#include <dualquaternion.hpp>

公開メンバ関数
	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.

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

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

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

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 in this dual quaternion $\sigma$ is applied before translation . The dual quaternion $\sigma$ is defined as [詳解]

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

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

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 [詳解]

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

DualQuat< _Tp >	power (const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of where p is a dual quaternion. This could be calculated as: [詳解]

DualQuat< _Tp >	power (const DualQuat< _Tp > &q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of where p and q are dual quaternions. This could be calculated as: [詳解]

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

DualQuat< _Tp >	log (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
	return the value of logarithm function value [詳解]

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

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().

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

DualQuat< _Tp >	operator- () const
	Return opposite dual quaternion which satisfies [詳解]

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 and is less than CV_DUAL_QUAT_EPS.

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 and . [詳解]

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. [詳解]

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 and . [詳解]

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. [詳解]

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. [詳解]

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. [詳解]

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

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. [詳解]

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

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. [詳解]

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. [詳解]

template<typename T >
DualQuat< T >	power (const T t, QuatAssumeType assumeUnit) const

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

template<int cn>
DualQuat< T >	gdqblend (const Vec< DualQuat< T >, cn > &_dualquat, InputArray _weight, QuatAssumeType assumeUnit)

静的公開メンバ関数
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: [詳解]

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 [詳解]

static DualQuat< _Tp >	createFromMat (InputArray _R)
	Transform this dual quaternion to an affine transformation matrix . Dual quaternion consists of a rotation and a translation $t=[\Delta x,\Delta y,\Delta z]$ . The affine transformation matrix has the form [詳解]

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 >	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 [詳解]

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: [詳解]

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 and and can be defined as: [詳解]

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 with convex weights , the generalized DQB is simply [詳解]

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 with convex weights , the generalized DQB is simply [詳解]

公開変数類
_Tp	w

_Tp	x

_Tp	y

_Tp	z

_Tp	w_

_Tp	x_

_Tp	y_

_Tp	z_

静的公開変数類
static constexpr _Tp	CV_DUAL_QUAT_EPS = (_Tp)1.e-6

フレンド
template<typename T >
DualQuat< T >	conjugate (const DualQuat< T > &dq)
	return the conjugate of a dual quaternion. [詳解]

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 [詳解]

template<typename T >
DualQuat< T >	power (const DualQuat< T > &dq, const T t, QuatAssumeType assumeUnit)
	return the value of where p is a dual quaternion. This could be calculated as: [詳解]

template<typename T >
DualQuat< T >	power (const DualQuat< T > &p, const DualQuat< T > &q, QuatAssumeType assumeUnit)
	return the value of where p and q are dual quaternions. This could be calculated as: [詳解]

template<typename T >
DualQuat< T >	exp (const DualQuat< T > &dq)
	return the value of exponential function value [詳解]

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

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. [詳解]

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. [詳解]

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. [詳解]

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. [詳解]

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. [詳解]

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. [詳解]

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

詳解

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:

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

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)

cv::DualQuat::sclerp

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 quat...

Definition: dualquaternion.inl.hpp:405

or dqblend().

DualQuatd::dqblend(q0, q1, t)

cv::DualQuat::dqblend

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 quater...

Definition: dualquaternion.inl.hpp:424

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

関数詳解

◆ conjugate()

template<typename T >

DualQuat< T > cv::DualQuat< T >::conjugate

inline

return the conjugate of a dual quaternion.

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

◆ createFromAngleAxisTrans()

template<typename _Tp >

DualQuat< T > cv::DualQuat< T >::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

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

引数

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

覚え書き: 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.

参照: Quat

◆ createFromMat()

template<typename T >

DualQuat< T > cv::DualQuat< T >::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.

引数

_R	4x4 matrix that represents rotations and translation.

覚え書き: 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 >

DualQuat< T > cv::DualQuat< T >::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.

引数

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.

覚え書き: 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 >

DualQuat< T > cv::DualQuat< T >::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.

引数

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

参照: Quat

◆ dot()

template<typename _Tp >

T cv::DualQuat< T >::dot ( DualQuat< _Tp > p ) const

inline

return the dot product of two dual quaternion.

引数

p	other dual quaternion.

◆ dqblend()

template<typename _Tp >

DualQuat< T > cv::DualQuat< T >::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

引数

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.

参照: gdqblend

◆ 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

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

引数

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 .
assumeUnit	if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

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

◆ gdqblend() [2/2]

template<typename T >

DualQuat< T > cv::DualQuat< T >::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

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

引数

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 .
assumeUnit	if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

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

◆ getDualPart()

template<typename T >

Quat< T > cv::DualQuat< T >::getDualPart

inline

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

参照: createFromQuat, getRealPart

◆ getRealPart()

template<typename T >

Quat< T > cv::DualQuat< T >::getRealPart

inline

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

参照: createFromQuat, getDualPart

◆ getTranslation()

template<typename T >

Vec< T, 3 > cv::DualQuat< T >::getTranslation ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

inline

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

$t = 2qp^*.$

引数

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

覚え書き: This dual quaternion's translation is applied after the rotation.

◆ inv()

template<typename T >

DualQuat< T > cv::DualQuat< T >::inv ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

inline

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,

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

引数

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 T >

DualQuat< T > cv::DualQuat< T >::log ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

return the value of logarithm function value

引数

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 T >

DualQuat< T > cv::DualQuat< T >::norm

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.

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

覚え書き: The data type of dual number is dual quaternion.

◆ normalize()

template<typename T >

DualQuat< T > cv::DualQuat< T >::normalize

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

inline

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]

覚え書き: 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< T > cv::DualQuat< T >::operator+ ( const DualQuat< _Tp > & ) const

inline

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< T > & cv::DualQuat< T >::operator+= ( const DualQuat< _Tp > & )

inline

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 T >

DualQuat< T > cv::DualQuat< T >::operator-

inline

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< T > & cv::DualQuat< T >::operator-= ( const DualQuat< _Tp > & )

inline

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]

覚え書き: 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]

覚え書き: 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;

◆ 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:

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

引数

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:

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

引数

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 >

DualQuat< T > cv::DualQuat< T >::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:

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

引数

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);

フレンドと関連関数の詳解

◆ conjugate

template<typename _Tp >

template<typename T >

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

friend

return the conjugate of a dual quaternion.

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

引数

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]

覚え書き: 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]

覚え書き: 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]

覚え書き: 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]

覚え書き: 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]

覚え書き: 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]

覚え書き: the type of scalar should be equal to the dual quaternion.

◆ exp

template<typename _Tp >

template<typename T >

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

friend

return the value of exponential function value

引数

dq	a dual quaternion.

◆ inv

template<typename _Tp >

template<typename T >

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

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,

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

引数

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 = `QUAT_ASSUME_NOT_UNIT`
	)

friend

return the value of logarithm function value

引数

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 = `QUAT_ASSUME_NOT_UNIT`
	)

friend

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

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

引数

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 = `QUAT_ASSUME_NOT_UNIT`
	)

friend

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

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

引数

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.

このクラス詳解は次のファイルから抽出されました:

公開メンバ関数

静的公開メンバ関数

公開変数類

静的公開変数類

フレンド

詳解

関数詳解

◆ conjugate()

◆ createFromAngleAxisTrans()

◆ createFromMat()

◆ createFromPitch()

◆ createFromQuat()

◆ dot()

◆ dqblend()

◆ gdqblend() [1/2]

◆ gdqblend() [2/2]

◆ getDualPart()

◆ getRealPart()

◆ getTranslation()

◆ inv()

◆ log()

◆ norm()

◆ normalize()

◆ operator*()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+()

◆ operator+=()

◆ operator-() [1/2]

◆ operator-() [2/2]

◆ operator-=()

◆ operator/() [1/2]

◆ operator/() [2/2]

◆ operator/=() [1/2]

◆ operator/=() [2/2]

◆ power() [1/2]

◆ power() [2/2]

◆ sclerp()

フレンドと関連関数の詳解

◆ conjugate

◆ cv::operator* [1/2]

◆ cv::operator* [2/2]

◆ cv::operator+ [1/2]

◆ cv::operator+ [2/2]

◆ cv::operator- [1/2]

◆ cv::operator- [2/2]

◆ exp

◆ inv

◆ log

◆ power [1/2]

◆ power [2/2]