ImathQuat.h

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//
// SPDX-License-Identifier: BSD-3-Clause
// Copyright Contributors to the OpenEXR Project.
//

//
// A quaternion
//
// "Quaternions came from Hamilton ... and have been an unmixed
// evil to those who have touched them in any way. Vector is a
// useless survival ... and has never been of the slightest use
// to any creature."
//
//   - Lord Kelvin
//

#ifndef INCLUDED_IMATHQUAT_H
#define INCLUDED_IMATHQUAT_H

#include "ImathExport.h"
#include "ImathNamespace.h"

#include "ImathMatrix.h"

#include <iostream>

IMATH_INTERNAL_NAMESPACE_HEADER_ENTER

#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER
// Disable MS VC++ warnings about conversion from double to float
#    pragma warning(push)
#    pragma warning(disable : 4244)
#endif

///
/// The Quat class implements the quaternion numerical type -- you
/// will probably want to use this class to represent orientations
/// in R3 and to convert between various euler angle reps. You
/// should probably use Imath::Euler<> for that.
///

template <class T> class IMATH_EXPORT_TEMPLATE_TYPE Quat
{
  public:

    /// @{
    /// @name Direct access to elements
    
    /// The real part
    T r;       

    /// The imaginary vector
    Vec3<T> v;

    /// @}

    /// Element access: q[0] is the real part, (q[1],q[2],q[3]) is the
    /// imaginary part.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 T& operator[] (int index) IMATH_NOEXCEPT; // as 4D vector

    /// Element access: q[0] is the real part, (q[1],q[2],q[3]) is the
    /// imaginary part.
    IMATH_HOSTDEVICE constexpr T operator[] (int index) const IMATH_NOEXCEPT;

    /// @{
    /// @name Constructors

    /// Default constructor is the identity quat
    IMATH_HOSTDEVICE constexpr Quat() IMATH_NOEXCEPT;

    /// Copy constructor
    IMATH_HOSTDEVICE constexpr Quat (const Quat& q) IMATH_NOEXCEPT;

    /// Construct from a quaternion of a another base type
    template <class S> IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat (const Quat<S>& q) IMATH_NOEXCEPT;

    /// Initialize with real part `s` and imaginary vector 1(i,j,k)`
    IMATH_HOSTDEVICE constexpr Quat (T s, T i, T j, T k) IMATH_NOEXCEPT;

    /// Initialize with real part `s` and imaginary vector `d`
    IMATH_HOSTDEVICE constexpr Quat (T s, Vec3<T> d) IMATH_NOEXCEPT;

    /// The identity quaternion
    IMATH_HOSTDEVICE constexpr static Quat<T> identity() IMATH_NOEXCEPT;

    /// Assignment
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator= (const Quat<T>& q) IMATH_NOEXCEPT;

    /// Destructor
    IMATH_HOSTDEVICE ~Quat() IMATH_NOEXCEPT = default;

    /// @}

    /// @{
    /// @name Basic Algebra 
    /// 
    /// Note that the operator return values are *NOT* normalized
    //

    /// Quaternion multiplication
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator*= (const Quat<T>& q) IMATH_NOEXCEPT;
    
    /// Scalar multiplication: multiply both real and imaginary parts
    /// by the given scalar.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator*= (T t) IMATH_NOEXCEPT;

    /// Quaterion division, using the inverse()
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator/= (const Quat<T>& q) IMATH_NOEXCEPT;

    /// Scalar division: multiply both real and imaginary parts
    /// by the given scalar.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator/= (T t) IMATH_NOEXCEPT;

    /// Quaternion addition
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator+= (const Quat<T>& q) IMATH_NOEXCEPT;

    /// Quaternion subtraction
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 const Quat<T>& operator-= (const Quat<T>& q) IMATH_NOEXCEPT;

    /// Equality
    template <class S> IMATH_HOSTDEVICE constexpr bool operator== (const Quat<S>& q) const IMATH_NOEXCEPT;

    /// Inequality
    template <class S> IMATH_HOSTDEVICE constexpr bool operator!= (const Quat<S>& q) const IMATH_NOEXCEPT;

    /// @}


    /// @{
    /// @name Query
    
    /// Return the R4 length
    IMATH_HOSTDEVICE constexpr T length() const IMATH_NOEXCEPT; // in R4

    /// Return the angle of the axis/angle representation
    IMATH_HOSTDEVICE constexpr T angle() const IMATH_NOEXCEPT;

    /// Return the axis of the axis/angle representation
    IMATH_HOSTDEVICE constexpr Vec3<T> axis() const IMATH_NOEXCEPT;

    /// Return a 3x3 rotation matrix
    IMATH_HOSTDEVICE constexpr Matrix33<T> toMatrix33() const IMATH_NOEXCEPT;

    /// Return a 4x4 rotation matrix
    IMATH_HOSTDEVICE constexpr Matrix44<T> toMatrix44() const IMATH_NOEXCEPT;

    /// Return the logarithm of the quaterion
    IMATH_HOSTDEVICE Quat<T> log() const IMATH_NOEXCEPT;

    /// Return the exponent of the quaterion
    IMATH_HOSTDEVICE Quat<T> exp() const IMATH_NOEXCEPT;

    /// @}

    /// @{
    /// @name Utility Methods
    
    /// Invert in place: this = 1 / this.
    /// @return const reference to this.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T>& invert() IMATH_NOEXCEPT;

    /// Return 1/this, leaving this unchanged.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T> inverse() const IMATH_NOEXCEPT;

    /// Normalize in place
    /// @return const reference to this.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T>& normalize() IMATH_NOEXCEPT;
    
    /// Return a normalized quaternion, leaving this unmodified.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T> normalized() const IMATH_NOEXCEPT;

    /// Rotate the given point by the quaterion.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Vec3<T> rotateVector (const Vec3<T>& original) const IMATH_NOEXCEPT;

    /// Return the Euclidean inner product.
    IMATH_HOSTDEVICE constexpr T euclideanInnerProduct (const Quat<T>& q) const IMATH_NOEXCEPT;

    /// Set the quaterion to be a rotation around the given axis by the
    /// given angle.
    /// @return const reference to this.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T>& setAxisAngle (const Vec3<T>& axis, T radians) IMATH_NOEXCEPT;

    /// Set the quaternion to be a rotation that transforms the
    /// direction vector `fromDirection` to `toDirection`
    /// @return const reference to this.
    IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T>&
    setRotation (const Vec3<T>& fromDirection, const Vec3<T>& toDirection) IMATH_NOEXCEPT;

    /// @}
    
    /// The base type: In templates that accept a parameter `V`, you
    /// can refer to `T` as `V::BaseType`
    typedef T BaseType;

  private:
    IMATH_HOSTDEVICE void setRotationInternal (const Vec3<T>& f0, const Vec3<T>& t0, Quat<T>& q) IMATH_NOEXCEPT;
};

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T> slerp (const Quat<T>& q1, const Quat<T>& q2, T t) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T> slerpShortestArc (const Quat<T>& q1, const Quat<T>& q2, T t) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Quat<T>
squad (const Quat<T>& q1, const Quat<T>& q2, const Quat<T>& qa, const Quat<T>& qb, T t) IMATH_NOEXCEPT;

///
/// From advanced Animation and Rendering Techniques by Watt and Watt,
/// Page 366:
///
/// computing the inner quadrangle points (qa and qb) to guarantee
/// tangent continuity.
template <class T>
IMATH_HOSTDEVICE void intermediate (const Quat<T>& q0,
                   const Quat<T>& q1,
                   const Quat<T>& q2,
                   const Quat<T>& q3,
                   Quat<T>& qa,
                   Quat<T>& qb) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Matrix33<T> operator* (const Matrix33<T>& M, const Quat<T>& q) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Matrix33<T> operator* (const Quat<T>& q, const Matrix33<T>& M) IMATH_NOEXCEPT;

template <class T> std::ostream& operator<< (std::ostream& o, const Quat<T>& q);

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator* (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator/ (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator/ (const Quat<T>& q, T t) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator* (const Quat<T>& q, T t) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator* (T t, const Quat<T>& q) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator+ (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator- (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator~ (const Quat<T>& q) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE constexpr Quat<T> operator- (const Quat<T>& q) IMATH_NOEXCEPT;

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 Vec3<T> operator* (const Vec3<T>& v, const Quat<T>& q) IMATH_NOEXCEPT;

/// Quaternion of type float
typedef Quat<float> Quatf;

/// Quaternion of type double
typedef Quat<double> Quatd;

//---------------
// Implementation
//---------------

template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>::Quat() IMATH_NOEXCEPT : r (1), v (0, 0, 0)
{
    // empty
}

template <class T>
template <class S>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>::Quat (const Quat<S>& q) IMATH_NOEXCEPT : r (q.r), v (q.v)
{
    // empty
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>::Quat (T s, T i, T j, T k) IMATH_NOEXCEPT : r (s), v (i, j, k)
{
    // empty
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>::Quat (T s, Vec3<T> d) IMATH_NOEXCEPT : r (s), v (d)
{
    // empty
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>::Quat (const Quat<T>& q) IMATH_NOEXCEPT : r (q.r), v (q.v)
{
    // empty
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
Quat<T>::identity() IMATH_NOEXCEPT
{
    return Quat<T>();
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator= (const Quat<T>& q) IMATH_NOEXCEPT
{
    r = q.r;
    v = q.v;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator*= (const Quat<T>& q) IMATH_NOEXCEPT
{
    T rtmp = r * q.r - (v ^ q.v);
    v      = r * q.v + v * q.r + v % q.v;
    r      = rtmp;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator*= (T t) IMATH_NOEXCEPT
{
    r *= t;
    v *= t;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator/= (const Quat<T>& q) IMATH_NOEXCEPT
{
    *this = *this * q.inverse();
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator/= (T t) IMATH_NOEXCEPT
{
    r /= t;
    v /= t;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator+= (const Quat<T>& q) IMATH_NOEXCEPT
{
    r += q.r;
    v += q.v;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline const Quat<T>&
Quat<T>::operator-= (const Quat<T>& q) IMATH_NOEXCEPT
{
    r -= q.r;
    v -= q.v;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline T&
Quat<T>::operator[] (int index) IMATH_NOEXCEPT
{
    return index ? v[index - 1] : r;
}

template <class T>
IMATH_HOSTDEVICE constexpr inline T
Quat<T>::operator[] (int index) const IMATH_NOEXCEPT
{
    return index ? v[index - 1] : r;
}

template <class T>
template <class S>
IMATH_HOSTDEVICE constexpr inline bool
Quat<T>::operator== (const Quat<S>& q) const IMATH_NOEXCEPT
{
    return r == q.r && v == q.v;
}

template <class T>
template <class S>
IMATH_HOSTDEVICE constexpr inline bool
Quat<T>::operator!= (const Quat<S>& q) const IMATH_NOEXCEPT
{
    return r != q.r || v != q.v;
}

/// 4D dot product
template <class T>
IMATH_HOSTDEVICE constexpr inline T
operator^ (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    return q1.r * q2.r + (q1.v ^ q2.v);
}

template <class T>
IMATH_HOSTDEVICE constexpr inline T
Quat<T>::length() const IMATH_NOEXCEPT
{
    return std::sqrt (r * r + (v ^ v));
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>&
Quat<T>::normalize() IMATH_NOEXCEPT
{
    if (T l = length())
    {
        r /= l;
        v /= l;
    }
    else
    {
        r = 1;
        v = Vec3<T> (0);
    }

    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
Quat<T>::normalized() const IMATH_NOEXCEPT
{
    if (T l = length())
        return Quat (r / l, v / l);

    return Quat();
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
Quat<T>::inverse() const IMATH_NOEXCEPT
{
    //
    // 1    Q*
    // - = ----   where Q* is conjugate (operator~)
    // Q   Q* Q   and (Q* Q) == Q ^ Q (4D dot)
    //

    T qdot = *this ^ *this;
    return Quat (r / qdot, -v / qdot);
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>&
Quat<T>::invert() IMATH_NOEXCEPT
{
    T qdot = (*this) ^ (*this);
    r /= qdot;
    v = -v / qdot;
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Vec3<T>
Quat<T>::rotateVector (const Vec3<T>& original) const IMATH_NOEXCEPT
{
    //
    // Given a vector p and a quaternion q (aka this),
    // calculate p' = qpq*
    //
    // Assumes unit quaternions (because non-unit
    // quaternions cannot be used to rotate vectors
    // anyway).
    //

    Quat<T> vec (0, original); // temporarily promote grade of original
    Quat<T> inv (*this);
    inv.v *= -1; // unit multiplicative inverse
    Quat<T> result = *this * vec * inv;
    return result.v;
}

template <class T>
IMATH_HOSTDEVICE constexpr inline T
Quat<T>::euclideanInnerProduct (const Quat<T>& q) const IMATH_NOEXCEPT
{
    return r * q.r + v.x * q.v.x + v.y * q.v.y + v.z * q.v.z;
}

///
/// Compute the angle between two quaternions,
/// interpreting the quaternions as 4D vectors.
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline T
angle4D (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    Quat<T> d = q1 - q2;
    T lengthD = std::sqrt (d ^ d);

    Quat<T> s = q1 + q2;
    T lengthS = std::sqrt (s ^ s);

    return 2 * std::atan2 (lengthD, lengthS);
}

///
/// Spherical linear interpolation.
/// Assumes q1 and q2 are normalized and that q1 != -q2.
///
/// This method does *not* interpolate along the shortest
/// arc between q1 and q2.  If you desire interpolation
/// along the shortest arc, and q1^q2 is negative, then
/// consider calling slerpShortestArc(), below, or flipping
/// the second quaternion explicitly.
///
/// The implementation of squad() depends on a slerp()
/// that interpolates as is, without the automatic
/// flipping.
///
/// Don Hatch explains the method we use here on his
/// web page, The Right Way to Calculate Stuff, at
/// http://www.plunk.org/~hatch/rightway.php
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
slerp (const Quat<T>& q1, const Quat<T>& q2, T t) IMATH_NOEXCEPT
{
    T a = angle4D (q1, q2);
    T s = 1 - t;

    Quat<T> q = sinx_over_x (s * a) / sinx_over_x (a) * s * q1 +
                sinx_over_x (t * a) / sinx_over_x (a) * t * q2;

    return q.normalized();
}

///
/// Spherical linear interpolation along the shortest
/// arc from q1 to either q2 or -q2, whichever is closer.
/// Assumes q1 and q2 are unit quaternions.
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
slerpShortestArc (const Quat<T>& q1, const Quat<T>& q2, T t) IMATH_NOEXCEPT
{
    if ((q1 ^ q2) >= 0)
        return slerp (q1, q2, t);
    else
        return slerp (q1, -q2, t);
}

///
/// Spherical Cubic Spline Interpolation - from Advanced Animation and
/// Rendering Techniques by Watt and Watt, Page 366:
///
/// A spherical curve is constructed using three spherical linear
/// interpolations of a quadrangle of unit quaternions: q1, qa, qb,
/// q2.  Given a set of quaternion keys: q0, q1, q2, q3, this routine
/// does the interpolation between q1 and q2 by constructing two
/// intermediate quaternions: qa and qb. The qa and qb are computed by
/// the intermediate function to guarantee the continuity of tangents
/// across adjacent cubic segments. The qa represents in-tangent for
/// q1 and the qb represents the out-tangent for q2.
///
/// The q1 q2 is the cubic segment being interpolated.
///
/// The q0 is from the previous adjacent segment and q3 is from the
/// next adjacent segment. The q0 and q3 are used in computing qa and
/// qb.
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
spline (const Quat<T>& q0, const Quat<T>& q1, const Quat<T>& q2, const Quat<T>& q3, T t) IMATH_NOEXCEPT
{
    Quat<T> qa     = intermediate (q0, q1, q2);
    Quat<T> qb     = intermediate (q1, q2, q3);
    Quat<T> result = squad (q1, qa, qb, q2, t);

    return result;
}

///
/// Spherical Quadrangle Interpolation - from Advanced Animation and
/// Rendering Techniques by Watt and Watt, Page 366:
///
/// It constructs a spherical cubic interpolation as a series of three
/// spherical linear interpolations of a quadrangle of unit
/// quaternions.
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
squad (const Quat<T>& q1, const Quat<T>& qa, const Quat<T>& qb, const Quat<T>& q2, T t) IMATH_NOEXCEPT
{
    Quat<T> r1     = slerp (q1, q2, t);
    Quat<T> r2     = slerp (qa, qb, t);
    Quat<T> result = slerp (r1, r2, 2 * t * (1 - t));

    return result;
}

/// Compute the intermediate point between three quaternions `q0`, `q1`,
/// and `q2`.
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>
intermediate (const Quat<T>& q0, const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    Quat<T> q1inv = q1.inverse();
    Quat<T> c1    = q1inv * q2;
    Quat<T> c2    = q1inv * q0;
    Quat<T> c3    = (T) (-0.25) * (c2.log() + c1.log());
    Quat<T> qa    = q1 * c3.exp();
    qa.normalize();
    return qa;
}

template <class T>
IMATH_HOSTDEVICE inline Quat<T>
Quat<T>::log() const IMATH_NOEXCEPT
{
    //
    // For unit quaternion, from Advanced Animation and
    // Rendering Techniques by Watt and Watt, Page 366:
    //

    T theta = std::acos (std::min (r, (T) 1.0));

    if (theta == 0)
        return Quat<T> (0, v);

    T sintheta = std::sin (theta);

    T k;
    if (std::abs(sintheta) < 1 && std::abs(theta) >= std::numeric_limits<T>::max() * std::abs(sintheta))
        k = 1;
    else
        k = theta / sintheta;

    return Quat<T> ((T) 0, v.x * k, v.y * k, v.z * k);
}

template <class T>
IMATH_HOSTDEVICE inline Quat<T>
Quat<T>::exp() const IMATH_NOEXCEPT
{
    //
    // For pure quaternion (zero scalar part):
    // from Advanced Animation and Rendering
    // Techniques by Watt and Watt, Page 366:
    //

    T theta    = v.length();
    T sintheta = std::sin (theta);

    T k;
    if (abs (theta) < 1 && abs (sintheta) >= std::numeric_limits<T>::max() * abs (theta))
        k = 1;
    else
        k = sintheta / theta;

    T costheta = std::cos (theta);

    return Quat<T> (costheta, v.x * k, v.y * k, v.z * k);
}

template <class T>
IMATH_HOSTDEVICE constexpr inline T
Quat<T>::angle() const IMATH_NOEXCEPT
{
    return 2 * std::atan2 (v.length(), r);
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Vec3<T>
Quat<T>::axis() const IMATH_NOEXCEPT
{
    return v.normalized();
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>&
Quat<T>::setAxisAngle (const Vec3<T>& axis, T radians) IMATH_NOEXCEPT
{
    r = std::cos (radians / 2);
    v = axis.normalized() * std::sin (radians / 2);
    return *this;
}

template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Quat<T>&
Quat<T>::setRotation (const Vec3<T>& from, const Vec3<T>& to) IMATH_NOEXCEPT
{
    //
    // Create a quaternion that rotates vector from into vector to,
    // such that the rotation is around an axis that is the cross
    // product of from and to.
    //
    // This function calls function setRotationInternal(), which is
    // numerically accurate only for rotation angles that are not much
    // greater than pi/2.  In order to achieve good accuracy for angles
    // greater than pi/2, we split large angles in half, and rotate in
    // two steps.
    //

    //
    // Normalize from and to, yielding f0 and t0.
    //

    Vec3<T> f0 = from.normalized();
    Vec3<T> t0 = to.normalized();

    if ((f0 ^ t0) >= 0)
    {
        //
        // The rotation angle is less than or equal to pi/2.
        //

        setRotationInternal (f0, t0, *this);
    }
    else
    {
        //
        // The angle is greater than pi/2.  After computing h0,
        // which is halfway between f0 and t0, we rotate first
        // from f0 to h0, then from h0 to t0.
        //

        Vec3<T> h0 = (f0 + t0).normalized();

        if ((h0 ^ h0) != 0)
        {
            setRotationInternal (f0, h0, *this);

            Quat<T> q;
            setRotationInternal (h0, t0, q);

            *this *= q;
        }
        else
        {
            //
            // f0 and t0 point in exactly opposite directions.
            // Pick an arbitrary axis that is orthogonal to f0,
            // and rotate by pi.
            //

            r = T (0);

            Vec3<T> f02 = f0 * f0;

            if (f02.x <= f02.y && f02.x <= f02.z)
                v = (f0 % Vec3<T> (1, 0, 0)).normalized();
            else if (f02.y <= f02.z)
                v = (f0 % Vec3<T> (0, 1, 0)).normalized();
            else
                v = (f0 % Vec3<T> (0, 0, 1)).normalized();
        }
    }

    return *this;
}

template <class T>
IMATH_HOSTDEVICE inline void
Quat<T>::setRotationInternal (const Vec3<T>& f0, const Vec3<T>& t0, Quat<T>& q) IMATH_NOEXCEPT
{
    //
    // The following is equivalent to setAxisAngle(n,2*phi),
    // where the rotation axis, n, is orthogonal to the f0 and
    // t0 vectors, and 2*phi is the angle between f0 and t0.
    //
    // This function is called by setRotation(), above; it assumes
    // that f0 and t0 are normalized and that the angle between
    // them is not much greater than pi/2.  This function becomes
    // numerically inaccurate if f0 and t0 point into nearly
    // opposite directions.
    //

    //
    // Find a normalized vector, h0, that is halfway between f0 and t0.
    // The angle between f0 and h0 is phi.
    //

    Vec3<T> h0 = (f0 + t0).normalized();

    //
    // Store the rotation axis and rotation angle.
    //

    q.r = f0 ^ h0; //  f0 ^ h0 == cos (phi)
    q.v = f0 % h0; // (f0 % h0).length() == sin (phi)
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Matrix33<T>
Quat<T>::toMatrix33() const IMATH_NOEXCEPT
{
    return Matrix33<T> (1 - 2 * (v.y * v.y + v.z * v.z),
                        2 * (v.x * v.y + v.z * r),
                        2 * (v.z * v.x - v.y * r),

                        2 * (v.x * v.y - v.z * r),
                        1 - 2 * (v.z * v.z + v.x * v.x),
                        2 * (v.y * v.z + v.x * r),

                        2 * (v.z * v.x + v.y * r),
                        2 * (v.y * v.z - v.x * r),
                        1 - 2 * (v.y * v.y + v.x * v.x));
}

template <class T>
IMATH_HOSTDEVICE constexpr inline Matrix44<T>
Quat<T>::toMatrix44() const IMATH_NOEXCEPT
{
    return Matrix44<T> (1 - 2 * (v.y * v.y + v.z * v.z),
                        2 * (v.x * v.y + v.z * r),
                        2 * (v.z * v.x - v.y * r),
                        0,
                        2 * (v.x * v.y - v.z * r),
                        1 - 2 * (v.z * v.z + v.x * v.x),
                        2 * (v.y * v.z + v.x * r),
                        0,
                        2 * (v.z * v.x + v.y * r),
                        2 * (v.y * v.z - v.x * r),
                        1 - 2 * (v.y * v.y + v.x * v.x),
                        0,
                        0,
                        0,
                        0,
                        1);
}

/// Transform the quaternion by the matrix
/// @return M * q
template <class T>
IMATH_HOSTDEVICE constexpr inline Matrix33<T>
operator* (const Matrix33<T>& M, const Quat<T>& q) IMATH_NOEXCEPT
{
    return M * q.toMatrix33();
}

/// Transform the matrix by the quaterion:
/// @return q * M
template <class T>
IMATH_HOSTDEVICE constexpr inline Matrix33<T>
operator* (const Quat<T>& q, const Matrix33<T>& M) IMATH_NOEXCEPT
{
    return q.toMatrix33() * M;
}

/// Stream output as "(r x y z)"
template <class T>
std::ostream&
operator<< (std::ostream& o, const Quat<T>& q)
{
    return o << "(" << q.r << " " << q.v.x << " " << q.v.y << " " << q.v.z << ")";
}

/// Quaterion multiplication
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator* (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    return Quat<T> (q1.r * q2.r - (q1.v ^ q2.v), q1.r * q2.v + q1.v * q2.r + q1.v % q2.v);
}

/// Quaterion division
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator/ (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    return q1 * q2.inverse();
}

/// Quaterion division
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator/ (const Quat<T>& q, T t) IMATH_NOEXCEPT
{
    return Quat<T> (q.r / t, q.v / t);
}

/// Quaterion*scalar multiplication
/// @return q * t
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator* (const Quat<T>& q, T t) IMATH_NOEXCEPT
{
    return Quat<T> (q.r * t, q.v * t);
}

/// Quaterion*scalar multiplication
/// @return q * t
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator* (T t, const Quat<T>& q) IMATH_NOEXCEPT
{
    return Quat<T> (q.r * t, q.v * t);
}

/// Quaterion addition
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator+ (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    return Quat<T> (q1.r + q2.r, q1.v + q2.v);
}

/// Quaterion subtraction
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator- (const Quat<T>& q1, const Quat<T>& q2) IMATH_NOEXCEPT
{
    return Quat<T> (q1.r - q2.r, q1.v - q2.v);
}

/// Compute the conjugate
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator~ (const Quat<T>& q) IMATH_NOEXCEPT
{
    return Quat<T> (q.r, -q.v);
}

/// Negate the quaterion
template <class T>
IMATH_HOSTDEVICE constexpr inline Quat<T>
operator- (const Quat<T>& q) IMATH_NOEXCEPT
{
    return Quat<T> (-q.r, -q.v);
}

/// Quaterion*vector multiplcation
/// @return v * q
template <class T>
IMATH_HOSTDEVICE IMATH_CONSTEXPR14 inline Vec3<T>
operator* (const Vec3<T>& v, const Quat<T>& q) IMATH_NOEXCEPT
{
    Vec3<T> a = q.v % v;
    Vec3<T> b = q.v % a;
    return v + T (2) * (q.r * a + b);
}

#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER
#    pragma warning(pop)
#endif

IMATH_INTERNAL_NAMESPACE_HEADER_EXIT

#endif // INCLUDED_IMATHQUAT_H