magnum/src/Magnum/Math/Quaternion.h

#ifndef Magnum_Math_Quaternion_h
#define Magnum_Math_Quaternion_h
/*
    This file is part of Magnum.

    Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018
              Vladimír Vondruš <mosra@centrum.cz>

    Permission is hereby granted, free of charge, to any person obtaining a
    copy of this software and associated documentation files (the "Software"),
    to deal in the Software without restriction, including without limitation
    the rights to use, copy, modify, merge, publish, distribute, sublicense,
    and/or sell copies of the Software, and to permit persons to whom the
    Software is furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
    THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
    FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
    DEALINGS IN THE SOFTWARE.
*/

/** @file
 * @brief Class @ref Magnum::Math::Quaternion, function @ref Magnum::Math::dot(), @ref Magnum::Math::angle(), @ref Magnum::Math::lerp(), @ref Magnum::Math::slerp()
 */

#include <cmath>
#include <Corrade/Utility/Assert.h>
#include <Corrade/Utility/Debug.h>

#include "Magnum/Math/Matrix.h"
#include "Magnum/Math/TypeTraits.h"
#include "Magnum/Math/Vector3.h"

namespace Magnum { namespace Math {

namespace Implementation {
    template<class, class> struct QuaternionConverter;
}

/** @relatesalso Quaternion
@brief Dot product between two quaternions

@f[
     p \cdot q = \boldsymbol p_V \cdot \boldsymbol q_V + p_S q_S
@f]
@see @ref Quaternion::dot() const
*/
template<class T> inline T dot(const Quaternion<T>& a, const Quaternion<T>& b) {
    return dot(a.vector(), b.vector()) + a.scalar()*b.scalar();
}

namespace Implementation {
    /* Used in angle() and slerp() (no assertions) */
    template<class T> inline T angle(const Quaternion<T>& normalizedA, const Quaternion<T>& normalizedB) {
        return std::acos(dot(normalizedA, normalizedB));
    }
}

/** @relatesalso Quaternion
@brief Angle between normalized quaternions

Expects that both quaternions are normalized. @f[
     \theta = acos \left( \frac{p \cdot q}{|p| |q|} \right) = acos(p \cdot q)
@f]
@see @ref Quaternion::isNormalized(),
    @ref angle(const Complex<T>&, const Complex<T>&),
    @ref angle(const Vector<size, T>&, const Vector<size, T>&)
 */
template<class T> inline Rad<T> angle(const Quaternion<T>& normalizedA, const Quaternion<T>& normalizedB) {
    CORRADE_ASSERT(normalizedA.isNormalized() && normalizedB.isNormalized(),
        "Math::angle(): quaternions must be normalized", {});
    return Rad<T>{Implementation::angle(normalizedA, normalizedB)};
}

/** @relatesalso Quaternion
@brief Linear interpolation of two quaternions
@param normalizedA  First quaternion
@param normalizedB  Second quaternion
@param t            Interpolation phase (from range @f$ [0; 1] @f$)

Expects that both quaternions are normalized. @f[
    q_{LERP} = \frac{(1 - t) q_A + t q_B}{|(1 - t) q_A + t q_B|}
@f]
@see @ref Quaternion::isNormalized(), @ref slerp(const Quaternion<T>&, const Quaternion<T>&, T),
    @ref lerp(const T&, const T&, U), @ref sclerp()
*/
template<class T> inline Quaternion<T> lerp(const Quaternion<T>& normalizedA, const Quaternion<T>& normalizedB, T t) {
    CORRADE_ASSERT(normalizedA.isNormalized() && normalizedB.isNormalized(),
        "Math::lerp(): quaternions must be normalized", {});
    return ((T(1) - t)*normalizedA + t*normalizedB).normalized();
}

/** @relatesalso Quaternion
@brief Spherical linear interpolation of two quaternions
@param normalizedA  First quaternion
@param normalizedB  Second quaternion
@param t            Interpolation phase (from range @f$ [0; 1] @f$)

Expects that both quaternions are normalized. If the quaternions are the same
or one is a negation of the other, returns the first argument. @f[
    q_{SLERP} = \frac{sin((1 - t) \theta) q_A + sin(t \theta) q_B}{sin \theta}
    ~ ~ ~ ~ ~ ~ ~
    \theta = acos \left( \frac{q_A \cdot q_B}{|q_A| \cdot |q_B|} \right) = acos(q_A \cdot q_B)
@f]
@see @ref Quaternion::isNormalized(), @ref lerp(const Quaternion<T>&, const Quaternion<T>&, T),
    @ref sclerp()
 */
template<class T> inline Quaternion<T> slerp(const Quaternion<T>& normalizedA, const Quaternion<T>& normalizedB, T t) {
    CORRADE_ASSERT(normalizedA.isNormalized() && normalizedB.isNormalized(),
        "Math::slerp(): quaternions must be normalized", {});
    const T cosHalfAngle = dot(normalizedA, normalizedB);

    /* Avoid division by zero */
    if(std::abs(cosHalfAngle) >= T(1)) return Quaternion<T>{normalizedA};

    const T a = std::acos(cosHalfAngle);
    return (std::sin((T(1) - t)*a)*normalizedA + std::sin(t*a)*normalizedB)/std::sin(a);
}

/**
@brief Quaternion
@tparam T   Underlying data type

Represents 3D rotation. See @ref transformations for brief introduction.
@see @ref Magnum::Quaternion, @ref Magnum::Quaterniond, @ref DualQuaternion,
    @ref Matrix4
*/
template<class T> class Quaternion {
    template<class> friend class Quaternion;

    public:
        typedef T Type; /**< @brief Underlying data type */

        /**
         * @brief Rotation quaternion
         * @param angle             Rotation angle (counterclockwise)
         * @param normalizedAxis    Normalized rotation axis
         *
         * Expects that the rotation axis is normalized. @f[
         *      q = [\boldsymbol a \cdot sin \frac \theta 2, cos \frac \theta 2]
         * @f]
         * @see @ref angle(), @ref axis(), @ref DualQuaternion::rotation(),
         *      @ref Matrix4::rotation(), @ref Complex::rotation(),
         *      @ref Vector3::xAxis(), @ref Vector3::yAxis(),
         *      @ref Vector3::zAxis(), @ref Vector::isNormalized()
         */
        static Quaternion<T> rotation(Rad<T> angle, const Vector3<T>& normalizedAxis);

        /**
         * @brief Create quaternion from rotation matrix
         *
         * Expects that the matrix is orthogonal (i.e. pure rotation).
         * @see @ref toMatrix(), @ref DualComplex::fromMatrix(),
         *      @ref Matrix::isOrthogonal()
         */
        static Quaternion<T> fromMatrix(const Matrix3x3<T>& matrix);

        /**
         * @brief Default constructor
         *
         * Creates unit quaternion. @f[
         *      q = [\boldsymbol 0, 1]
         * @f]
         */
        constexpr /*implicit*/ Quaternion(IdentityInitT = IdentityInit) noexcept: _scalar{T(1)} {}

        /** @brief Construct zero-initialized quaternion */
        constexpr explicit Quaternion(ZeroInitT) noexcept: _vector{ZeroInit}, _scalar{T{0}} {}

        /** @brief Construct without initializing the contents */
        explicit Quaternion(NoInitT) noexcept: _vector{NoInit} {}

        /**
         * @brief Construct quaternion from vector and scalar
         *
         * @f[
         *      q = [\boldsymbol v, s]
         * @f]
         */
        constexpr /*implicit*/ Quaternion(const Vector3<T>& vector, T scalar) noexcept: _vector(vector), _scalar(scalar) {}

        /**
         * @brief Construct quaternion from vector
         *
         * To be used in transformations later. @f[
         *      q = [\boldsymbol v, 0]
         * @f]
         * @see @ref transformVector(), @ref transformVectorNormalized()
         */
        constexpr explicit Quaternion(const Vector3<T>& vector) noexcept: _vector(vector), _scalar(T(0)) {}

        /**
         * @brief Construct dual complex number from another of different type
         *
         * Performs only default casting on the values, no rounding or anything
         * else.
         */
        template<class U> constexpr explicit Quaternion(const Quaternion<U>& other) noexcept: _vector{other._vector}, _scalar{T(other._scalar)} {}

        /** @brief Construct quaternion from external representation */
        template<class U, class V = decltype(Implementation::QuaternionConverter<T, U>::from(std::declval<U>()))> constexpr explicit Quaternion(const U& other): Quaternion{Implementation::QuaternionConverter<T, U>::from(other)} {}

        /** @brief Copy constructor */
        constexpr /*implicit*/ Quaternion(const Quaternion<T>&) noexcept = default;

        /** @brief Convert quaternion to external representation */
        template<class U, class V = decltype(Implementation::QuaternionConverter<T, U>::to(std::declval<Quaternion<T>>()))> constexpr explicit operator U() const {
            return Implementation::QuaternionConverter<T, U>::to(*this);
        }

        /** @brief Equality comparison */
        bool operator==(const Quaternion<T>& other) const {
            return _vector == other._vector && TypeTraits<T>::equals(_scalar, other._scalar);
        }

        /** @brief Non-equality comparison */
        bool operator!=(const Quaternion<T>& other) const {
            return !operator==(other);
        }

        /**
         * @brief Whether the quaternion is normalized
         *
         * Quaternion is normalized if it has unit length: @f[
         *      |q \cdot q - 1| < 2 \epsilon + \epsilon^2 \cong 2 \epsilon
         * @f]
         * @see @ref dot(), @ref normalized()
         */
        bool isNormalized() const {
            return Implementation::isNormalizedSquared(dot());
        }

        /** @brief Vector part */
        constexpr const Vector3<T> vector() const { return _vector; }

        /** @brief Scalar part */
        constexpr T scalar() const { return _scalar; }

        /**
         * @brief Rotation angle of unit quaternion
         *
         * Expects that the quaternion is normalized. @f[
         *      \theta = 2 \cdot acos q_S
         * @f]
         * @see @ref isNormalized(), @ref axis(), @ref rotation()
         */
        Rad<T> angle() const;

        /**
         * @brief Rotation axis of unit quaternion
         *
         * Expects that the quaternion is normalized. Returns either unit-length
         * vector for valid rotation quaternion or NaN vector for
         * default-constructed quaternion. @f[
         *      \boldsymbol a = \frac{\boldsymbol q_V}{\sqrt{1 - q_S^2}}
         * @f]
         * @see @ref isNormalized(), @ref angle(), @ref rotation()
         */
        Vector3<T> axis() const;

        /**
         * @brief Convert quaternion to rotation matrix
         *
         * @see @ref fromMatrix(), @ref DualQuaternion::toMatrix(),
         *      @ref Matrix4::from(const Matrix3x3<T>&, const Vector3<T>&)
         */
        Matrix3x3<T> toMatrix() const;

        /**
         * @brief Add and assign quaternion
         *
         * The computation is done in-place. @f[
         *      p + q = [\boldsymbol p_V + \boldsymbol q_V, p_S + q_S]
         * @f]
         */
        Quaternion<T>& operator+=(const Quaternion<T>& other) {
            _vector += other._vector;
            _scalar += other._scalar;
            return *this;
        }

        /**
         * @brief Add quaternion
         *
         * @see @ref operator+=()
         */
        Quaternion<T> operator+(const Quaternion<T>& other) const {
            return Quaternion<T>(*this) += other;
        }

        /**
         * @brief Negated quaternion
         *
         * @f[
         *      -q = [-\boldsymbol q_V, -q_S]
         * @f]
         */
        Quaternion<T> operator-() const { return {-_vector, -_scalar}; }

        /**
         * @brief Subtract and assign quaternion
         *
         * The computation is done in-place. @f[
         *      p - q = [\boldsymbol p_V - \boldsymbol q_V, p_S - q_S]
         * @f]
         */
        Quaternion<T>& operator-=(const Quaternion<T>& other) {
            _vector -= other._vector;
            _scalar -= other._scalar;
            return *this;
        }

        /**
         * @brief Subtract quaternion
         *
         * @see @ref operator-=()
         */
        Quaternion<T> operator-(const Quaternion<T>& other) const {
            return Quaternion<T>(*this) -= other;
        }

        /**
         * @brief Multiply with scalar and assign
         *
         * The computation is done in-place. @f[
         *      q \cdot a = [\boldsymbol q_V \cdot a, q_S \cdot a]
         * @f]
         */
        Quaternion<T>& operator*=(T scalar) {
            _vector *= scalar;
            _scalar *= scalar;
            return *this;
        }

        /**
         * @brief Multiply with scalar
         *
         * @see @ref operator*=(T)
         */
        Quaternion<T> operator*(T scalar) const {
            return Quaternion<T>(*this) *= scalar;
        }

        /**
         * @brief Divide with scalar and assign
         *
         * The computation is done in-place. @f[
         *      \frac q a = [\frac {\boldsymbol q_V} a, \frac {q_S} a]
         * @f]
         */
        Quaternion<T>& operator/=(T scalar) {
            _vector /= scalar;
            _scalar /= scalar;
            return *this;
        }

        /**
         * @brief Divide with scalar
         *
         * @see @ref operator/=(T)
         */
        Quaternion<T> operator/(T scalar) const {
            return Quaternion<T>(*this) /= scalar;
        }

        /**
         * @brief Multiply with quaternion
         *
         * @f[
         *      p q = [p_S \boldsymbol q_V + q_S \boldsymbol p_V + \boldsymbol p_V \times \boldsymbol q_V,
         *             p_S q_S - \boldsymbol p_V \cdot \boldsymbol q_V]
         * @f]
         */
        Quaternion<T> operator*(const Quaternion<T>& other) const;

        /**
         * @brief Dot product of the quaternion
         *
         * Should be used instead of @ref length() for comparing quaternion
         * length with other values, because it doesn't compute the square
         * root. @f[
         *      q \cdot q = \boldsymbol q_V \cdot \boldsymbol q_V + q_S^2
         * @f]
         * @see @ref isNormalized(),
         *      @ref dot(const Quaternion<T>&, const Quaternion<T>&)
         */
        T dot() const { return Math::dot(*this, *this); }

        /**
         * @brief Quaternion length
         *
         * See also @ref dot() const which is faster for comparing length with
         * other values. @f[
         *      |q| = \sqrt{q \cdot q}
         * @f]
         * @see @ref isNormalized()
         */
        T length() const { return std::sqrt(dot()); }

        /**
         * @brief Normalized quaternion (of unit length)
         *
         * @see @ref isNormalized()
         */
        Quaternion<T> normalized() const { return (*this)/length(); }

        /**
         * @brief Conjugated quaternion
         *
         * @f[
         *      q^* = [-\boldsymbol q_V, q_S]
         * @f]
         */
        Quaternion<T> conjugated() const { return {-_vector, _scalar}; }

        /**
         * @brief Inverted quaternion
         *
         * See @ref invertedNormalized() which is faster for normalized
         * quaternions. @f[
         *      q^{-1} = \frac{q^*}{|q|^2} = \frac{q^*}{q \cdot q}
         * @f]
         */
        Quaternion<T> inverted() const { return conjugated()/dot(); }

        /**
         * @brief Inverted normalized quaternion
         *
         * Equivalent to @ref conjugated(). Expects that the quaternion is
         * normalized. @f[
         *      q^{-1} = \frac{q^*}{|q|^2} = q^*
         * @f]
         * @see @ref isNormalized(), @ref inverted()
         */
        Quaternion<T> invertedNormalized() const;

        /**
         * @brief Rotate vector with quaternion
         *
         * See @ref transformVectorNormalized(), which is faster for normalized
         * quaternions. @f[
         *      v' = qvq^{-1} = q [\boldsymbol v, 0] q^{-1}
         * @f]
         * @see @ref Quaternion(const Vector3<T>&), @ref vector(),
         *      @ref Matrix4::transformVector(),
         *      @ref DualQuaternion::transformPoint(),
         *      @ref Complex::transformVector()
         */
        Vector3<T> transformVector(const Vector3<T>& vector) const {
            return ((*this)*Quaternion<T>(vector)*inverted()).vector();
        }

        /**
         * @brief Rotate vector with normalized quaternion
         *
         * Faster alternative to @ref transformVector(), expects that the
         * quaternion is normalized. Done using the following equation: @f[
         *      \begin{array}{rcl}
         *          \boldsymbol t & = & 2 (\boldsymbol q_V \times \boldsymbol v) \\
         *          \boldsymbol v' & = & \boldsymbol v + q_S \boldsymbol t + \boldsymbol q_V \times \boldsymbol t
         *      \end{array}
         * @f]
         * Which is equivalent to the common equation (source:
         * https://molecularmusings.wordpress.com/2013/05/24/a-faster-quaternion-vector-multiplication/): @f[
         *      v' = qvq^{-1} = qvq^* = q [\boldsymbol v, 0] q^*
         * @f]
         * @see @ref isNormalized(), @ref Quaternion(const Vector3<T>&),
         *      @ref vector(), @ref Matrix4::transformVector(),
         *      @ref DualQuaternion::transformPointNormalized(),
         *      @ref Complex::transformVector()
         */
        Vector3<T> transformVectorNormalized(const Vector3<T>& vector) const;

    private:
        /* Used to avoid including Functions.h */
        constexpr static T pow2(T value) {
            return value*value;
        }

        Vector3<T> _vector;
        T _scalar;
};

/** @relates Quaternion
@brief Multiply scalar with quaternion

Same as @ref Quaternion::operator*(T) const.
*/
template<class T> inline Quaternion<T> operator*(T scalar, const Quaternion<T>& quaternion) {
    return quaternion*scalar;
}

/** @relates Quaternion
@brief Divide quaternion with number and invert

@f[
    \frac a q = [\frac a {\boldsymbol q_V}, \frac a {q_S}]
@f]
@see @ref Quaternion::operator/()
*/
template<class T> inline Quaternion<T> operator/(T scalar, const Quaternion<T>& quaternion) {
    return {scalar/quaternion.vector(), scalar/quaternion.scalar()};
}

/** @debugoperator{Quaternion} */
template<class T> Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug& debug, const Quaternion<T>& value) {
    return debug << "Quaternion({" << Corrade::Utility::Debug::nospace
        << value.vector().x() << Corrade::Utility::Debug::nospace << ","
        << value.vector().y() << Corrade::Utility::Debug::nospace << ","
        << value.vector().z() << Corrade::Utility::Debug::nospace << "},"
        << value.scalar() << Corrade::Utility::Debug::nospace << ")";
}

/* Explicit instantiation for commonly used types */
#ifndef DOXYGEN_GENERATING_OUTPUT
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const Quaternion<Float>&);
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const Quaternion<Double>&);
#endif

namespace Implementation {

/* No assertions fired, for internal use. Not private member because used from
   outside the class. */
template<class T> Quaternion<T> quaternionFromMatrix(const Matrix3x3<T>& m) {
    const Vector<3, T> diagonal = m.diagonal();
    const T trace = diagonal.sum();

    /* Diagonal is positive */
    if(trace > T(0)) {
        const T s = std::sqrt(trace + T(1));
        const T t = T(0.5)/s;
        return {Vector3<T>(m[1][2] - m[2][1],
                           m[2][0] - m[0][2],
                           m[0][1] - m[1][0])*t, s*T(0.5)};
    }

    /* Diagonal is negative */
    std::size_t i = 0;
    if(diagonal[1] > diagonal[0]) i = 1;
    if(diagonal[2] > diagonal[i]) i = 2;

    const std::size_t j = (i + 1) % 3;
    const std::size_t k = (i + 2) % 3;

    const T s = std::sqrt(diagonal[i] - diagonal[j] - diagonal[k] + T(1));
    const T t = (s == T(0) ? T(0) : T(0.5)/s);

    Vector3<T> vec;
    vec[i] = s*T(0.5);
    vec[j] = (m[i][j] + m[j][i])*t;
    vec[k] = (m[i][k] + m[k][i])*t;

    return {vec, (m[j][k] - m[k][j])*t};
}

}

template<class T> inline Quaternion<T> Quaternion<T>::rotation(const Rad<T> angle, const Vector3<T>& normalizedAxis) {
    CORRADE_ASSERT(normalizedAxis.isNormalized(),
        "Math::Quaternion::rotation(): axis must be normalized", {});
    return {normalizedAxis*std::sin(T(angle)/2), std::cos(T(angle)/2)};
}

template<class T> inline Quaternion<T> Quaternion<T>::fromMatrix(const Matrix3x3<T>& matrix) {
    CORRADE_ASSERT(matrix.isOrthogonal(), "Math::Quaternion::fromMatrix(): the matrix is not orthogonal", {});
    return Implementation::quaternionFromMatrix(matrix);
}

template<class T> inline Rad<T> Quaternion<T>::angle() const {
    CORRADE_ASSERT(isNormalized(), "Math::Quaternion::angle(): quaternion must be normalized", {});
    return Rad<T>(T(2)*std::acos(_scalar));
}

template<class T> inline Vector3<T> Quaternion<T>::axis() const {
    CORRADE_ASSERT(isNormalized(), "Math::Quaternion::axis(): quaternion must be normalized", {});
    return _vector/std::sqrt(1-pow2(_scalar));
}

template<class T> Matrix3x3<T> Quaternion<T>::toMatrix() const {
    return {
        Vector<3, T>(T(1) - 2*pow2(_vector.y()) - 2*pow2(_vector.z()),
            2*_vector.x()*_vector.y() + 2*_vector.z()*_scalar,
                2*_vector.x()*_vector.z() - 2*_vector.y()*_scalar),
        Vector<3, T>(2*_vector.x()*_vector.y() - 2*_vector.z()*_scalar,
            T(1) - 2*pow2(_vector.x()) - 2*pow2(_vector.z()),
                2*_vector.y()*_vector.z() + 2*_vector.x()*_scalar),
        Vector<3, T>(2*_vector.x()*_vector.z() + 2*_vector.y()*_scalar,
            2*_vector.y()*_vector.z() - 2*_vector.x()*_scalar,
                T(1) - 2*pow2(_vector.x()) - 2*pow2(_vector.y()))
    };
}

template<class T> inline Quaternion<T> Quaternion<T>::operator*(const Quaternion<T>& other) const {
    return {_scalar*other._vector + other._scalar*_vector + Math::cross(_vector, other._vector),
            _scalar*other._scalar - Math::dot(_vector, other._vector)};
}

template<class T> inline Quaternion<T> Quaternion<T>::invertedNormalized() const {
    CORRADE_ASSERT(isNormalized(), "Math::Quaternion::invertedNormalized(): quaternion must be normalized", {});
    return conjugated();
}

template<class T> inline Vector3<T> Quaternion<T>::transformVectorNormalized(const Vector3<T>& vector) const {
    CORRADE_ASSERT(isNormalized(), "Math::Quaternion::transformVectorNormalized(): quaternion must be normalized", {});
    const Vector3<T> t = T(2)*Math::cross(_vector, vector);
    return vector + _scalar*t + Math::cross(_vector, t);
}

}}

#endif