) \]

Parameters

t	index of power function.
assumeUnit	if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

power() [2/2]

template<typename _Tp >

DualQuat< _Tp > cv::DualQuat< _Tp >::power	(	const DualQuat< _Tp > &	q,
		QuatAssumeType	assumeUnit = QUAT_ASSUME_NOT_UNIT
	)		const

return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:

\[ p^q = \exp(q\ln p) \]

Parameters

q	a dual quaternion
assumeUnit	if QUAT_ASSUME_UNIT, this dual quaternion assume to be a dual unit quaternion and this function will save some computations.

sclerp()

template<typename _Tp >

static DualQuat< _Tp > cv::DualQuat< _Tp >::sclerp ( const DualQuat< _Tp > &  q1, const DualQuat< _Tp > &  q2, const _Tp  t, bool  directChange = true, QuatAssumeType  assumeUnit = QUAT_ASSUME_NOT_UNIT  )

static

The screw linear interpolation(ScLERP) is an extension of spherical linear interpolation of dual quaternion. If \(\sigma_1\) and \(\sigma_2\) are two dual quaternions representing the initial and final pose. The interpolation of ScLERP function can be defined as:

\[ ScLERP(t;\sigma_1,\sigma_2) = \sigma_1 * (\sigma_1^{-1} * \sigma_2)^t, t\in[0,1] \]

Parameters

q1	a dual quaternion represents a initial pose.
q2	a dual quaternion represents a final pose.
t	interpolation parameter
directChange	if true, it always return the shortest path.
assumeUnit	if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

For example

double angle1 = CV_PI / 2;
Vec3d axis{0, 0, 1};
Vec3d t(0, 0, 3);
DualQuatd initial = DualQuatd::createFromAngleAxisTrans(angle1, axis, t);
double angle2 = CV_PI;
DualQuatd final = DualQuatd::createFromAngleAxisTrans(angle2, axis, t);
DualQuatd inter = DualQuatd::sclerp(initial, final, 0.5);

toAffine3()

template<typename _Tp >

Affine3< _Tp > cv::DualQuat< _Tp >::toAffine3

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

Transform this dual quaternion to a instance of Affine3.

toMat()

template<typename _Tp >

Matx< _Tp, 4, 4 > cv::DualQuat< _Tp >::toMat

(

QuatAssumeType

assumeUnit = QUAT_ASSUME_NOT_UNIT

)

const

Transform this dual quaternion to a affine transformation matrix the form of matrix, see createFromMat().

toVec()

template<typename _Tp >

Vec< _Tp, 8 > cv::DualQuat< _Tp >::toVec

(

)

const

Transform this dual quaternion to a vector.

Friends And Related Symbol Documentation

conjugate

template<typename _Tp >

template<typename T >

DualQuat< T > conjugate ( const DualQuat< T > &  dq )

friend

return the conjugate of a dual quaternion.

\[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \]

Parameters

dq	a dual quaternion.

cv::operator* [1/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator* ( const DualQuat< T > &  , const T  s  )

friend

Multiplication operator of a dual quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication with a scalar:

\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
std::cout << p * s << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator* [2/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator* ( const T  s, const DualQuat< T > &    )

friend

Multiplication operator of a scalar and a dual quaternions. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication with a scalar:

\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
std::cout << s * p << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator+ [1/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator+ ( const DualQuat< T > &  , const T  s  )

friend

Addition operator of a dual quaternions and a scalar. Adds right hand operand from left hand operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double scalar = 2.0;
std::cout << p + scalar << std::endl; //[3.0, 2, 3, 4, 5, 6, 7, 8]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator+ [2/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator+ ( const T  s, const DualQuat< T > &    )

friend

Addition operator of a scalar and a dual quaternions. Adds right hand operand from left hand operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double scalar = 2.0;
std::cout << scalar + p << std::endl; //[3.0, 2, 3, 4, 5, 6, 7, 8]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator- [1/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator- ( const DualQuat< T > &  , const T  s  )

friend

Subtraction operator of a dual quaternion and a scalar. Subtracts right hand operand from left hand operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double scalar = 2.0;
std::cout << p - scalar << std::endl; //[-1, 2, 3, 4, 5, 6, 7, 8]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator- [2/2]

template<typename _Tp >

template<typename T >

DualQuat< T > cv::operator- ( const T  s, const DualQuat< T > &    )

friend

Subtraction operator of a scalar and a dual quaternions. Subtracts right hand operand from left hand operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double scalar = 2.0;
std::cout << scalar - p << std::endl; //[1.0, -2, -3, -4, -5, -6, -7, -8]

Note

the type of scalar should be equal to the dual quaternion.

cv::operator<<

template<typename _Tp >

template<typename S >

std::ostream & cv::operator<< ( std::ostream &  , const DualQuat< S > &    )

friend

exp

template<typename _Tp >

template<typename T >

DualQuat< T > exp ( const DualQuat< T > &  dq )

friend

return the value of exponential function value

Parameters

dq	a dual quaternion.

inv

template<typename _Tp >

template<typename T >

DualQuat< T > inv ( const DualQuat< T > &  dq, QuatAssumeType  assumeUnit  )

friend

if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is

\[\sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \]

or equivalentlly,

\[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\]

Parameters

dq	a dual quaternion.
assumeUnit	if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations.

log

template<typename _Tp >

template<typename T >

DualQuat< T > log ( const DualQuat< T > &  dq, QuatAssumeType  assumeUnit  )

friend

return the value of logarithm function value

Parameters

dq	a dual quaternion.
assumeUnit	if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations.

power [1/2]

template<typename _Tp >

template<typename T >

DualQuat< T > power ( const DualQuat< T > &  dq, const T  t, QuatAssumeType  assumeUnit  )

friend

return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:

\[ p^t = \exp(t\ln p) \]

Parameters

dq	a dual quaternion.
t	index of power function.
assumeUnit	if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations.

power [2/2]

template<typename _Tp >

template<typename T >

DualQuat< T > power ( const DualQuat< T > &  p, const DualQuat< T > &  q, QuatAssumeType  assumeUnit  )

friend

return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:

\[ p^q = \exp(q\ln p) \]

Parameters

p	a dual quaternion.
q	a dual quaternion.
assumeUnit	if QUAT_ASSUME_UNIT, dual quaternion p assume to be a dual unit quaternion and this function will save some computations.

Member Data Documentation

CV_DUAL_QUAT_EPS

template<typename _Tp >

constexpr _Tp cv::DualQuat< _Tp >::CV_DUAL_QUAT_EPS = (_Tp)1.e-6

staticconstexpr

w

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::w

w_

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::w_

x

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::x

x_

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::x_

y

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::y

y_

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::y_

z

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::z

z_

template<typename _Tp >

_Tp cv::DualQuat< _Tp >::z_

---

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

• opencv2/core/dualquaternion.hpp