quaternion.inl.hpp

// This file is part of OpenCV project.
// It is subject to the license terms in the LICENSE file found in the top-level directory
// of this distribution and at http://opencv.org/license.html.
//
//
//                          License Agreement
//                For Open Source Computer Vision Library
//
// Copyright (C) 2020, Huawei Technologies Co., Ltd. All rights reserved.
// Third party copyrights are property of their respective owners.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//       http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//
// Author: Liangqian Kong <chargerKong@126.com>
//         Longbu Wang <riskiest@gmail.com>
 
#ifndef OPENCV_CORE_QUATERNION_INL_HPP
#define OPENCV_CORE_QUATERNION_INL_HPP
 
#ifndef OPENCV_CORE_QUATERNION_HPP
#error This is not a standalone header. Include quaternion.hpp instead.
#endif
 
//@cond IGNORE
//Implementation
namespace cv {
 
template <typename T>
Quat<T>::Quat() : w(0), x(0), y(0), z(0) {}
 
template <typename T>
Quat<T>::Quat(const Vec<T, 4> &coeff):w(coeff[0]), x(coeff[1]), y(coeff[2]), z(coeff[3]){}
 
template <typename T>
Quat<T>::Quat(const T qw, const T qx, const T qy, const T qz):w(qw), x(qx), y(qy), z(qz){}
 
template <typename T>
Quat<T> Quat<T>::createFromAngleAxis(const T angle, const Vec<T, 3> &axis)
{
    T w, x, y, z;
    T vNorm = std::sqrt(axis.dot(axis));
    if (vNorm < CV_QUAT_EPS)
    {
        CV_Error(Error::StsBadArg, "this quaternion does not represent a rotation");
    }
    const T angle_half = angle * T(0.5);
    w = std::cos(angle_half);
    const T sin_v = std::sin(angle_half);
    const T sin_norm = sin_v / vNorm;
    x = sin_norm * axis[0];
    y = sin_norm * axis[1];
    z = sin_norm * axis[2];
    return Quat<T>(w, x, y, z);
}
 
template <typename T>
Quat<T> Quat<T>::createFromRotMat(InputArray _R)
{
    CV_CheckTypeEQ(_R.type(), cv::traits::Type<T>::value, "");
    if (_R.rows() != 3 || _R.cols() != 3)
    {
        CV_Error(Error::StsBadArg, "Cannot convert matrix to quaternion: rotation matrix should be a 3x3 matrix");
    }
    Matx<T, 3, 3> R;
    _R.copyTo(R);
 
    T S, w, x, y, z;
    T trace = R(0, 0) + R(1, 1) + R(2, 2);
    if (trace > 0)
    {
        S = std::sqrt(trace + 1) * T(2);
        x = (R(1, 2) - R(2, 1)) / S;
        y = (R(2, 0) - R(0, 2)) / S;
        z = (R(0, 1) - R(1, 0)) / S;
        w = -T(0.25) * S;
    }
    else if (R(0, 0) > R(1, 1) && R(0, 0) > R(2, 2))
    {
 
        S = std::sqrt(T(1.0) + R(0, 0) - R(1, 1) - R(2, 2)) * T(2);
        x = -T(0.25) * S;
        y = -(R(1, 0) + R(0, 1)) / S;
        z = -(R(0, 2) + R(2, 0)) / S;
        w = (R(1, 2) - R(2, 1)) / S;
    }
    else if (R(1, 1) > R(2, 2))
    {
        S = std::sqrt(T(1.0) - R(0, 0) + R(1, 1) - R(2, 2)) * T(2);
        x = (R(0, 1) + R(1, 0)) / S;
        y = T(0.25) * S;
        z = (R(1, 2) + R(2, 1)) / S;
        w = (R(0, 2) - R(2, 0)) / S;
    }
    else
    {
        S = std::sqrt(T(1.0) - R(0, 0) - R(1, 1) + R(2, 2)) * T(2);
        x = (R(0, 2) + R(2, 0)) / S;
        y = (R(1, 2) + R(2, 1)) / S;
        z = T(0.25) * S;
        w = -(R(0, 1) - R(1, 0)) / S;
    }
    return Quat<T> (w, x, y, z);
}
 
template <typename T>
Quat<T> Quat<T>::createFromRvec(InputArray _rvec)
{
    if (!((_rvec.cols() == 1 && _rvec.rows() == 3) || (_rvec.cols() == 3 && _rvec.rows() == 1))) {
        CV_Error(Error::StsBadArg, "Cannot convert rotation vector to quaternion: The length of rotation vector should be 3");
    }
    Vec<T, 3> rvec;
    _rvec.copyTo(rvec);
    T psi = std::sqrt(rvec.dot(rvec));
    if (abs(psi) < CV_QUAT_EPS) {
        return Quat<T> (1, 0, 0, 0);
    }
    Vec<T, 3> axis = rvec / psi;
    return createFromAngleAxis(psi, axis);
}
 
template <typename T>
inline Quat<T> Quat<T>::operator-() const
{
    return Quat<T>(-w, -x, -y, -z);
}
 
 
template <typename T>
inline bool Quat<T>::operator==(const Quat<T> &q) const
{
    return (abs(w - q.w) < CV_QUAT_EPS && abs(x - q.x) < CV_QUAT_EPS && abs(y - q.y) < CV_QUAT_EPS && abs(z - q.z) < CV_QUAT_EPS);
}
 
template <typename T>
inline Quat<T> Quat<T>::operator+(const Quat<T> &q1) const
{
    return Quat<T>(w + q1.w, x + q1.x, y + q1.y, z + q1.z);
}
 
template <typename T>
inline Quat<T> operator+(const T a, const Quat<T>& q)
{
    return Quat<T>(q.w + a, q.x, q.y, q.z);
}
 
template <typename T>
inline Quat<T> operator+(const Quat<T>& q, const T a)
{
    return Quat<T>(q.w + a, q.x, q.y, q.z);
}
 
template <typename T>
inline Quat<T> operator-(const T a, const Quat<T>& q)
{
    return Quat<T>(a - q.w, -q.x, -q.y, -q.z);
}
 
template <typename T>
inline Quat<T> operator-(const Quat<T>& q, const T a)
{
    return Quat<T>(q.w - a, q.x, q.y, q.z);
}
 
template <typename T>
inline Quat<T> Quat<T>::operator-(const Quat<T> &q1) const
{
    return Quat<T>(w - q1.w, x - q1.x, y - q1.y, z - q1.z);
}
 
template <typename T>
inline Quat<T>& Quat<T>::operator+=(const Quat<T> &q1)
{
    w += q1.w;
    x += q1.x;
    y += q1.y;
    z += q1.z;
    return *this;
}
 
template <typename T>
inline Quat<T>& Quat<T>::operator-=(const Quat<T> &q1)
{
    w -= q1.w;
    x -= q1.x;
    y -= q1.y;
    z -= q1.z;
    return *this;
}
 
template <typename T>
inline Quat<T> Quat<T>::operator*(const Quat<T> &q1) const
{
    Vec<T, 4> q{w, x, y, z};
    Vec<T, 4> q2{q1.w, q1.x, q1.y, q1.z};
    return Quat<T>(q * q2);
}
 
 
template <typename T>
Quat<T> operator*(const Quat<T> &q1, const T a)
{
    return Quat<T>(a * q1.w, a * q1.x, a * q1.y, a * q1.z);
}
 
template <typename T>
Quat<T> operator*(const T a, const Quat<T> &q1)
{
    return Quat<T>(a * q1.w, a * q1.x, a * q1.y, a * q1.z);
}
 
template <typename T>
inline Quat<T>& Quat<T>::operator*=(const Quat<T> &q1)
{
    T qw, qx, qy, qz;
    qw = w * q1.w - x * q1.x - y * q1.y - z * q1.z;
    qx = x * q1.w + w * q1.x + y * q1.z - z * q1.y;
    qy = y * q1.w + w * q1.y + z * q1.x - x * q1.z;
    qz = z * q1.w + w * q1.z + x * q1.y - y * q1.x;
    w = qw;
    x = qx;
    y = qy;
    z = qz;
    return *this;
}
 
template <typename T>
inline Quat<T>& Quat<T>::operator/=(const Quat<T> &q1)
{
    Quat<T> q(*this * q1.inv());
    w = q.w;
    x = q.x;
    y = q.y;
    z = q.z;
    return *this;
}
template <typename T>
Quat<T>& Quat<T>::operator*=(const T q1)
{
    w *= q1;
    x *= q1;
    y *= q1;
    z *= q1;
    return *this;
}
 
template <typename T>
inline Quat<T>& Quat<T>::operator/=(const T a)
{
    const T a_inv = 1.0 / a;
    w *= a_inv;
    x *= a_inv;
    y *= a_inv;
    z *= a_inv;
    return *this;
}
 
template <typename T>
inline Quat<T> Quat<T>::operator/(const T a) const
{
    const T a_inv = T(1.0) / a;
    return Quat<T>(w * a_inv, x * a_inv, y * a_inv, z * a_inv);
}
 
template <typename T>
inline Quat<T> Quat<T>::operator/(const Quat<T> &q) const
{
    return *this * q.inv();
}
 
template <typename T>
inline const T& Quat<T>::operator[](std::size_t n) const
{
    switch (n) {
        case 0:
            return w;
        case 1:
            return x;
        case 2:
            return y;
        case 3:
            return z;
        default:
            CV_Error(Error::StsOutOfRange, "subscript exceeds the index range");
    }
}
 
template <typename T>
inline T& Quat<T>::operator[](std::size_t n)
{
    switch (n) {
        case 0:
            return w;
        case 1:
            return x;
        case 2:
            return y;
        case 3:
            return z;
        default:
            CV_Error(Error::StsOutOfRange, "subscript exceeds the index range");
    }
}
 
template <typename T>
std::ostream & operator<<(std::ostream &os, const Quat<T> &q)
{
    os << "Quat " << Vec<T, 4>{q.w, q.x, q.y, q.z};
    return os;
}
 
template <typename T>
inline T Quat<T>::at(size_t index) const
{
    return (*this)[index];
}
 
template <typename T>
inline Quat<T> Quat<T>::conjugate() const
{
    return Quat<T>(w, -x, -y, -z);
}
 
template <typename T>
inline T Quat<T>::norm() const
{
    return std::sqrt(dot(*this));
}
 
template <typename T>
Quat<T> exp(const Quat<T> &q)
{
    return q.exp();
}
 
template <typename T>
Quat<T> Quat<T>::exp() const
{
    Vec<T, 3> v{x, y, z};
    T normV = std::sqrt(v.dot(v));
    T k = normV < CV_QUAT_EPS ? 1 : std::sin(normV) / normV;
    return std::exp(w) * Quat<T>(std::cos(normV), v[0] * k, v[1] * k, v[2] * k);
}
 
template <typename T>
Quat<T> log(const Quat<T> &q, QuatAssumeType assumeUnit)
{
    return q.log(assumeUnit);
}
 
template <typename T>
Quat<T> Quat<T>::log(QuatAssumeType assumeUnit) const
{
    Vec<T, 3> v{x, y, z};
    T vNorm = std::sqrt(v.dot(v));
    if (assumeUnit)
    {
        T k = vNorm < CV_QUAT_EPS ? 1 : std::acos(w) / vNorm;
        return Quat<T>(0, v[0] * k, v[1] * k, v[2] * k);
    }
    T qNorm = norm();
    if (qNorm < CV_QUAT_EPS)
    {
        CV_Error(Error::StsBadArg, "Cannot apply this quaternion to log function: undefined");
    }
    T k = vNorm < CV_QUAT_EPS ? 1 : std::acos(w / qNorm) / vNorm;
    return Quat<T>(std::log(qNorm), v[0] * k, v[1] * k, v[2] *k);
}
 
template <typename T>
inline Quat<T> power(const Quat<T> &q1, const T alpha, QuatAssumeType assumeUnit)
{
    return q1.power(alpha, assumeUnit);
}
 
template <typename T>
inline Quat<T> Quat<T>::power(const T alpha, QuatAssumeType assumeUnit) const
{
    if (x * x + y * y + z * z > CV_QUAT_EPS)
    {
        T angle = getAngle(assumeUnit);
        Vec<T, 3> axis = getAxis(assumeUnit);
        if (assumeUnit)
        {
            return createFromAngleAxis(alpha * angle, axis);
        }
        return std::pow(norm(), alpha) * createFromAngleAxis(alpha * angle, axis);
    }
    else
    {
        return std::pow(norm(), alpha) * Quat<T>(w, x, y, z);
    }
}
 
 
template <typename T>
inline Quat<T> sqrt(const Quat<T> &q, QuatAssumeType assumeUnit)
{
    return q.sqrt(assumeUnit);
}
 
template <typename T>
inline Quat<T> Quat<T>::sqrt(QuatAssumeType assumeUnit) const
{
    return power(0.5, assumeUnit);
}
 
 
template <typename T>
inline Quat<T> power(const Quat<T> &p, const Quat<T> &q, QuatAssumeType assumeUnit)
{
    return p.power(q, assumeUnit);
}
 
 
template <typename T>
inline Quat<T> Quat<T>::power(const Quat<T> &q, QuatAssumeType assumeUnit) const
{
    return cv::exp(q * log(assumeUnit));
}
 
template <typename T>
inline T Quat<T>::dot(Quat<T> q1) const
{
    return w * q1.w + x * q1.x + y * q1.y + z * q1.z;
}
 
 
template <typename T>
inline Quat<T> crossProduct(const Quat<T> &p, const Quat<T> &q)
{
    return p.crossProduct(q);
}
 
 
template <typename T>
inline Quat<T> Quat<T>::crossProduct(const Quat<T> &q) const
{
    return Quat<T> (0, y * q.z - z * q.y, z * q.x - x * q.z, x * q.y - q.x * y);
}
 
template <typename T>
inline Quat<T> Quat<T>::normalize() const
{
    T normVal = norm();
    if (normVal < CV_QUAT_EPS)
    {
        CV_Error(Error::StsBadArg, "Cannot normalize this quaternion: the norm is too small.");
    }
    return Quat<T>(w / normVal, x / normVal, y / normVal, z / normVal) ;
}
 
template <typename T>
inline Quat<T> inv(const Quat<T> &q, QuatAssumeType assumeUnit)
{
    return q.inv(assumeUnit);
}
 
 
template <typename T>
inline Quat<T> Quat<T>::inv(QuatAssumeType assumeUnit) const
{
    if (assumeUnit)
    {
        return conjugate();
    }
    T norm2 = dot(*this);
    if (norm2 < CV_QUAT_EPS)
    {
        CV_Error(Error::StsBadArg, "This quaternion do not have inverse quaternion");
    }
    return conjugate() / norm2;
}
 
template <typename T>
inline Quat<T> sinh(const Quat<T> &q)
{
    return q.sinh();
}
 
 
template <typename T>
inline Quat<T> Quat<T>::sinh() const
{
    Vec<T, 3> v{x, y ,z};
    T vNorm = std::sqrt(v.dot(v));
    T k = vNorm < CV_QUAT_EPS ? 1 : std::cosh(w) * std::sin(vNorm) / vNorm;
    return Quat<T>(std::sinh(w) * std::cos(vNorm), v[0] * k, v[1] * k, v[2] * k);
}
 
 
template <typename T>
inline Quat<T> cosh(const Quat<T> &q)
{
    return q.cosh();
}
 
 
template <typename T>
inline Quat<T> Quat<T>::cosh() const
{
    Vec<T, 3> v{x, y ,z};
    T vNorm = std::sqrt(v.dot(v));
    T k = vNorm < CV_QUAT_EPS ? 1 : std::sinh(w) * std::sin(vNorm) / vNorm;
    return Quat<T>(std::cosh(w) * std::cos(vNorm), v[0] * k, v[1] * k, v[2] * k);
}
 
template <typename T>
inline Quat<T> tanh(const Quat<T> &q)
{
    return q.tanh();
}
 
template <typename T>
inline Quat<T> Quat<T>::tanh() const
{
    return sinh() * cosh().inv();
}
 
 
template <typename T>
inline Quat<T> sin(const Quat<T> &q)
{
    return q.sin();
}
 
 
template <typename T>
inline Quat<T> Quat<T>::sin() const
{
    Vec<T, 3> v{x, y ,z};
    T vNorm = std::sqrt(v.dot(v));
    T k = vNorm < CV_QUAT_EPS ? 1 : std::cos(w) * std::sinh(vNorm) / vNorm;
    return Quat<T>(std::sin(w) * std::cosh(vNorm), v[0] * k, v[1] * k, v[2] * k);
}
 
template <typename T>
inline Quat<T> cos(const Quat<T> &q)
{
    return q.cos();
}
 
template <typename T>
inline Quat<T> Quat<T>::cos() const
{
    Vec<T, 3> v{x, y ,z};
    T vNorm = std::sqrt(v.dot(v));
    T k = vNorm < CV_QUAT_EPS ? 1 : std::sin(w) * std::sinh(vNorm) / vNorm;
    return Quat<T>(std::cos(w) * std::cosh(vNorm), -v[0] * k, -v[1] * k, -v[2] * k);
}
 
template <typename T>
inline Quat<T> tan(const Quat<T> &q)
{
    return q.tan();
}
 
template <typename T>
inline Quat<T> Quat<T>::tan() const
{
    return sin() * cos().inv();
}
 
template <typename T>
inline Quat<T> asinh(const Quat<T> &q)
{
    return q.asinh();
}
 
template <typename T>
inline Quat<T> Quat<T>::asinh() const
{
    return cv::log(*this + cv::power(*this * *this + Quat<T>(1, 0, 0, 0), 0.5));
}
 
template <typename T>
inline Quat<T> acosh(const Quat<T> &q)
{
    return q.acosh();
}
 
template <typename T>
inline Quat<T> Quat<T>::acosh() const
{
    return cv::log(*this + cv::power(*this * *this - Quat<T>(1,0,0,0), 0.5));
}
 
template <typename T>
inline Quat<T> atanh(const Quat<T> &q)
{
    return q.atanh();
}
 
template <typename T>
inline Quat<T> Quat<T>::atanh() const
{
    Quat<T> ident(1, 0, 0, 0);
    Quat<T> c1 = (ident + *this).log();
    Quat<T> c2 = (ident - *this).log();
    return 0.5 * (c1 - c2);
}
 
template <typename T>
inline Quat<T> asin(const Quat<T> &q)
{
    return q.asin();
}
 
template <typename T>
inline Quat<T> Quat<T>::asin() const
{
    Quat<T> v(0, x, y, z);
    T vNorm = v.norm();
    T k = vNorm < CV_QUAT_EPS ? 1 : vNorm;
    return -v / k * (*this * v / k).asinh();
}
 
template <typename T>
inline Quat<T> acos(const Quat<T> &q)
{
    return q.acos();
}
 
template <typename T>
inline Quat<T> Quat<T>::acos() const
{
    Quat<T> v(0, x, y, z);
    T vNorm = v.norm();
    T k = vNorm < CV_QUAT_EPS ? 1 : vNorm;
    return -v / k * acosh();
}
 
template <typename T>
inline Quat<T> atan(const Quat<T> &q)
{
    return q.atan();
}
 
template <typename T>
inline Quat<T> Quat<T>::atan() const
{
    Quat<T> v(0, x, y, z);
    T vNorm = v.norm();
    T k = vNorm < CV_QUAT_EPS ? 1 : vNorm;
    return -v / k * (*this * v / k).atanh();
}
 
template <typename T>
inline T Quat<T>::getAngle(QuatAssumeType assumeUnit) const
{
    if (assumeUnit)
    {
        return 2 * std::acos(w);
    }
    if (norm() < CV_QUAT_EPS)
    {
        CV_Error(Error::StsBadArg, "This quaternion does not represent a rotation");
    }
    return 2 * std::acos(w / norm());
}
 
template <typename T>
inline Vec<T, 3> Quat<T>::getAxis(QuatAssumeType assumeUnit) const
{
    T angle = getAngle(assumeUnit);
    const T sin_v = std::sin(angle * 0.5);
    if (assumeUnit)
    {
        return Vec<T, 3>{x, y, z} / sin_v;
    }
    return Vec<T, 3> {x, y, z} / (norm() * sin_v);
}
 
template <typename T>
Matx<T, 4, 4> Quat<T>::toRotMat4x4(QuatAssumeType assumeUnit) const
{
    T a = w, b = x, c = y, d = z;
    if (!assumeUnit)
    {
        Quat<T> qTemp = normalize();
        a = qTemp.w;
        b = qTemp.x;
        c = qTemp.y;
        d = qTemp.z;
    }
    Matx<T, 4, 4> R{
        1 - 2 * (c * c + d * d), 2 * (b * c - a * d)    , 2 * (b * d + a * c)    , 0,
        2 * (b * c + a * d)    , 1 - 2 * (b * b + d * d), 2 * (c * d - a * b)    , 0,
        2 * (b * d - a * c)    , 2 * (c * d + a * b)    , 1 - 2 * (b * b + c * c), 0,
        0                      , 0                      , 0                      , 1,
    };
    return R;
}
 
template <typename T>
Matx<T, 3, 3> Quat<T>::toRotMat3x3(QuatAssumeType assumeUnit) const
{
    T a = w, b = x, c = y, d = z;
    if (!assumeUnit)
    {
        Quat<T> qTemp = normalize();
        a = qTemp.w;
        b = qTemp.x;
        c = qTemp.y;
        d = qTemp.z;
    }
    Matx<T, 3, 3> R{
        1 - 2 * (c * c + d * d), 2 * (b * c - a * d)    , 2 * (b * d + a * c),
        2 * (b * c + a * d)    , 1 - 2 * (b * b + d * d), 2 * (c * d - a * b),
        2 * (b * d - a * c)    , 2 * (c * d + a * b)    , 1 - 2 * (b * b + c * c)
    };
    return R;
}
 
template <typename T>
Vec<T, 3> Quat<T>::toRotVec(QuatAssumeType assumeUnit) const
{
    T angle = getAngle(assumeUnit);
    Vec<T, 3> axis = getAxis(assumeUnit);
    return angle * axis;
}
 
template <typename T>
Vec<T, 4> Quat<T>::toVec() const
{
    return Vec<T, 4>{w, x, y, z};
}
 
template <typename T>
Quat<T> Quat<T>::lerp(const Quat<T> &q0, const Quat<T> &q1, const T t)
{
    return (1 - t) * q0 + t * q1;
}
 
template <typename T>
Quat<T> Quat<T>::slerp(const Quat<T> &q0, const Quat<T> &q1, const T t, QuatAssumeType assumeUnit, bool directChange)
{
    Quat<T> v0(q0);
    Quat<T> v1(q1);
    if (!assumeUnit)
    {
        v0 = v0.normalize();
        v1 = v1.normalize();
    }
    T cosTheta = v0.dot(v1);
    constexpr T DOT_THRESHOLD = 0.995;
    if (std::abs(cosTheta) > DOT_THRESHOLD)
    {
        return nlerp(v0, v1, t, QUAT_ASSUME_UNIT);
    }
 
    if (directChange && cosTheta < 0)
    {
        v0 = -v0;
        cosTheta = -cosTheta;
    }
    T sinTheta = std::sqrt(1 - cosTheta * cosTheta);
    T angle = atan2(sinTheta, cosTheta);
    return (std::sin((1 - t) * angle) / (sinTheta) * v0 + std::sin(t * angle) / (sinTheta) * v1).normalize();
}
 
 
template <typename T>
inline Quat<T> Quat<T>::nlerp(const Quat<T> &q0, const Quat<T> &q1, const T t, QuatAssumeType assumeUnit)
{
    Quat<T> v0(q0), v1(q1);
    if (v1.dot(v0) < 0)
    {
        v0 = -v0;
    }
    if (assumeUnit)
    {
        return ((1 - t) * v0 + t * v1).normalize();
    }
    v0 = v0.normalize();
    v1 = v1.normalize();
    return ((1 - t) * v0 + t * v1).normalize();
}
 
 
template <typename T>
inline bool Quat<T>::isNormal(T eps) const
{
 
    double normVar = norm();
    if ((normVar > 1 - eps) && (normVar < 1 + eps))
        return true;
    return false;
}
 
template <typename T>
inline void Quat<T>::assertNormal(T eps) const
{
    if (!isNormal(eps))
        CV_Error(Error::StsBadArg, "Quaternion should be normalized");
}
 
 
template <typename T>
inline Quat<T> Quat<T>::squad(const Quat<T> &q0, const Quat<T> &q1,
                            const Quat<T> &q2, const Quat<T> &q3,
                            const T t, QuatAssumeType assumeUnit,
                            bool directChange)
{
    Quat<T> v0(q0), v1(q1), v2(q2), v3(q3);
    if (!assumeUnit)
    {
        v0 = v0.normalize();
        v1 = v1.normalize();
        v2 = v2.normalize();
        v3 = v3.normalize();
    }
 
    Quat<T> c0 = slerp(v0, v3, t, assumeUnit, directChange);
    Quat<T> c1 = slerp(v1, v2, t, assumeUnit, directChange);
    return slerp(c0, c1, 2 * t * (1 - t), assumeUnit, directChange);
}
 
template <typename T>
Quat<T> Quat<T>::interPoint(const Quat<T> &q0, const Quat<T> &q1,
                            const Quat<T> &q2, QuatAssumeType assumeUnit)
{
    Quat<T> v0(q0), v1(q1), v2(q2);
    if (!assumeUnit)
    {
        v0 = v0.normalize();
        v1 = v1.normalize();
        v2 = v2.normalize();
    }
    return v1 * cv::exp(-(cv::log(v1.conjugate() * v0, assumeUnit) + (cv::log(v1.conjugate() * v2, assumeUnit))) / 4);
}
 
template <typename T>
Quat<T> Quat<T>::spline(const Quat<T> &q0, const Quat<T> &q1, const Quat<T> &q2, const Quat<T> &q3, const T t, QuatAssumeType assumeUnit)
{
    Quat<T> v0(q0), v1(q1), v2(q2), v3(q3);
    if (!assumeUnit)
    {
        v0 = v0.normalize();
        v1 = v1.normalize();
        v2 = v2.normalize();
        v3 = v3.normalize();
    }
    T cosTheta;
    std::vector<Quat<T>> vec{v0, v1, v2, v3};
    for (size_t i = 0; i < 3; ++i)
    {
        cosTheta = vec[i].dot(vec[i + 1]);
        if (cosTheta < 0)
        {
            vec[i + 1] = -vec[i + 1];
        }
    }
    Quat<T> s1 = interPoint(vec[0], vec[1], vec[2], QUAT_ASSUME_UNIT);
    Quat<T> s2 = interPoint(vec[1], vec[2], vec[3], QUAT_ASSUME_UNIT);
    return squad(vec[1], s1, s2, vec[2], t, assumeUnit, QUAT_ASSUME_NOT_UNIT);
}
 
namespace detail {
 
template <typename T> static
Quat<T> createFromAxisRot(int axis, const T theta)
{
    if (axis == 0)
        return Quat<T>::createFromXRot(theta);
    if (axis == 1)
        return Quat<T>::createFromYRot(theta);
    if (axis == 2)
        return Quat<T>::createFromZRot(theta);
    CV_Assert(0);
}
 
inline bool isIntAngleType(QuatEnum::EulerAnglesType eulerAnglesType)
{
    return eulerAnglesType < QuatEnum::EXT_XYZ;
}
 
inline bool isTaitBryan(QuatEnum::EulerAnglesType eulerAnglesType)
{
    return eulerAnglesType/6 == 1 || eulerAnglesType/6 == 3;
}
}  // namespace detail
 
template <typename T>
Quat<T> Quat<T>::createFromYRot(const T theta)
{
    return Quat<T>{std::cos(theta * 0.5f), 0, std::sin(theta * 0.5f), 0};
}
 
template <typename T>
Quat<T> Quat<T>::createFromXRot(const T theta){
    return Quat<T>{std::cos(theta * 0.5f), std::sin(theta * 0.5f), 0, 0};
}
 
template <typename T>
Quat<T> Quat<T>::createFromZRot(const T theta){
    return Quat<T>{std::cos(theta * 0.5f), 0, 0, std::sin(theta * 0.5f)};
}
 
 
template <typename T>
Quat<T> Quat<T>::createFromEulerAngles(const Vec<T, 3> &angles, QuatEnum::EulerAnglesType eulerAnglesType) {
    CV_Assert(eulerAnglesType < QuatEnum::EulerAnglesType::EULER_ANGLES_MAX_VALUE);
    static const int rotationAxis[24][3] = {
        {0, 1, 2}, 
        {0, 2, 1}, 
        {1, 0, 2}, 
        {1, 2, 0}, 
        {2, 0, 1}, 
        {2, 1, 0}, 
        {0, 1, 0}, 
        {0, 2, 0}, 
        {1, 0, 1}, 
        {1, 2, 1}, 
        {2, 0, 2}, 
        {2, 1, 2}, 
        {0, 1, 2}, 
        {0, 2, 1}, 
        {1, 0, 2}, 
        {1, 2, 0}, 
        {2, 0, 1}, 
        {2, 1, 0}, 
        {0, 1, 0}, 
        {0, 2, 0}, 
        {1, 0, 1}, 
        {1, 2, 1}, 
        {2, 0, 2}, 
        {2, 1, 2}  
    };
    Quat<T> q1 = detail::createFromAxisRot(rotationAxis[eulerAnglesType][0], angles(0));
    Quat<T> q2 = detail::createFromAxisRot(rotationAxis[eulerAnglesType][1], angles(1));
    Quat<T> q3 = detail::createFromAxisRot(rotationAxis[eulerAnglesType][2], angles(2));
    if (detail::isIntAngleType(eulerAnglesType))
    {
        return q1 * q2 * q3;
    }
    else // (!detail::isIntAngleType<T>(eulerAnglesType))
    {
        return q3 * q2 * q1;
    }
}
 
template <typename T>
Vec<T, 3> Quat<T>::toEulerAngles(QuatEnum::EulerAnglesType eulerAnglesType){
    CV_Assert(eulerAnglesType < QuatEnum::EulerAnglesType::EULER_ANGLES_MAX_VALUE);
    Matx33d R = toRotMat3x3();
    enum {
        C_ZERO,
        C_PI,
        C_PI_2,
        N_CONSTANTS,
        R_0_0 = N_CONSTANTS, R_0_1, R_0_2,
        R_1_0, R_1_1, R_1_2,
        R_2_0, R_2_1, R_2_2
    };
    static const T constants_[N_CONSTANTS] = {
        0,  // C_ZERO
        (T)CV_PI,  // C_PI
        (T)(CV_PI * 0.5)  // C_PI_2, -C_PI_2
    };
    static const int rotationR_[24][12] = {
        {+R_0_2,    +R_1_0, +R_1_1, C_PI_2,     +R_2_1, +R_1_1, -C_PI_2,    -R_1_2, +R_2_2,    +R_0_2,    -R_0_1, +R_0_0},  // INT_XYZ
        {+R_0_1,    -R_1_2, +R_2_2, -C_PI_2,    +R_2_0, +R_2_2, C_PI_2,     +R_2_1, +R_1_1,    -R_0_1,    +R_0_2, +R_0_0},  // INT_XZY
        {+R_1_2,    -R_0_1, +R_0_0, -C_PI_2,    +R_0_1, +R_0_0, C_PI_2,     +R_0_2, +R_2_2,    -R_1_2,    +R_1_0, +R_1_1},  // INT_YXZ
        {+R_1_0,    +R_0_2, +R_2_2, C_PI_2,     +R_0_2, +R_0_1, -C_PI_2,    -R_2_0, +R_0_0,    +R_1_0,    -R_1_2, +R_1_1},  // INT_YZX
        {+R_2_1,    +R_1_0, +R_0_0, C_PI_2,     +R_1_0, +R_0_0, -C_PI_2,    -R_0_1, +R_1_1,    +R_2_1,    -R_2_0, +R_2_2},  // INT_ZXY
        {+R_2_0,    -R_0_1, +R_1_1, -C_PI_2,    +R_1_2, +R_1_1, C_PI_2,     +R_1_0, +R_0_0,    -R_2_0,    +R_2_1, +R_2_2},  // INT_ZYX
        {+R_0_0,    +R_2_1, +R_2_2, C_ZERO,     +R_1_2, +R_1_1, C_PI,       +R_1_0, -R_2_0,    +R_0_0,    +R_0_1, +R_0_2},  // INT_XYX
        {+R_0_0,    +R_2_1, +R_2_2, C_ZERO,     -R_2_1, +R_2_2, C_PI,       +R_2_0, +R_1_0,    +R_0_0,    +R_0_2, -R_0_1},  // INT_XZX
        {+R_1_1,    +R_0_2, +R_0_0, C_ZERO,     -R_2_0, +R_0_0, C_PI,       +R_0_1, +R_2_1,    +R_1_1,    +R_1_0, -R_1_2},  // INT_YXY
        {+R_1_1,    +R_0_2, +R_0_0, C_ZERO,     +R_0_2, -R_0_0, C_PI,       +R_2_1, -R_0_1,    +R_1_1,    +R_1_2, +R_1_0},  // INT_YZY
        {+R_2_2,    +R_1_0, +R_1_1, C_ZERO,     +R_1_0, +R_0_0, C_PI,       +R_0_2, -R_1_2,    +R_2_2,    +R_2_0, +R_2_1},  // INT_ZXZ
        {+R_2_2,    +R_1_0, +R_0_0, C_ZERO,     +R_1_0, +R_0_0, C_PI,       +R_1_2, +R_0_2,    +R_2_2,    +R_2_1, -R_2_0},  // INT_ZYZ
 
        {+R_2_0,    -C_PI_2, -R_0_1, +R_1_1,    C_PI_2,  +R_1_2, +R_1_1,    +R_2_1, +R_2_2,    -R_2_0,    +R_1_0, +R_0_0},  // EXT_XYZ
        {+R_1_0,    C_PI_2,  +R_0_2, +R_2_2,    -C_PI_2, +R_0_2, +R_0_1,    -R_1_2, +R_1_1,    +R_1_0,    -R_2_0, +R_0_0},  // EXT_XZY
        {+R_2_1,    C_PI_2,  +R_1_0, +R_0_0,    -C_PI_2, +R_1_0, +R_0_0,    -R_2_0, +R_2_2,    +R_2_1,    -R_0_1, +R_1_1},  // EXT_YXZ
        {+R_0_2,    -C_PI_2, -R_1_2, +R_2_2,    C_PI_2,  +R_2_0, +R_2_2,    +R_0_2, +R_0_0,    -R_0_1,    +R_2_1, +R_1_1},  // EXT_YZX
        {+R_1_2,    -C_PI_2, -R_0_1, +R_0_0,    C_PI_2,  +R_0_1, +R_0_0,    +R_1_0, +R_1_1,    -R_1_2,    +R_0_2, +R_2_2},  // EXT_ZXY
        {+R_0_2,    C_PI_2,  +R_1_0, +R_1_1,    -C_PI_2, +R_2_1, +R_1_1,    -R_0_1, +R_0_0,    +R_0_2,    -R_1_2, +R_2_2},  // EXT_ZYX
        {+R_0_0,    C_ZERO,  +R_2_1, +R_2_2,    C_PI,    +R_1_2, +R_1_1,    +R_0_1, +R_0_2,    +R_0_0,    +R_1_0, -R_2_0},  // EXT_XYX
        {+R_0_0,    C_ZERO,  +R_2_1, +R_2_2,    C_PI,    +R_2_1, +R_2_2,    +R_0_2, -R_0_1,    +R_0_0,    +R_2_0, +R_1_0},  // EXT_XZX
        {+R_1_1,    C_ZERO,  +R_0_2, +R_0_0,    C_PI,    -R_2_0, +R_0_0,    +R_1_0, -R_1_2,    +R_1_1,    +R_0_1, +R_2_1},  // EXT_YXY
        {+R_1_1,    C_ZERO,  +R_0_2, +R_0_0,    C_PI,    +R_0_2, -R_0_0,    +R_1_2, +R_1_0,    +R_1_1,    +R_2_1, -R_0_1},  // EXT_YZY
        {+R_2_2,    C_ZERO,  +R_1_0, +R_1_1,    C_PI,    +R_1_0, +R_0_0,    +R_2_0, +R_2_1,    +R_2_2,    +R_0_2, -R_1_2},  // EXT_ZXZ
        {+R_2_2,    C_ZERO,  +R_1_0, +R_0_0,    C_PI,    +R_1_0, +R_0_0,    +R_2_1, -R_2_0,    +R_2_2,    +R_1_2, +R_0_2},  // EXT_ZYZ
    };
    T rotationR[12];
    for (int i = 0; i < 12; i++)
    {
        int id = rotationR_[eulerAnglesType][i];
        unsigned idx = std::abs(id);
        T value = 0.0f;
        if (idx < N_CONSTANTS)
        {
            value = constants_[idx];
        }
        else
        {
            unsigned r_idx = idx - N_CONSTANTS;
            CV_DbgAssert(r_idx < 9);
            value = R.val[r_idx];
        }
        bool isNegative = id < 0;
        if (isNegative)
            value = -value;
        rotationR[i] = value;
    }
    Vec<T, 3> angles;
    if (detail::isIntAngleType(eulerAnglesType))
    {
        if (abs(rotationR[0] - 1) < CV_QUAT_CONVERT_THRESHOLD)
        {
            CV_LOG_WARNING(NULL,"Gimbal Lock occurs. Euler angles are non-unique, we set the third angle to 0");
            angles = {std::atan2(rotationR[1], rotationR[2]), rotationR[3], 0};
            return angles;
        }
        else if(abs(rotationR[0] + 1) < CV_QUAT_CONVERT_THRESHOLD)
        {
            CV_LOG_WARNING(NULL,"Gimbal Lock occurs. Euler angles are non-unique, we set the third angle to 0");
            angles = {std::atan2(rotationR[4], rotationR[5]), rotationR[6], 0};
            return angles;
        }
    }
    else // (!detail::isIntAngleType<T>(eulerAnglesType))
    {
        if (abs(rotationR[0] - 1) < CV_QUAT_CONVERT_THRESHOLD)
        {
            CV_LOG_WARNING(NULL,"Gimbal Lock occurs. Euler angles are non-unique, we set the first angle to 0");
            angles = {0, rotationR[1], std::atan2(rotationR[2], rotationR[3])};
            return angles;
        }
        else if (abs(rotationR[0] + 1) < CV_QUAT_CONVERT_THRESHOLD)
        {
            CV_LOG_WARNING(NULL,"Gimbal Lock occurs. Euler angles are non-unique, we set the first angle to 0");
            angles = {0, rotationR[4], std::atan2(rotationR[5], rotationR[6])};
            return angles;
        }
    }
 
    angles(0) = std::atan2(rotationR[7], rotationR[8]);
    if (detail::isTaitBryan(eulerAnglesType))
        angles(1) = std::acos(rotationR[9]);
    else
        angles(1) = std::asin(rotationR[9]);
    angles(2) = std::atan2(rotationR[10], rotationR[11]);
    return angles;
}
 
}  // namepsace
 
#endif /*OPENCV_CORE_QUATERNION_INL_HPP*/