OpenCV  4.5.5 Open Source Computer Vision
cv::DualQuat< _Tp > Class Template Reference

#include <opencv2/core/dualquaternion.hpp>

## 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. More...

DualQuat (const Vec< _Tp, 8 > &q)
create from a double or float vector. More...

DualQuat< _Tp > conjugate () const
return the conjugate of a dual quaternion.

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

More...

_Tp dot (DualQuat< _Tp > p) const
return the dot product of two dual quaternion. More...

DualQuat< _Tp > exp () const
return the value of exponential function value More...

Quat< _Tp > getDualPart () const
return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat(). More...

Quat< _Tp > getRealPart () const
return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat(). More...

Quat< _Tp > getRotation (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return the rotation in quaternion form. More...

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

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

Thus, the translation can be obtained as follows

$t = 2qp^*.$

More...

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,

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

More...

DualQuat< _Tp > log (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return the value of logarithm function value More...

DualQuat< _Tp > norm () const
return the norm $$||\sigma||$$ of dual quaternion $$\sigma = p + \epsilon q$$.

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

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.

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

More...

DualQuat< _Tp > normalize () const
return a normalized dual quaternion. A dual quaternion can be expressed as

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

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

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

DualQuat< _Tp > operator* (const DualQuat< _Tp > &) const
Multiplication operator of two dual quaternions q and p. Multiplies values on either side of the operator. More...

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. More...

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. More...

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 $$q_i$$. More...

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. More...

DualQuat< _Tp > operator- () const
Return opposite dual quaternion $$-p$$ which satisfies $$p + (-p) = 0.$$. More...

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 $$-q_i$$. More...

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. More...

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. More...

DualQuat< _Tp > operator/ (const DualQuat< _Tp > &) const
Division operator of two dual quaternions p and q. Divides left hand operand by right hand operand. More...

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. More...

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. More...

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. More...

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:

$p^t = \exp(t\ln p)$

More...

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:

$p^q = \exp(q\ln p)$

More...

Affine3< _Tp > toAffine3 (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
Transform this dual quaternion to a instance of Affine3. More...

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(). More...

Vec< _Tp, 8 > toVec () const
Transform this dual quaternion to a vector. More...

## 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() More...

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

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

More...

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

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

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. More...

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

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

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. More...

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:

$\sigma = p + \epsilon q$

where p and q are defined as follows:

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

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

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:

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

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. More...

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

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

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

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

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:

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

_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.

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

More...

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. More...

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. More...

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. More...

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. More...

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. More...

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. More...

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 More...

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

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

or equivalentlly,

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

More...

template<typename T >
DualQuat< T > log (const DualQuat< T > &dq, QuatAssumeType assumeUnit)
return the value of logarithm function value More...

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:

$p^t = \exp(t\ln p)$

More...

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:

$p^q = \exp(q\ln p)$

More...

## 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

A unit dual quaternion can be classically represented as:

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

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);
// create from an angle, an axis and a translation
Vec3d axis{0, 0, 1};
Vec3d trans{3, 4, 5};
// If you already have an instance of class Affine3, then you can use
Affine3d R = dq3.toAffine3();
// or create directly by affine transformation matrix Rt
// see createFromMat() in detail for the form of Rt
Matx44d Rt = dq3.toMat();
// 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()

or dqblend().

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

## ◆ 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.

## ◆ conjugate()

template<typename _Tp>
 DualQuat<_Tp> cv::DualQuat< _Tp >::conjugate ( ) const

return the conjugate of a dual quaternion.

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

## ◆ 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

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

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.
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

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

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:

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

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.
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.
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().

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().

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

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

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 $$\sigma = p + \epsilon q$$.

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

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

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

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

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

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

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 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

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

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*=() [2/2]

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:

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

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+()

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:

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

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:

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

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>
 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:

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

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/=() [2/2]

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:

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

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==()

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);
double angle2 = CV_PI;
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.

## ◆ conjugate

template<typename _Tp>
template<typename T >
 DualQuat conjugate ( const DualQuat< T > & dq )
friend

return the conjugate of a dual quaternion.

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

Parameters
 dq a dual quaternion.

## ◆ cv::operator* [1/2]

template<typename _Tp>
template<typename T >
 DualQuat 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:

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

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* [2/2]

template<typename _Tp>
template<typename T >
 DualQuat 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:

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

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+ [1/2]

template<typename _Tp>
template<typename T >
 DualQuat 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+ [2/2]

template<typename _Tp>
template<typename T >
 DualQuat 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- [1/2]

template<typename _Tp>
template<typename T >
 DualQuat 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 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 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 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 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 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 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.

## ◆ CV_DUAL_QUAT_EPS

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

## ◆ 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: