OpenCV  4.5.4 Open Source Computer Vision
cv::Quat< _Tp > Class Template Reference

#include <opencv2/core/quaternion.hpp>

## Public Member Functions

Quat ()

Quat (const Vec< _Tp, 4 > &coeff)
From Vec4d or Vec4f. More...

Quat (_Tp w, _Tp x, _Tp y, _Tp z)
from four numbers. More...

Quat< _Tp > acos () const
return arccos value of this quaternion, arccos could be calculated as:

$\arccos(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arccosh(q)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > acosh () const
return arccosh value of this quaternion, arccosh could be calculated as:

$arcosh(q) = \ln(q + \sqrt{q^2 - 1})$

Quat< _Tp > asin () const
return arcsin value of this quaternion, arcsin could be calculated as:

$\arcsin(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arcsinh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > asinh () const
return arcsinh value of this quaternion, arcsinh could be calculated as:

$arcsinh(q) = \ln(q + \sqrt{q^2 + 1})$

void assertNormal (_Tp eps=CV_QUAT_EPS) const
to throw an error if this quaternion is not a unit quaternion. More...

_Tp at (size_t index) const
a way to get element. More...

Quat< _Tp > atan () const
return arctan value of this quaternion, arctan could be calculated as:

$\arctan(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arctanh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > atanh () const
return arctanh value of this quaternion, arctanh could be calculated as:

$arcsinh(q) = \frac{\ln(q + 1) - \ln(1 - q)}{2}$

Quat< _Tp > conjugate () const
return the conjugate of this quaternion.

$q.conjugate() = (w, -x, -y, -z).$

More...

Quat< _Tp > cos () const
return cos value of this quaternion, cos could be calculated as:

$\cos(p) = \cos(w) * \cosh(||\boldsymbol{v}||) - \sin(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > cosh () const
return cosh value of this quaternion, cosh could be calculated as:

$\cosh(p) = \cosh(w) * \cos(||\boldsymbol{v}||) + \sinh(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}sin(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > crossProduct (const Quat< _Tp > &q) const
return the crossProduct between $$p = (a, b, c, d) = (a, \boldsymbol{u})$$ and $$q = (w, x, y, z) = (w, \boldsymbol{v})$$.

$p \times q = \frac{pq- qp}{2}.$

$p \times q = \boldsymbol{u} \times \boldsymbol{v}.$

$p \times q = (cz-dy)i + (dx-bz)j + (by-xc)k.$

More...

_Tp dot (Quat< _Tp > q) const
return the dot between quaternion $$q$$ and this quaternion. More...

Quat< _Tp > exp () const
return the value of exponential value.

$\exp(q) = e^w (\cos||\boldsymbol{v}||+ \frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

_Tp getAngle (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
get the angle of quaternion, it returns the rotation angle. More...

Vec< _Tp, 3 > getAxis (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
get the axis of quaternion, it returns a vector of length 3. More...

Quat< _Tp > inv (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return $$q^{-1}$$ which is an inverse of $$q$$ satisfying $$q * q^{-1} = 1$$. More...

bool isNormal (_Tp eps=CV_QUAT_EPS) const
return true if this quaternion is a unit quaternion. More...

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

$\ln(q) = \ln||q|| + \frac{\boldsymbol{v}}{||\boldsymbol{v}||}\arccos\frac{w}{||q||}$

. where $$\boldsymbol{v} = [x, y, z].$$ More...

_Tp norm () const
return the norm of quaternion.

$||q|| = \sqrt{w^2 + x^2 + y^2 + z^2}.$

More...

Quat< _Tp > normalize () const
return a normalized $$p$$.

$p = \frac{q}{||q||}$

where $$p$$ satisfies $$(p.x)^2 + (p.y)^2 + (p.z)^2 + (p.w)^2 = 1.$$ More...

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

Quat< _Tp > & operator*= (const Quat< _Tp > &)
Multiplication assignment operator of two quaternions q and p. It multiplies right operand with the left operand and assign the result to left operand. More...

Quat< _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...

Quat< _Tp > operator+ (const Quat< _Tp > &) const
Addition operator of two quaternions p and q. It returns a new quaternion that each value is the sum of $$p_i$$ and $$q_i$$. More...

Quat< _Tp > & operator+= (const Quat< _Tp > &)
Addition assignment operator of two quaternions p and q. It adds right operand to the left operand and assign the result to left operand. More...

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

Quat< _Tp > operator- (const Quat< _Tp > &) const
Subtraction operator of two quaternions p and q. It returns a new quaternion that each value is the sum of $$p_i$$ and $$-q_i$$. More...

Quat< _Tp > & operator-= (const Quat< _Tp > &)
Subtraction assignment operator of two quaternions p and q. It subtracts right operand from the left operand and assign the result to left operand. More...

Quat< _Tp > operator/ (const _Tp s) const
Division operator of a quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand. More...

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

Quat< _Tp > & operator/= (const _Tp s)
Division assignment operator of a quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand. More...

Quat< _Tp > & operator/= (const Quat< _Tp > &)
Division assignment operator of two quaternions p and q; It divides left operand with the right operand and assign the result to left operand. More...

bool operator== (const Quat< _Tp > &) const
return true if two quaternions p and q are nearly equal, i.e. when the absolute value of each $$p_i$$ and $$q_i$$ is less than CV_QUAT_EPS. More...

_Tp & operator[] (std::size_t n)

const _Tp & operator[] (std::size_t n) const

Quat< _Tp > power (const _Tp x, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return the value of power function with index $$x$$.

$q^x = ||q||(\cos(x\theta) + \boldsymbol{u}\sin(x\theta))).$

More...

Quat< _Tp > power (const Quat< _Tp > &q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return the value of power function with quaternion $$q$$.

$p^q = e^{q\ln(p)}.$

More...

Quat< _Tp > sin () const
return sin value of this quaternion, sin could be calculated as:

$\sin(p) = \sin(w) * \cosh(||\boldsymbol{v}||) + \cos(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > sinh () const
return sinh value of this quaternion, sinh could be calculated as: $$\sinh(p) = \sin(w)\cos(||\boldsymbol{v}||) + \cosh(w)\frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||$$ where $$\boldsymbol{v} = [x, y, z].$$ More...

Quat< _Tp > sqrt (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
return $$\sqrt{q}$$. More...

Quat< _Tp > tan () const
return tan value of this quaternion, tan could be calculated as:

$\tan(q) = \frac{\sin(q)}{\cos(q)}.$

More...

Quat< _Tp > tanh () const
return tanh value of this quaternion, tanh could be calculated as:

$\tanh(q) = \frac{\sinh(q)}{\cosh(q)}.$

More...

Vec< _Tp, 3 > toEulerAngles (QuatEnum::EulerAnglesType eulerAnglesType)
Transform a quaternion q to Euler angles. More...

Matx< _Tp, 3, 3 > toRotMat3x3 (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
transform a quaternion to a 3x3 rotation matrix. More...

Matx< _Tp, 4, 4 > toRotMat4x4 (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
transform a quaternion to a 4x4 rotation matrix. More...

Vec< _Tp, 3 > toRotVec (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const
transform this quaternion to a Rotation vector. More...

Vec< _Tp, 4 > toVec () const
transform the this quaternion to a Vec<T, 4>. More...

## Static Public Member Functions

static Quat< _Tp > createFromAngleAxis (const _Tp angle, const Vec< _Tp, 3 > &axis)
from an angle, axis. Axis will be normalized in this function. And it generates

$q = [\cos\psi, u_x\sin\psi,u_y\sin\psi, u_z\sin\psi].$

where $$\psi = \frac{\theta}{2}$$, $$\theta$$ is the rotation angle. More...

static Quat< _Tp > createFromEulerAngles (const Vec< _Tp, 3 > &angles, QuatEnum::EulerAnglesType eulerAnglesType)
from Euler angles More...

static Quat< _Tp > createFromRotMat (InputArray R)
from a 3x3 rotation matrix. More...

static Quat< _Tp > createFromRvec (InputArray rvec)
from a rotation vector $$r$$ has the form $$\theta \cdot \boldsymbol{u}$$, where $$\theta$$ represents rotation angle and $$\boldsymbol{u}$$ represents normalized rotation axis. More...

static Quat< _Tp > createFromXRot (const _Tp theta)
get a quaternion from a rotation about the X-axis by $$\theta$$ .

$q = \cos(\theta/2)+sin(\theta/2) i +0 j +0 k$

More...

static Quat< _Tp > createFromYRot (const _Tp theta)
get a quaternion from a rotation about the Y-axis by $$\theta$$ .

$q = \cos(\theta/2)+0 i+ sin(\theta/2) j +0k$

More...

static Quat< _Tp > createFromZRot (const _Tp theta)
get a quaternion from a rotation about the Z-axis by $$\theta$$.

$q = \cos(\theta/2)+0 i +0 j +sin(\theta/2) k$

More...

static Quat< _Tp > interPoint (const Quat< _Tp > &q0, const Quat< _Tp > &q1, const Quat< _Tp > &q2, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
This is the part calculation of squad. To calculate the intermedia quaternion $$s_i$$ between each three quaternion

$s_i = q_i\exp(-\frac{\log(q^*_iq_{i+1}) + \log(q^*_iq_{i-1})}{4}).$

static Quat< _Tp > lerp (const Quat< _Tp > &q0, const Quat &q1, const _Tp t)
To calculate the interpolation from $$q_0$$ to $$q_1$$ by Linear Interpolation(Nlerp) For two quaternions, this interpolation curve can be displayed as:

$Lerp(q_0, q_1, t) = (1 - t)q_0 + tq_1.$

Obviously, the lerp will interpolate along a straight line if we think of $$q_0$$ and $$q_1$$ as a vector in a two-dimensional space. When $$t = 0$$, it returns $$q_0$$ and when $$t= 1$$, it returns $$q_1$$. $$t$$ should to be ranged in $$[0, 1]$$ normally. More...

static Quat< _Tp > nlerp (const Quat< _Tp > &q0, const Quat &q1, const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
To calculate the interpolation from $$q_0$$ to $$q_1$$ by Normalized Linear Interpolation(Nlerp). it returns a normalized quaternion of Linear Interpolation(Lerp).

$Nlerp(q_0, q_1, t) = \frac{(1 - t)q_0 + tq_1}{||(1 - t)q_0 + tq_1||}.$

The interpolation will always choose the shortest path but the constant speed is not guaranteed. More...

static Quat< _Tp > slerp (const Quat< _Tp > &q0, const Quat &q1, const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT, bool directChange=true)
To calculate the interpolation between $$q_0$$ and $$q_1$$ by Spherical Linear Interpolation(Slerp), which can be defined as:

$Slerp(q_0, q_1, t) = \frac{\sin((1-t)\theta)}{\sin(\theta)}q_0 + \frac{\sin(t\theta)}{\sin(\theta)}q_1$

where $$\theta$$ can be calculated as:

$\theta=cos^{-1}(q_0\cdot q_1)$

resulting from the both of their norm is unit. More...

static Quat< _Tp > spline (const Quat< _Tp > &q0, const Quat< _Tp > &q1, const Quat< _Tp > &q2, const Quat< _Tp > &q3, const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
to calculate a quaternion which is the result of a $$C^1$$ continuous spline curve constructed by squad at the ratio t. Here, the interpolation values are between $$q_1$$ and $$q_2$$. $$q_0$$ and $$q_2$$ are used to ensure the $$C^1$$ continuity. if t = 0, it returns $$q_1$$, if t = 1, it returns $$q_2$$. More...

static Quat< _Tp > squad (const Quat< _Tp > &q0, const Quat< _Tp > &s0, const Quat< _Tp > &s1, const Quat< _Tp > &q1, const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT, bool directChange=true)
To calculate the interpolation between $$q_0$$, $$q_1$$, $$q_2$$, $$q_3$$ by Spherical and quadrangle(Squad). This could be defined as:

$Squad(q_i, s_i, s_{i+1}, q_{i+1}, t) = Slerp(Slerp(q_i, q_{i+1}, t), Slerp(s_i, s_{i+1}, t), 2t(1-t))$

where

$s_i = q_i\exp(-\frac{\log(q^*_iq_{i+1}) + \log(q^*_iq_{i-1})}{4})$

_Tp w

_Tp x

_Tp y

_Tp z

## Static Public Attributes

static constexpr _Tp CV_QUAT_CONVERT_THRESHOLD = (_Tp)1.e-6

static constexpr _Tp CV_QUAT_EPS = (_Tp)1.e-6

## Friends

template<typename T >
Quat< T > acos (const Quat< T > &q)
return arccos value of quaternion q, arccos could be calculated as:

$\arccos(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arccosh(q)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > acosh (const Quat< T > &q)
return arccosh value of quaternion q, arccosh could be calculated as:

$arccosh(q) = \ln(q + \sqrt{q^2 - 1})$

template<typename T >
Quat< T > asin (const Quat< T > &q)
return arcsin value of quaternion q, arcsin could be calculated as:

$\arcsin(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arcsinh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > asinh (const Quat< T > &q)
return arcsinh value of quaternion q, arcsinh could be calculated as:

$arcsinh(q) = \ln(q + \sqrt{q^2 + 1})$

template<typename T >
Quat< T > atan (const Quat< T > &q)
return arctan value of quaternion q, arctan could be calculated as:

$\arctan(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arctanh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > atanh (const Quat< T > &q)
return arctanh value of quaternion q, arctanh could be calculated as:

$arctanh(q) = \frac{\ln(q + 1) - \ln(1 - q)}{2}$

template<typename T >
Quat< T > cos (const Quat< T > &q)
return sin value of quaternion q, cos could be calculated as:

$\cos(p) = \cos(w) * \cosh(||\boldsymbol{v}||) - \sin(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > cosh (const Quat< T > &q)
return cosh value of quaternion q, cosh could be calculated as:

$\cosh(p) = \cosh(w) * \cos(||\boldsymbol{v}||) + \sinh(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sin(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > crossProduct (const Quat< T > &p, const Quat< T > &q)
return the crossProduct between $$p = (a, b, c, d) = (a, \boldsymbol{u})$$ and $$q = (w, x, y, z) = (w, \boldsymbol{v})$$.

$p \times q = \frac{pq- qp}{2}$

$p \times q = \boldsymbol{u} \times \boldsymbol{v}$

$p \times q = (cz-dy)i + (dx-bz)j + (by-xc)k$

More...

template<typename T >
Quat< T > cv::operator* (const T s, const Quat< T > &)
Multiplication operator of a scalar and a quaternions. It multiplies right operand with the left operand and assign the result to left operand. More...

template<typename T >
Quat< T > cv::operator* (const Quat< T > &, const T s)
Multiplication operator of a quaternion and a scalar. It multiplies right operand with the left operand and assign the result to left operand. More...

template<typename T >
Quat< T > cv::operator+ (const T s, const Quat< T > &)
Addition operator of a quaternions and a scalar. Adds right hand operand from left hand operand. More...

template<typename T >
Quat< T > cv::operator+ (const Quat< T > &, const T s)
Addition operator of a quaternions and a scalar. Adds right hand operand from left hand operand. More...

template<typename T >
Quat< T > cv::operator- (const T s, const Quat< T > &)
Subtraction operator of a scalar and a quaternions. Subtracts right hand operand from left hand operand. More...

template<typename T >
Quat< T > cv::operator- (const Quat< T > &, const T s)
Subtraction operator of a quaternions and a scalar. Subtracts right hand operand from left hand operand. More...

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

template<typename T >
Quat< T > exp (const Quat< T > &q)
return the value of exponential value.

$\exp(q) = e^w (\cos||\boldsymbol{v}||+ \frac{v}{||\boldsymbol{v}||})\sin||\boldsymbol{v}||$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > inv (const Quat< T > &q, QuatAssumeType assumeUnit)
return $$q^{-1}$$ which is an inverse of $$q$$ which satisfies $$q * q^{-1} = 1$$. More...

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

$\ln(q) = \ln||q|| + \frac{\boldsymbol{v}}{||\boldsymbol{v}||}\arccos\frac{w}{||q||}.$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > power (const Quat< T > &q, const T x, QuatAssumeType assumeUnit)
return the value of power function with index $$x$$.

$q^x = ||q||(cos(x\theta) + \boldsymbol{u}sin(x\theta))).$

More...

template<typename T >
Quat< T > power (const Quat< T > &p, const Quat< T > &q, QuatAssumeType assumeUnit)
return the value of power function with quaternion $$q$$.

$p^q = e^{q\ln(p)}.$

More...

template<typename T >
Quat< T > sin (const Quat< T > &q)
return tanh value of quaternion q, sin could be calculated as:

$\sin(p) = \sin(w) * \cosh(||\boldsymbol{v}||) + \cos(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > sinh (const Quat< T > &q)
return sinh value of quaternion q, sinh could be calculated as:

$\sinh(p) = \sin(w)\cos(||\boldsymbol{v}||) + \cosh(w)\frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||$

where $$\boldsymbol{v} = [x, y, z].$$ More...

template<typename T >
Quat< T > sqrt (const Quat< T > &q, QuatAssumeType assumeUnit)
return $$\sqrt{q}$$. More...

template<typename T >
Quat< T > tan (const Quat< T > &q)
return tan value of quaternion q, tan could be calculated as:

$\tan(q) = \frac{\sin(q)}{\cos(q)}.$

More...

template<typename T >
Quat< T > tanh (const Quat< T > &q)
return tanh value of quaternion q, tanh could be calculated as:

$\tanh(q) = \frac{\sinh(q)}{\cosh(q)}.$

More...

## Detailed Description

### template<typename _Tp> class cv::Quat< _Tp >

Quaternion is a number system that extends the complex numbers. It can be expressed as a rotation in three-dimensional space. A quaternion is generally represented in the form:

$q = w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}$

$q = [w, x, y, z]$

$q = [w, \boldsymbol{v}]$

$q = ||q||[\cos\psi, u_x\sin\psi,u_y\sin\psi, u_z\sin\psi].$

$q = ||q||[\cos\psi, \boldsymbol{u}\sin\psi]$

where $$\psi = \frac{\theta}{2}$$, $$\theta$$ represents rotation angle, $$\boldsymbol{u} = [u_x, u_y, u_z]$$ represents normalized rotation axis, and $$||q||$$ represents the norm of $$q$$.

A unit quaternion is usually represents rotation, which has the form:

$q = [\cos\psi, u_x\sin\psi,u_y\sin\psi, u_z\sin\psi].$

To create a quaternion representing the rotation around the axis $$\boldsymbol{u}$$ with angle $$\theta$$, you can use

using namespace cv;
double angle = CV_PI;
Vec3d axis = {0, 0, 1};

You can simply use four same type number to create a quaternion

Quatd q(1, 2, 3, 4);

Or use a Vec4d or Vec4f vector.

Vec4d vec{1, 2, 3, 4};
Quatd q(vec);
Vec4f vec{1, 2, 3, 4};
Quatf q(vec);

If you already have a 3x3 rotation matrix R, then you can use

If you already have a rotation vector rvec which has the form of angle * axis, then you can use

To extract the rotation matrix from quaternion, see toRotMat3x3()

To extract the Vec4d or Vec4f, see toVec()

To extract the rotation vector, see toRotVec()

If there are two quaternions $$q_0, q_1$$ are needed to interpolate, you can use nlerp(), slerp() or spline()

Quatd::nlerp(q0, q1, t)
Quatd::slerp(q0, q1, t)
Quatd::spline(q0, q0, q1, q1, t)

spline can smoothly connect rotations of multiple quaternions

Three ways to get an element in Quaternion

Quatf q(1,2,3,4);
std::cout << q.w << std::endl; // w=1, x=2, y=3, z=4
std::cout << q[0] << std::endl; // q[0]=1, q[1]=2, q[2]=3, q[3]=4
std::cout << q.at(0) << std::endl;

## ◆ Quat() [1/3]

template<typename _Tp>
 cv::Quat< _Tp >::Quat ( )

## ◆ Quat() [2/3]

template<typename _Tp>
 cv::Quat< _Tp >::Quat ( const Vec< _Tp, 4 > & coeff )
explicit

From Vec4d or Vec4f.

## ◆ Quat() [3/3]

template<typename _Tp>
 cv::Quat< _Tp >::Quat ( _Tp w, _Tp x, _Tp y, _Tp z )

from four numbers.

## ◆ acos()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::acos ( ) const

return arccos value of this quaternion, arccos could be calculated as:

$\arccos(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arccosh(q)$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.acos();

## ◆ acosh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::acosh ( ) const

return arccosh value of this quaternion, arccosh could be calculated as:

$arcosh(q) = \ln(q + \sqrt{q^2 - 1})$

.

For example

Quatd q(1,2,3,4);
q.acosh();

## ◆ asin()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::asin ( ) const

return arcsin value of this quaternion, arcsin could be calculated as:

$\arcsin(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arcsinh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.asin();

## ◆ asinh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::asinh ( ) const

return arcsinh value of this quaternion, arcsinh could be calculated as:

$arcsinh(q) = \ln(q + \sqrt{q^2 + 1})$

.

For example

Quatd q(1,2,3,4);
q.asinh();

## ◆ assertNormal()

template<typename _Tp>
 void cv::Quat< _Tp >::assertNormal ( _Tp eps = CV_QUAT_EPS ) const

to throw an error if this quaternion is not a unit quaternion.

Parameters
 eps tolerance scope of normalization.
isNormal

## ◆ at()

template<typename _Tp>
 _Tp cv::Quat< _Tp >::at ( size_t index ) const

a way to get element.

Parameters
 index over a range [0, 3].

A quaternion q

q.at(0) is equivalent to q.w,

q.at(1) is equivalent to q.x,

q.at(2) is equivalent to q.y,

q.at(3) is equivalent to q.z.

## ◆ atan()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::atan ( ) const

return arctan value of this quaternion, arctan could be calculated as:

$\arctan(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arctanh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.atan();

## ◆ atanh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::atanh ( ) const

return arctanh value of this quaternion, arctanh could be calculated as:

$arcsinh(q) = \frac{\ln(q + 1) - \ln(1 - q)}{2}$

.

For example

Quatd q(1,2,3,4);
q.atanh();

## ◆ conjugate()

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

return the conjugate of this quaternion.

$q.conjugate() = (w, -x, -y, -z).$

## ◆ cos()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::cos ( ) const

return cos value of this quaternion, cos could be calculated as:

$\cos(p) = \cos(w) * \cosh(||\boldsymbol{v}||) - \sin(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.cos();

## ◆ cosh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::cosh ( ) const

return cosh value of this quaternion, cosh could be calculated as:

$\cosh(p) = \cosh(w) * \cos(||\boldsymbol{v}||) + \sinh(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}sin(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.cosh();

## ◆ createFromAngleAxis()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromAngleAxis ( const _Tp angle, const Vec< _Tp, 3 > & axis )
static

from an angle, axis. Axis will be normalized in this function. And it generates

$q = [\cos\psi, u_x\sin\psi,u_y\sin\psi, u_z\sin\psi].$

where $$\psi = \frac{\theta}{2}$$, $$\theta$$ is the rotation angle.

## ◆ createFromEulerAngles()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromEulerAngles ( const Vec< _Tp, 3 > & angles, QuatEnum::EulerAnglesType eulerAnglesType )
static

from Euler angles

A quaternion can be generated from Euler angles by combining the quaternion representations of the Euler rotations.

For example, if we use intrinsic rotations in the order of X-Y-Z, $$\theta_1$$ is rotation around the X-axis, $$\theta_2$$ is rotation around the Y-axis, $$\theta_3$$ is rotation around the Z-axis. The final quaternion q can be calculated by

${q} = q_{X, \theta_1} q_{Y, \theta_2} q_{Z, \theta_3}$

where $$q_{X, \theta_1}$$ is created from createFromXRot, $$q_{Y, \theta_2}$$ is created from createFromYRot, $$q_{Z, \theta_3}$$ is created from createFromZRot.

Parameters
 angles the Euler angles in a vector of length 3 eulerAnglesType the convertion Euler angles type

## ◆ createFromRotMat()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromRotMat ( InputArray R )
static

from a 3x3 rotation matrix.

## ◆ createFromRvec()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromRvec ( InputArray rvec )
static

from a rotation vector $$r$$ has the form $$\theta \cdot \boldsymbol{u}$$, where $$\theta$$ represents rotation angle and $$\boldsymbol{u}$$ represents normalized rotation axis.

Angle and axis could be easily derived as:

$$$\begin{split} \psi &= ||r||\\ \boldsymbol{u} &= \frac{r}{\theta} \end{split}$$$

Then a quaternion can be calculated by

$q = [\cos\psi, \boldsymbol{u}\sin\psi]$

where $$\psi = \theta / 2$$

## ◆ createFromXRot()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromXRot ( const _Tp theta )
static

get a quaternion from a rotation about the X-axis by $$\theta$$ .

$q = \cos(\theta/2)+sin(\theta/2) i +0 j +0 k$

## ◆ createFromYRot()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromYRot ( const _Tp theta )
static

get a quaternion from a rotation about the Y-axis by $$\theta$$ .

$q = \cos(\theta/2)+0 i+ sin(\theta/2) j +0k$

## ◆ createFromZRot()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::createFromZRot ( const _Tp theta )
static

get a quaternion from a rotation about the Z-axis by $$\theta$$.

$q = \cos(\theta/2)+0 i +0 j +sin(\theta/2) k$

## ◆ crossProduct()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::crossProduct ( const Quat< _Tp > & q ) const

return the crossProduct between $$p = (a, b, c, d) = (a, \boldsymbol{u})$$ and $$q = (w, x, y, z) = (w, \boldsymbol{v})$$.

$p \times q = \frac{pq- qp}{2}.$

$p \times q = \boldsymbol{u} \times \boldsymbol{v}.$

$p \times q = (cz-dy)i + (dx-bz)j + (by-xc)k.$

For example

Quatd q{1,2,3,4};
Quatd p{5,6,7,8};
p.crossProduct(q)

## ◆ dot()

template<typename _Tp>
 _Tp cv::Quat< _Tp >::dot ( Quat< _Tp > q ) const

return the dot between quaternion $$q$$ and this quaternion.

dot(p, q) is a good metric of how close the quaternions are. Indeed, consider the unit quaternion difference $$p^{-1} * q$$, its real part is dot(p, q). At the same time its real part is equal to $$\cos(\beta/2)$$ where $$\beta$$ is an angle of rotation between p and q, i.e., Therefore, the closer dot(p, q) to 1, the smaller rotation between them.

$p \cdot q = p.w \cdot q.w + p.x \cdot q.x + p.y \cdot q.y + p.z \cdot q.z$

Parameters
 q the other quaternion.

For example

Quatd q(1,2,3,4);
Quatd p(5,6,7,8);
p.dot(q);

## ◆ exp()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::exp ( ) const

return the value of exponential value.

$\exp(q) = e^w (\cos||\boldsymbol{v}||+ \frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q{1,2,3,4};
cout << q.exp() << endl;

## ◆ getAngle()

template<typename _Tp>
 _Tp cv::Quat< _Tp >::getAngle ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

get the angle of quaternion, it returns the rotation angle.

Parameters
 assumeUnit if QUAT_ASSUME_UNIT, this quaternion assume to be a unit quaternion and this function will save some computations. $\psi = 2 *arccos(\frac{w}{||q||})$

For example

Quatd q(1,2,3,4);
q.getAngle();
q.normalize().getAngle(assumeUnit);//same as q.getAngle().
Note
It always return the value between $$[0, 2\pi]$$.

## ◆ getAxis()

template<typename _Tp>
 Vec<_Tp, 3> cv::Quat< _Tp >::getAxis ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

get the axis of quaternion, it returns a vector of length 3.

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

the unit axis $$\boldsymbol{u}$$ is defined by

$$$\begin{split} \boldsymbol{v} &= \boldsymbol{u} ||\boldsymbol{v}||\\ &= \boldsymbol{u}||q||sin(\frac{\theta}{2}) \end{split}$$$

where $$v=[x, y ,z]$$ and $$\theta$$ represents rotation angle.

For example

Quatd q(1,2,3,4);
q.getAxis();
q.normalize().getAxis(assumeUnit);//same as q.getAxis()

## ◆ interPoint()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::interPoint ( const Quat< _Tp > & q0, const Quat< _Tp > & q1, const Quat< _Tp > & q2, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )
static

This is the part calculation of squad. To calculate the intermedia quaternion $$s_i$$ between each three quaternion

$s_i = q_i\exp(-\frac{\log(q^*_iq_{i+1}) + \log(q^*_iq_{i-1})}{4}).$

.

Parameters
 q0 the first quaternion. q1 the second quaternion. q2 the third quaternion. assumeUnit if QUAT_ASSUME_UNIT, all input quaternions assume to be unit quaternion. Otherwise, all input quaternions will be normalized inside the function.

## ◆ inv()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::inv ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

return $$q^{-1}$$ which is an inverse of $$q$$ satisfying $$q * q^{-1} = 1$$.

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

For example

Quatd q(1,2,3,4);
q.inv();
q = q.normalize();
q.inv(assumeUnit); //assumeUnit means p is a unit quaternion

## ◆ isNormal()

template<typename _Tp>
 bool cv::Quat< _Tp >::isNormal ( _Tp eps = CV_QUAT_EPS ) const

return true if this quaternion is a unit quaternion.

Parameters
 eps tolerance scope of normalization. The eps could be defined as

$eps = |1 - dotValue|$

where

$dotValue = (this.w^2 + this.x^2 + this,y^2 + this.z^2).$

And this function will consider it is normalized when the dotValue over a range $$[1-eps, 1+eps]$$.

## ◆ lerp()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::lerp ( const Quat< _Tp > & q0, const Quat< _Tp > & q1, const _Tp t )
static

To calculate the interpolation from $$q_0$$ to $$q_1$$ by Linear Interpolation(Nlerp) For two quaternions, this interpolation curve can be displayed as:

$Lerp(q_0, q_1, t) = (1 - t)q_0 + tq_1.$

Obviously, the lerp will interpolate along a straight line if we think of $$q_0$$ and $$q_1$$ as a vector in a two-dimensional space. When $$t = 0$$, it returns $$q_0$$ and when $$t= 1$$, it returns $$q_1$$. $$t$$ should to be ranged in $$[0, 1]$$ normally.

Parameters
 q0 a quaternion used in linear interpolation. q1 a quaternion used in linear interpolation. t percent of vector $$\overrightarrow{q_0q_1}$$ over a range [0, 1].
Note
it returns a non-unit quaternion.

## ◆ log()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::log ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

return the value of logarithm function.

$\ln(q) = \ln||q|| + \frac{\boldsymbol{v}}{||\boldsymbol{v}||}\arccos\frac{w}{||q||}$

. where $$\boldsymbol{v} = [x, y, z].$$

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

For example

Quatd q(1,2,3,4);
q.log();
Quatd q1(1,2,3,4);
q1.normalize().log(assumeUnit);

## ◆ nlerp()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::nlerp ( const Quat< _Tp > & q0, const Quat< _Tp > & q1, const _Tp t, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )
static

To calculate the interpolation from $$q_0$$ to $$q_1$$ by Normalized Linear Interpolation(Nlerp). it returns a normalized quaternion of Linear Interpolation(Lerp).

$Nlerp(q_0, q_1, t) = \frac{(1 - t)q_0 + tq_1}{||(1 - t)q_0 + tq_1||}.$

The interpolation will always choose the shortest path but the constant speed is not guaranteed.

Parameters
 q0 a quaternion used in normalized linear interpolation. q1 a quaternion used in normalized linear interpolation. t percent of vector $$\overrightarrow{q_0q_1}$$ over a range [0, 1]. assumeUnit if QUAT_ASSUME_UNIT, all input quaternions assume to be unit quaternion. Otherwise, all inputs quaternion will be normalized inside the function.
lerp

## ◆ norm()

template<typename _Tp>
 _Tp cv::Quat< _Tp >::norm ( ) const

return the norm of quaternion.

$||q|| = \sqrt{w^2 + x^2 + y^2 + z^2}.$

## ◆ normalize()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::normalize ( ) const

return a normalized $$p$$.

$p = \frac{q}{||q||}$

where $$p$$ satisfies $$(p.x)^2 + (p.y)^2 + (p.z)^2 + (p.w)^2 = 1.$$

## ◆ operator*()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator* ( const Quat< _Tp > & ) const

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

Rule of quaternion multiplication:

$$$\begin{split} p * q &= [p_0, \boldsymbol{u}]*[q_0, \boldsymbol{v}]\\ &=[p_0q_0 - \boldsymbol{u}\cdot \boldsymbol{v}, p_0\boldsymbol{v} + q_0\boldsymbol{u}+ \boldsymbol{u}\times \boldsymbol{v}]. \end{split}$$$

where $$\cdot$$ means dot product and $$\times$$ means cross product.

For example

Quatd p{1, 2, 3, 4};
Quatd q{5, 6, 7, 8};
std::cout << p * q << std::endl; //[-60, 12, 30, 24]

## ◆ operator*=() [1/2]

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _Tp >::operator*= ( const Quat< _Tp > & )

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

Rule of quaternion multiplication:

$$$\begin{split} p * q &= [p_0, \boldsymbol{u}]*[q_0, \boldsymbol{v}]\\ &=[p_0q_0 - \boldsymbol{u}\cdot \boldsymbol{v}, p_0\boldsymbol{v} + q_0\boldsymbol{u}+ \boldsymbol{u}\times \boldsymbol{v}]. \end{split}$$$

where $$\cdot$$ means dot product and $$\times$$ means cross product.

For example

Quatd p{1, 2, 3, 4};
Quatd q{5, 6, 7, 8};
p *= q; // equivalent to p = p * q
std::cout << p << std::endl; //[-60, 12, 30, 24]

## ◆ operator*=() [2/2]

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _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 quaternion multiplication with a scalar:

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

For example

Quatd p{1, 2, 3, 4};
double s = 2.0;
p *= s; // equivalent to p = p * s
std::cout << p << std::endl; //[2.0, 4.0, 6.0, 8.0]
Note
the type of scalar should be equal to the quaternion.

## ◆ operator+()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator+ ( const Quat< _Tp > & ) const

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

For example

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

## ◆ operator+=()

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _Tp >::operator+= ( const Quat< _Tp > & )

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

For example

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

## ◆ operator-() [1/2]

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator- ( ) const

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

For example

Quatd q{1, 2, 3, 4};
std::cout << -q << std::endl; // [-1, -2, -3, -4]

## ◆ operator-() [2/2]

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator- ( const Quat< _Tp > & ) const

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

For example

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

## ◆ operator-=()

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _Tp >::operator-= ( const Quat< _Tp > & )

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

For example

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

## ◆ operator/() [1/2]

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator/ ( const _Tp s ) const

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

Rule of quaternion division with a scalar:

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

For example

Quatd p{1, 2, 3, 4};
double s = 2.0;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1, 1.5, 2]
Note
the type of scalar should be equal to this quaternion.

## ◆ operator/() [2/2]

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::operator/ ( const Quat< _Tp > & ) const

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

Rule of quaternion division with a scalar:

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

For example

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

## ◆ operator/=() [1/2]

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _Tp >::operator/= ( const _Tp s )

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

Rule of quaternion division with a scalar:

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

For example

Quatd p{1, 2, 3, 4};
double s = 2.0;;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1.0, 1.5, 2.0]
Note
the type of scalar should be equal to the quaternion.

## ◆ operator/=() [2/2]

template<typename _Tp>
 Quat<_Tp>& cv::Quat< _Tp >::operator/= ( const Quat< _Tp > & )

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

Rule of quaternion division with a quaternion:

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

For example

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

## ◆ operator==()

template<typename _Tp>
 bool cv::Quat< _Tp >::operator== ( const Quat< _Tp > & ) const

return true if two quaternions p and q are nearly equal, i.e. when the absolute value of each $$p_i$$ and $$q_i$$ is less than CV_QUAT_EPS.

## ◆ operator[]() [1/2]

template<typename _Tp>
 _Tp& cv::Quat< _Tp >::operator[] ( std::size_t n )

## ◆ operator[]() [2/2]

template<typename _Tp>
 const _Tp& cv::Quat< _Tp >::operator[] ( std::size_t n ) const

## ◆ power() [1/2]

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::power ( const _Tp x, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

return the value of power function with index $$x$$.

$q^x = ||q||(\cos(x\theta) + \boldsymbol{u}\sin(x\theta))).$

Parameters
 x index of exponentiation. assumeUnit if QUAT_ASSUME_UNIT, this quaternion assume to be a unit quaternion and this function will save some computations.

For example

Quatd q(1,2,3,4);
q.power(2.0);
double angle = CV_PI;
Vec3d axis{0, 0, 1};
Quatd q1 = Quatd::createFromAngleAxis(angle, axis); //generate a unit quat by axis and angle
q1.power(2.0, assumeUnit); //This assumeUnt means q1 is a unit quaternion

## ◆ power() [2/2]

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

return the value of power function with quaternion $$q$$.

$p^q = e^{q\ln(p)}.$

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

For example

Quatd p(1,2,3,4);
Quatd q(5,6,7,8);
p.power(q);
p = p.normalize();
p.power(q, assumeUnit); //This assumeUnit means p is a unit quaternion

## ◆ sin()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::sin ( ) const

return sin value of this quaternion, sin could be calculated as:

$\sin(p) = \sin(w) * \cosh(||\boldsymbol{v}||) + \cos(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.sin();

## ◆ sinh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::sinh ( ) const

return sinh value of this quaternion, sinh could be calculated as: $$\sinh(p) = \sin(w)\cos(||\boldsymbol{v}||) + \cosh(w)\frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||$$ where $$\boldsymbol{v} = [x, y, z].$$

For example

Quatd q(1,2,3,4);
q.sinh();

## ◆ slerp()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::slerp ( const Quat< _Tp > & q0, const Quat< _Tp > & q1, const _Tp t, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT, bool directChange = true )
static

To calculate the interpolation between $$q_0$$ and $$q_1$$ by Spherical Linear Interpolation(Slerp), which can be defined as:

$Slerp(q_0, q_1, t) = \frac{\sin((1-t)\theta)}{\sin(\theta)}q_0 + \frac{\sin(t\theta)}{\sin(\theta)}q_1$

where $$\theta$$ can be calculated as:

$\theta=cos^{-1}(q_0\cdot q_1)$

resulting from the both of their norm is unit.

Parameters
 q0 a quaternion used in Slerp. q1 a quaternion used in Slerp. t percent of angle between $$q_0$$ and $$q_1$$ over a range [0, 1]. assumeUnit if QUAT_ASSUME_UNIT, all input quaternions assume to be unit quaternions. Otherwise, all input quaternions will be normalized inside the function. directChange if QUAT_ASSUME_UNIT, the interpolation will choose the nearest path.
Note
If the interpolation angle is small, the error between Nlerp and Slerp is not so large. To improve efficiency and avoid zero division error, we use Nlerp instead of Slerp.

## ◆ spline()

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::spline ( const Quat< _Tp > & q0, const Quat< _Tp > & q1, const Quat< _Tp > & q2, const Quat< _Tp > & q3, const _Tp t, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )
static

to calculate a quaternion which is the result of a $$C^1$$ continuous spline curve constructed by squad at the ratio t. Here, the interpolation values are between $$q_1$$ and $$q_2$$. $$q_0$$ and $$q_2$$ are used to ensure the $$C^1$$ continuity. if t = 0, it returns $$q_1$$, if t = 1, it returns $$q_2$$.

Parameters
 q0 the first input quaternion to ensure $$C^1$$ continuity. q1 the second input quaternion. q2 the third input quaternion. q3 the fourth input quaternion the same use of $$q1$$. t ratio over a range [0, 1]. assumeUnit if QUAT_ASSUME_UNIT, $$q_0, q_1, q_2, q_3$$ assume to be unit quaternion. Otherwise, all input quaternions will be normalized inside the function.

For example:

If there are three double quaternions $$v_0, v_1, v_2$$ waiting to be interpolated.

Interpolation between $$v_0$$ and $$v_1$$ with a ratio $$t_0$$ could be calculated as

Quatd::spline(v0, v0, v1, v2, t0);

Interpolation between $$v_1$$ and $$v_2$$ with a ratio $$t_0$$ could be calculated as

Quatd::spline(v0, v1, v2, v2, t0);

## ◆ sqrt()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::sqrt ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

return $$\sqrt{q}$$.

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

For example

Quatf q(1,2,3,4);
q.sqrt();
q = {1,0,0,0};
q.sqrt(assumeUnit); //This assumeUnit means q is a unit quaternion

template<typename _Tp>
 static Quat<_Tp> cv::Quat< _Tp >::squad ( const Quat< _Tp > & q0, const Quat< _Tp > & s0, const Quat< _Tp > & s1, const Quat< _Tp > & q1, const _Tp t, QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT, bool directChange = true )
static

To calculate the interpolation between $$q_0$$, $$q_1$$, $$q_2$$, $$q_3$$ by Spherical and quadrangle(Squad). This could be defined as:

$Squad(q_i, s_i, s_{i+1}, q_{i+1}, t) = Slerp(Slerp(q_i, q_{i+1}, t), Slerp(s_i, s_{i+1}, t), 2t(1-t))$

where

$s_i = q_i\exp(-\frac{\log(q^*_iq_{i+1}) + \log(q^*_iq_{i-1})}{4})$

.

The Squad expression is analogous to the $$B\acute{e}zier$$ curve, but involves spherical linear interpolation instead of simple linear interpolation. Each $$s_i$$ needs to be calculated by three quaternions.

Parameters
 q0 the first quaternion. s0 the second quaternion. s1 the third quaternion. q1 thr fourth quaternion. t interpolation parameter of quadratic and linear interpolation over a range $$[0, 1]$$. assumeUnit if QUAT_ASSUME_UNIT, all input quaternions assume to be unit quaternion. Otherwise, all input quaternions will be normalized inside the function. directChange if QUAT_ASSUME_UNIT, squad will find the nearest path to interpolate.
interPoint, spline

## ◆ tan()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::tan ( ) const

return tan value of this quaternion, tan could be calculated as:

$\tan(q) = \frac{\sin(q)}{\cos(q)}.$

For example

Quatd q(1,2,3,4);
q.tan();

## ◆ tanh()

template<typename _Tp>
 Quat<_Tp> cv::Quat< _Tp >::tanh ( ) const

return tanh value of this quaternion, tanh could be calculated as:

$\tanh(q) = \frac{\sinh(q)}{\cosh(q)}.$

For example

Quatd q(1,2,3,4);
q.tanh();
sinh, cosh

## ◆ toEulerAngles()

template<typename _Tp>
 Vec<_Tp, 3> cv::Quat< _Tp >::toEulerAngles ( QuatEnum::EulerAnglesType eulerAnglesType )

Transform a quaternion q to Euler angles.

When transforming a quaternion $$q = w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}$$ to Euler angles, rotation matrix M can be calculated by:

\begin{aligned} {M} &={\begin{bmatrix}1-2(y^{2}+z^{2})&2(xy-zx)&2(xz+yw)\\2(xy+zw)&1-2(x^{2}+z^{2})&2(yz-xw)\\2(xz-yw)&2(yz+xw)&1-2(x^{2}+y^{2})\end{bmatrix}}\end{aligned}.

On the other hand, the rotation matrix can be obtained from Euler angles. Using intrinsic rotations with Euler angles type XYZ as an example, $$\theta_1$$, $$\theta_2$$, $$\theta_3$$ are three angles for Euler angles, the rotation matrix R can be calculated by:

$R =X(\theta_1)Y(\theta_2)Z(\theta_3) ={\begin{bmatrix}\cos\theta_{2}\cos\theta_{3}&-\cos\theta_{2}\sin\theta_{3}&\sin\theta_{2}\\\cos\theta_{1}\sin\theta_{3}+\cos\theta_{3}\sin\theta_{1}\sin\theta_{2}&\cos\theta_{1}\cos\theta_{3}-\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}&-\cos\theta_{2}\sin\theta_{1}\\\sin\theta_{1}\sin\theta_{3}-\cos\theta_{1}\cos\theta_{3}\sin\theta_{2}&\cos\theta_{3}\sin\theta_{1}+\cos\theta_{1}\sin\theta_{2}\sin\theta_{3}&\cos\theta_{1}\cos_{2}\end{bmatrix}}$

Rotation matrix M and R are equal. As long as $$s_{2} \neq 1$$, by comparing each element of two matrices ,the solution is $$\begin{cases} \theta_1 = \arctan2(-m_{23},m_{33})\\\theta_2 = arcsin(m_{13}) \\\theta_3 = \arctan2(-m_{12},m_{11}) \end{cases}$$.

When $$s_{2}=1$$ or $$s_{2}=-1$$, the gimbal lock occurs. The function will prompt "WARNING: Gimbal Lock will occur. Euler angles is non-unique. For intrinsic rotations, we set the third angle to 0, and for external rotation, we set the first angle to 0.".

When $$s_{2}=1$$ , The rotation matrix R is $$R = {\begin{bmatrix}0&0&1\\\sin(\theta_1+\theta_3)&\cos(\theta_1+\theta_3)&0\\-\cos(\theta_1+\theta_3)&\sin(\theta_1+\theta_3)&0\end{bmatrix}}$$.

The number of solutions is infinite with the condition $$\begin{cases} \theta_1+\theta_3 = \arctan2(m_{21},m_{22})\\ \theta_2=\pi/2 \end{cases}\$$.

We set $$\theta_3 = 0$$, the solution is $$\begin{cases} \theta_1=\arctan2(m_{21},m_{22})\\ \theta_2=\pi/2\\ \theta_3=0 \end{cases}$$.

When $$s_{2}=-1$$, The rotation matrix R is $$X_{1}Y_{2}Z_{3}={\begin{bmatrix}0&0&-1\\-\sin(\theta_1-\theta_3)&\cos(\theta_1-\theta_3)&0\\\cos(\theta_1-\theta_3)&\sin(\theta_1-\theta_3)&0\end{bmatrix}}$$.

The number of solutions is infinite with the condition $$\begin{cases} \theta_1+\theta_3 = \arctan2(m_{32},m_{22})\\ \theta_2=\pi/2 \end{cases}\$$.

We set $$\theta_3 = 0$$, the solution is $$\begin{cases}\theta_1=\arctan2(m_{32},m_{22}) \\ \theta_2=-\pi/2\\ \theta_3=0\end{cases}$$.

Since $$sin \theta\in [-1,1]$$ and $$cos \theta \in [-1,1]$$, the unnormalized quaternion will cause computational troubles. For this reason, this function will normalize the quaternion at first and QuatAssumeType is not needed.

When the gimbal lock occurs, we set $$\theta_3 = 0$$ for intrinsic rotations or $$\theta_1 = 0$$ for extrinsic rotations.

As a result, for every Euler angles type, we can get solution as shown in the following table.

EulerAnglesType Ordinary $$\theta_2 = Ï€/2$$ $$\theta_2 = -Ï€/2$$
INT_XYZ$$\theta_1 = \arctan2(-m_{23},m_{33})\\\theta_2 = \arcsin(m_{13}) \\\theta_3= \arctan2(-m_{12},m_{11})$$$$\theta_1=\arctan2(m_{21},m_{22})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(m_{32},m_{22})\\ \theta_2=-\pi/2\\ \theta_3=0$$
INT_XZY$$\theta_1 = \arctan2(m_{32},m_{22})\\\theta_2 = -\arcsin(m_{12}) \\\theta_3= \arctan2(m_{13},m_{11})$$$$\theta_1=\arctan2(m_{31},m_{33})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(-m_{23},m_{33})\\ \theta_2=-\pi/2\\ \theta_3=0$$
INT_YXZ$$\theta_1 = \arctan2(m_{13},m_{33})\\\theta_2 = -\arcsin(m_{23}) \\\theta_3= \arctan2(m_{21},m_{22})$$$$\theta_1=\arctan2(m_{12},m_{11})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(-m_{12},m_{11})\\ \theta_2=-\pi/2\\ \theta_3=0$$
INT_YZX$$\theta_1 = \arctan2(-m_{31},m_{11})\\\theta_2 = \arcsin(m_{21}) \\\theta_3= \arctan2(-m_{23},m_{22})$$$$\theta_1=\arctan2(m_{13},m_{33})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(m_{13},m_{12})\\ \theta_2=-\pi/2\\ \theta_3=0$$
INT_ZXY$$\theta_1 = \arctan2(-m_{12},m_{22})\\\theta_2 = \arcsin(m_{32}) \\\theta_3= \arctan2(-m_{31},m_{33})$$$$\theta_1=\arctan2(m_{21},m_{11})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(m_{21},m_{11})\\ \theta_2=-\pi/2\\ \theta_3=0$$
INT_ZYX$$\theta_1 = \arctan2(m_{21},m_{11})\\\theta_2 = \arcsin(-m_{31}) \\\theta_3= \arctan2(m_{32},m_{33})$$$$\theta_1=\arctan2(m_{23},m_{22})\\ \theta_2=\pi/2\\ \theta_3=0$$$$\theta_1=\arctan2(-m_{12},m_{22})\\ \theta_2=-\pi/2\\ \theta_3=0$$
EXT_XYZ$$\theta_1 = \arctan2(m_{32},m_{33})\\\theta_2 = \arcsin(-m_{31}) \\\ \theta_3 = \arctan2(m_{21},m_{11})$$$$\theta_1= 0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{23},m_{22})$$$$\theta_1=0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(-m_{12},m_{22})$$
EXT_XZY$$\theta_1 = \arctan2(-m_{23},m_{22})\\\theta_2 = \arcsin(m_{21}) \\\theta_3= \arctan2(-m_{31},m_{11})$$$$\theta_1= 0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{13},m_{33})$$$$\theta_1=0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(m_{13},m_{12})$$
EXT_YXZ$$\theta_1 = \arctan2(-m_{31},m_{33}) \\\theta_2 = \arcsin(m_{32}) \\\theta_3= \arctan2(-m_{12},m_{22})$$$$\theta_1= 0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{21},m_{11})$$$$\theta_1=0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(m_{21},m_{11})$$
EXT_YZX$$\theta_1 = \arctan2(m_{13},m_{11})\\\theta_2 = -\arcsin(m_{12}) \\\theta_3= \arctan2(m_{32},m_{22})$$$$\theta_1= 0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{31},m_{33})$$$$\theta_1=0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(-m_{23},m_{33})$$
EXT_ZXY$$\theta_1 = \arctan2(m_{21},m_{22})\\\theta_2 = -\arcsin(m_{23}) \\\theta_3= \arctan2(m_{13},m_{33})$$$$\theta_1= 0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{12},m_{11})$$$$\theta_1= 0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(-m_{12},m_{11})$$
EXT_ZYX$$\theta_1 = \arctan2(-m_{12},m_{11})\\\theta_2 = \arcsin(m_{13}) \\\theta_3= \arctan2(-m_{23},m_{33})$$$$\theta_1=0\\ \theta_2=\pi/2\\ \theta_3=\arctan2(m_{21},m_{22})$$$$\theta_1=0\\ \theta_2=-\pi/2\\ \theta_3=\arctan2(m_{32},m_{22})$$
EulerAnglesType Ordinary $$\theta_2 = 0$$ $$\theta_2 = Ï€$$
INT_XYX$$\theta_1 = \arctan2(m_{21},-m_{31})\\\theta_2 =\arccos(m_{11}) \\\theta_3 = \arctan2(m_{12},m_{13})$$$$\theta_1=\arctan2(m_{32},m_{33})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(m_{23},m_{22})\\ \theta_2=\pi\\ \theta_3=0$$
INT_XZX$$\theta_1 = \arctan2(m_{31},m_{21})\\\theta_2 = \arccos(m_{11}) \\\theta_3 = \arctan2(m_{13},-m_{12})$$$$\theta_1=\arctan2(m_{32},m_{33})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(-m_{32},m_{33})\\ \theta_2=\pi\\ \theta_3=0$$
INT_YXY$$\theta_1 = \arctan2(m_{12},m_{32})\\\theta_2 = \arccos(m_{22}) \\\theta_3 = \arctan2(m_{21},-m_{23})$$$$\theta_1=\arctan2(m_{13},m_{11})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(-m_{31},m_{11})\\ \theta_2=\pi\\ \theta_3=0$$
INT_YZY$$\theta_1 = \arctan2(m_{32},-m_{12})\\\theta_2 = \arccos(m_{22}) \\\theta_3 =\arctan2(m_{23},m_{21})$$$$\theta_1=\arctan2(m_{13},m_{11})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(m_{13},-m_{11})\\ \theta_2=\pi\\ \theta_3=0$$
INT_ZXZ$$\theta_1 = \arctan2(-m_{13},m_{23})\\\theta_2 = \arccos(m_{33}) \\\theta_3 =\arctan2(m_{31},m_{32})$$$$\theta_1=\arctan2(m_{21},m_{22})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(m_{21},m_{11})\\ \theta_2=\pi\\ \theta_3=0$$
INT_ZYZ$$\theta_1 = \arctan2(m_{23},m_{13})\\\theta_2 = \arccos(m_{33}) \\\theta_3 = \arctan2(m_{32},-m_{31})$$$$\theta_1=\arctan2(m_{21},m_{11})\\ \theta_2=0\\ \theta_3=0$$$$\theta_1=\arctan2(m_{21},m_{11})\\ \theta_2=\pi\\ \theta_3=0$$
EXT_XYX$$\theta_1 = \arctan2(m_{12},m_{13}) \\\theta_2 = \arccos(m_{11}) \\\theta_3 = \arctan2(m_{21},-m_{31})$$$$\theta_1=0\\ \theta_2=0\\ \theta_3=\arctan2(m_{32},m_{33})$$$$\theta_1= 0\\ \theta_2=\pi\\ \theta_3= \arctan2(m_{23},m_{22})$$
EXT_XZX$$\theta_1 = \arctan2(m_{13},-m_{12})\\\theta_2 = \arccos(m_{11}) \\\theta_3 = \arctan2(m_{31},m_{21})$$$$\theta_1= 0\\ \theta_2=0\\ \theta_3=\arctan2(m_{32},m_{33})$$$$\theta_1= 0\\ \theta_2=\pi\\ \theta_3=\arctan2(-m_{32},m_{33})$$
EXT_YXY$$\theta_1 = \arctan2(m_{21},-m_{23})\\\theta_2 = \arccos(m_{22}) \\\theta_3 = \arctan2(m_{12},m_{32})$$$$\theta_1= 0\\ \theta_2=0\\ \theta_3=\arctan2(m_{13},m_{11})$$$$\theta_1= 0\\ \theta_2=\pi\\ \theta_3=\arctan2(-m_{31},m_{11})$$
EXT_YZY$$\theta_1 = \arctan2(m_{23},m_{21}) \\\theta_2 = \arccos(m_{22}) \\\theta_3 = \arctan2(m_{32},-m_{12})$$$$\theta_1= 0\\ \theta_2=0\\ \theta_3=\arctan2(m_{13},m_{11})$$$$\theta_1=0\\ \theta_2=\pi\\ \theta_3=\arctan2(m_{13},-m_{11})$$
EXT_ZXZ$$\theta_1 = \arctan2(m_{31},m_{32}) \\\theta_2 = \arccos(m_{33}) \\\theta_3 = \arctan2(-m_{13},m_{23})$$$$\theta_1=0\\ \theta_2=0\\ \theta_3=\arctan2(m_{21},m_{22})$$$$\theta_1= 0\\ \theta_2=\pi\\ \theta_3=\arctan2(m_{21},m_{11})$$
EXT_ZYZ$$\theta_1 = \arctan2(m_{32},-m_{31})\\\theta_2 = \arccos(m_{33}) \\\theta_3 = \arctan2(m_{23},m_{13})$$$$\theta_1=0\\ \theta_2=0\\ \theta_3=\arctan2(m_{21},m_{11})$$$$\theta_1= 0\\ \theta_2=\pi\\ \theta_3=\arctan2(m_{21},m_{11})$$
Parameters
 eulerAnglesType the convertion Euler angles type

## ◆ toRotMat3x3()

template<typename _Tp>
 Matx<_Tp, 3, 3> cv::Quat< _Tp >::toRotMat3x3 ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

transform a quaternion to a 3x3 rotation matrix.

Parameters
 assumeUnit if QUAT_ASSUME_UNIT, this quaternion assume to be a unit quaternion and this function will save some computations. Otherwise, this function will normalize this quaternion at first then do the transformation.
Note
Matrix A which is to be rotated should have the form

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

where the same subscript represents a point. The shape of A assume to be [3, n] The points matrix A can be rotated by toRotMat3x3() * A. The result has 3 rows and n columns too.

For example

double angle = CV_PI;
Vec3d axis{0,0,1};
Quatd q_unit = Quatd::createFromAngleAxis(angle, axis); //quaternion could also be get by interpolation by two or more quaternions.
//assume there is two points (1,0,0) and (1,0,1) to be rotated
Mat pointsA = (Mat_<double>(2, 3) << 1,0,0,1,0,1);
//change the shape
pointsA = pointsA.t();
// rotate 180 degrees around the z axis
Mat new_point = q_unit.toRotMat3x3() * pointsA;
// print two points
cout << new_point << endl;

## ◆ toRotMat4x4()

template<typename _Tp>
 Matx<_Tp, 4, 4> cv::Quat< _Tp >::toRotMat4x4 ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

transform a quaternion to a 4x4 rotation matrix.

Parameters
 assumeUnit if QUAT_ASSUME_UNIT, this quaternion assume to be a unit quaternion and this function will save some computations. Otherwise, this function will normalize this quaternion at first then do the transformation.

The operations is similar as toRotMat3x3 except that the points matrix should have the form

$\begin{bmatrix} x_0& x_1& x_2&...&x_n\\ y_0& y_1& y_2&...&y_n\\ z_0& z_1& z_2&...&z_n\\ 0&0&0&...&0 \end{bmatrix}$

toRotMat3x3

## ◆ toRotVec()

template<typename _Tp>
 Vec<_Tp, 3> cv::Quat< _Tp >::toRotVec ( QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT ) const

transform this quaternion to a Rotation vector.

Parameters
 assumeUnit if QUAT_ASSUME_UNIT, this quaternion assume to be a unit quaternion and this function will save some computations. Rotation vector rVec is defined as: $rVec = [\theta v_x, \theta v_y, \theta v_z]$ where $$\theta$$ represents rotation angle, and $$\boldsymbol{v}$$ represents the normalized rotation axis.

For example

Quatd q(1,2,3,4);
q.toRotVec();
q.normalize().toRotVec(assumeUnit); //answer is same as q.toRotVec().

## ◆ toVec()

template<typename _Tp>
 Vec<_Tp, 4> cv::Quat< _Tp >::toVec ( ) const

transform the this quaternion to a Vec<T, 4>.

For example

Quatd q(1,2,3,4);
q.toVec();

## ◆ acos

template<typename _Tp>
template<typename T >
 Quat acos ( const Quat< T > & q )
friend

return arccos value of quaternion q, arccos could be calculated as:

$\arccos(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arccosh(q)$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
acos(q);

## ◆ acosh

template<typename _Tp>
template<typename T >
 Quat acosh ( const Quat< T > & q )
friend

return arccosh value of quaternion q, arccosh could be calculated as:

$arccosh(q) = \ln(q + \sqrt{q^2 - 1})$

.

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
acosh(q);

## ◆ asin

template<typename _Tp>
template<typename T >
 Quat asin ( const Quat< T > & q )
friend

return arcsin value of quaternion q, arcsin could be calculated as:

$\arcsin(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arcsinh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
asin(q);

## ◆ asinh

template<typename _Tp>
template<typename T >
 Quat asinh ( const Quat< T > & q )
friend

return arcsinh value of quaternion q, arcsinh could be calculated as:

$arcsinh(q) = \ln(q + \sqrt{q^2 + 1})$

.

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
asinh(q);

## ◆ atan

template<typename _Tp>
template<typename T >
 Quat atan ( const Quat< T > & q )
friend

return arctan value of quaternion q, arctan could be calculated as:

$\arctan(q) = -\frac{\boldsymbol{v}}{||\boldsymbol{v}||}arctanh(q\frac{\boldsymbol{v}}{||\boldsymbol{v}||})$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
atan(q);

## ◆ atanh

template<typename _Tp>
template<typename T >
 Quat atanh ( const Quat< T > & q )
friend

return arctanh value of quaternion q, arctanh could be calculated as:

$arctanh(q) = \frac{\ln(q + 1) - \ln(1 - q)}{2}$

.

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
atanh(q);

## ◆ cos

template<typename _Tp>
template<typename T >
 Quat cos ( const Quat< T > & q )
friend

return sin value of quaternion q, cos could be calculated as:

$\cos(p) = \cos(w) * \cosh(||\boldsymbol{v}||) - \sin(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
cos(q);

## ◆ cosh

template<typename _Tp>
template<typename T >
 Quat cosh ( const Quat< T > & q )
friend

return cosh value of quaternion q, cosh could be calculated as:

$\cosh(p) = \cosh(w) * \cos(||\boldsymbol{v}||) + \sinh(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sin(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
cosh(q);

## ◆ crossProduct

template<typename _Tp>
template<typename T >
 Quat crossProduct ( const Quat< T > & p, const Quat< T > & q )
friend

return the crossProduct between $$p = (a, b, c, d) = (a, \boldsymbol{u})$$ and $$q = (w, x, y, z) = (w, \boldsymbol{v})$$.

$p \times q = \frac{pq- qp}{2}$

$p \times q = \boldsymbol{u} \times \boldsymbol{v}$

$p \times q = (cz-dy)i + (dx-bz)j + (by-xc)k$

For example

Quatd q{1,2,3,4};
Quatd p{5,6,7,8};

## ◆ cv::operator* [1/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator* ( const T s, const Quat< T > & )
friend

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

Rule of quaternion multiplication with a scalar:

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

For example

Quatd p{1, 2, 3, 4};
double s = 2.0;
std::cout << s * p << std::endl; //[2.0, 4.0, 6.0, 8.0]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator* [2/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator* ( const Quat< T > & , const T s )
friend

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

Rule of quaternion multiplication with a scalar:

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

For example

Quatd p{1, 2, 3, 4};
double s = 2.0;
std::cout << p * s << std::endl; //[2.0, 4.0, 6.0, 8.0]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator+ [1/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator+ ( const T s, const Quat< T > & )
friend

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

For example

Quatd p{1, 2, 3, 4};
double scalar = 2.0;
std::cout << scalar + p << std::endl; //[3.0, 2, 3, 4]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator+ [2/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator+ ( const Quat< T > & , const T s )
friend

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

For example

Quatd p{1, 2, 3, 4};
double scalar = 2.0;
std::cout << p + scalar << std::endl; //[3.0, 2, 3, 4]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator- [1/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator- ( const T s, const Quat< T > & )
friend

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

For example

Quatd p{1, 2, 3, 4};
double scalar = 2.0;
std::cout << scalar - p << std::endl; //[1.0, -2, -3, -4]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator- [2/2]

template<typename _Tp>
template<typename T >
 Quat cv::operator- ( const Quat< T > & , const T s )
friend

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

For example

Quatd p{1, 2, 3, 4};
double scalar = 2.0;
std::cout << p - scalar << std::endl; //[-1.0, 2, 3, 4]
Note
the type of scalar should be equal to the quaternion.

## ◆ cv::operator<<

template<typename _Tp>
template<typename S >
 std::ostream& cv::operator<< ( std::ostream & , const Quat< S > & )
friend

## ◆ exp

template<typename _Tp>
template<typename T >
 Quat exp ( const Quat< T > & q )
friend

return the value of exponential value.

$\exp(q) = e^w (\cos||\boldsymbol{v}||+ \frac{v}{||\boldsymbol{v}||})\sin||\boldsymbol{v}||$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example:

Quatd q{1,2,3,4};
cout << exp(q) << endl;

## ◆ inv

template<typename _Tp>
template<typename T >
 Quat inv ( const Quat< T > & q, QuatAssumeType assumeUnit )
friend

return $$q^{-1}$$ which is an inverse of $$q$$ which satisfies $$q * q^{-1} = 1$$.

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

For example

Quatd q(1,2,3,4);
inv(q);
q = q.normalize();
inv(q, assumeUnit);//This assumeUnit means p is a unit quaternion

## ◆ log

template<typename _Tp>
template<typename T >
 Quat log ( const Quat< T > & q, QuatAssumeType assumeUnit )
friend

return the value of logarithm function.

$\ln(q) = \ln||q|| + \frac{\boldsymbol{v}}{||\boldsymbol{v}||}\arccos\frac{w}{||q||}.$

where $$\boldsymbol{v} = [x, y, z].$$

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

For example

Quatd q1{1,2,3,4};
cout << log(q1) << endl;

## ◆ power [1/2]

template<typename _Tp>
template<typename T >
 Quat power ( const Quat< T > & q, const T x, QuatAssumeType assumeUnit )
friend

return the value of power function with index $$x$$.

$q^x = ||q||(cos(x\theta) + \boldsymbol{u}sin(x\theta))).$

Parameters
 q a quaternion. x index of exponentiation. assumeUnit if QUAT_ASSUME_UNIT, quaternion q assume to be a unit quaternion and this function will save some computations.

For example

Quatd q(1,2,3,4);
power(q, 2.0);
double angle = CV_PI;
Vec3d axis{0, 0, 1};
Quatd q1 = Quatd::createFromAngleAxis(angle, axis); //generate a unit quat by axis and angle
power(q1, 2.0, assumeUnit);//This assumeUnit means q1 is a unit quaternion.
Note
the type of the index should be the same as the quaternion.

## ◆ power [2/2]

template<typename _Tp>
template<typename T >
 Quat power ( const Quat< T > & p, const Quat< T > & q, QuatAssumeType assumeUnit )
friend

return the value of power function with quaternion $$q$$.

$p^q = e^{q\ln(p)}.$

Parameters
 p base quaternion of power function. q index quaternion of power function. assumeUnit if QUAT_ASSUME_UNIT, quaternion $$p$$ assume to be a unit quaternion and this function will save some computations.

For example

Quatd p(1,2,3,4);
Quatd q(5,6,7,8);
power(p, q);
p = p.normalize();
power(p, q, assumeUnit); //This assumeUnit means p is a unit quaternion

## ◆ sin

template<typename _Tp>
template<typename T >
 Quat sin ( const Quat< T > & q )
friend

return tanh value of quaternion q, sin could be calculated as:

$\sin(p) = \sin(w) * \cosh(||\boldsymbol{v}||) + \cos(w)\frac{\boldsymbol{v}}{||\boldsymbol{v}||}\sinh(||\boldsymbol{v}||)$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
sin(q);

## ◆ sinh

template<typename _Tp>
template<typename T >
 Quat sinh ( const Quat< T > & q )
friend

return sinh value of quaternion q, sinh could be calculated as:

$\sinh(p) = \sin(w)\cos(||\boldsymbol{v}||) + \cosh(w)\frac{v}{||\boldsymbol{v}||}\sin||\boldsymbol{v}||$

where $$\boldsymbol{v} = [x, y, z].$$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
sinh(q);

## ◆ sqrt

template<typename _Tp>
template<typename T >
 Quat sqrt ( const Quat< T > & q, QuatAssumeType assumeUnit )
friend

return $$\sqrt{q}$$.

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

For example

Quatf q(1,2,3,4);
sqrt(q);
q = {1,0,0,0};
sqrt(q, assumeUnit); //This assumeUnit means q is a unit quaternion.

## ◆ tan

template<typename _Tp>
template<typename T >
 Quat tan ( const Quat< T > & q )
friend

return tan value of quaternion q, tan could be calculated as:

$\tan(q) = \frac{\sin(q)}{\cos(q)}.$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
tan(q);

## ◆ tanh

template<typename _Tp>
template<typename T >
 Quat tanh ( const Quat< T > & q )
friend

return tanh value of quaternion q, tanh could be calculated as:

$\tanh(q) = \frac{\sinh(q)}{\cosh(q)}.$

Parameters
 q a quaternion.

For example

Quatd q(1,2,3,4);
tanh(q);
sinh, cosh

## ◆ CV_QUAT_CONVERT_THRESHOLD

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

## ◆ CV_QUAT_EPS

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

## ◆ w

template<typename _Tp>
 _Tp cv::Quat< _Tp >::w

## ◆ x

template<typename _Tp>
 _Tp cv::Quat< _Tp >::x

## ◆ y

template<typename _Tp>
 _Tp cv::Quat< _Tp >::y

## ◆ z

template<typename _Tp>
 _Tp cv::Quat< _Tp >::z

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