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magnum/src/Magnum/Math/DualQuaternion.h

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#ifndef Magnum_Math_DualQuaternion_h
#define Magnum_Math_DualQuaternion_h
/*
This file is part of Magnum.
Copyright © 2010, 2011, 2012, 2013, 2014, 2015
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::DualQuaternion
*/
#include <cmath>
#include "Magnum/Math/Dual.h"
#include "Magnum/Math/Matrix4.h"
#include "Magnum/Math/Quaternion.h"
namespace Magnum { namespace Math {
namespace Implementation {
template<class, class> struct DualQuaternionConverter;
}
/** @relatesalso DualQuaternion
@brief Screw linear interpolation of two dual quaternions
@param normalizedA First dual quaternion
@param normalizedB Second dual quaternion
@param t Interpolation phase (from range @f$ [0; 1] @f$)
Expects that both dual quaternions are normalized. @f[
\begin{array}{rcl}
l + \epsilon m & = & \hat q_A^* \hat q_B \\
\frac{\hat a} 2 & = & acos \left( l_S \right) - \epsilon m_S \frac 1 {|l_V|} \\
\hat {\boldsymbol n} & = & \boldsymbol n_0 + \epsilon \boldsymbol n_\epsilon
~~~~~~~~ \boldsymbol n_0 = l_V \frac 1 {|l_V|}
~~~~~~~~ \boldsymbol n_\epsilon = \left( m_V - {\boldsymbol n}_0 \frac {a_\epsilon} 2 l_S \right)\frac 1 {|l_V|} \\
{\hat q}_{ScLERP} & = & \hat q_A (\hat q_A^* \hat q_B)^t =
\hat q_A \left[ \hat {\boldsymbol n} sin \left( t \frac {\hat a} 2 \right), cos \left( t \frac {\hat a} 2 \right) \right] \\
\end{array}
@f]
@see @ref DualQuaternion::isNormalized(),
@ref slerp(const Quaternion<T>&, const Quaternion<T>&, T),
@ref lerp(const T&, const T&, U)
*/
template<class T> inline DualQuaternion<T> sclerp(const DualQuaternion<T>& normalizedA, const DualQuaternion<T>& normalizedB, const T t) {
CORRADE_ASSERT(normalizedA.isNormalized() && normalizedB.isNormalized(),
"Math::sclerp(): dual quaternions must be normalized", {});
const T dotResult = dot(normalizedA.real().vector(), normalizedB.real().vector());
/* l + εm = q_A^**q_B, multiplying with -1 ensures shortest path when dot < 0 */
const DualQuaternion<T> diff = normalizedA.quaternionConjugated()*(dotResult < T(0) ? -normalizedB : normalizedB);
const Quaternion<T>& l = diff.real();
const Quaternion<T>& m = diff.dual();
/* a/2 = acos(l_S) - εm_S/|l_V| */
const T invr = l.vector().lengthInverted();
const Dual<T> aHalf{std::acos(l.scalar()), -m.scalar()*invr};
/* direction = n_0 = l_V/|l_V|
moment = n_ε = (m_V - n_0*(a_ε/2)*l_S)/|l_V| */
const Vector3<T> direction = l.vector()*invr;
const Vector3<T> moment = (m.vector() - direction*(aHalf.dual()*l.scalar()))*invr;
const Dual<Vector3<T>> n{direction, moment};
/* q_ScLERP = q_A*(cos(t*a/2) + n*sin(t*a/2)) */
Dual<T> sin, cos;
std::tie(sin, cos) = Math::sincos(t*Dual<Rad<T>>(aHalf));
return normalizedA*DualQuaternion<T>{n*sin, cos};
}
/**
@brief Dual quaternion
@tparam T Underlying data type
Represents 3D rotation and translation. See @ref transformations for brief
introduction.
@see @ref Magnum::DualQuaternion, @ref Magnum::DualQuaterniond, @ref Dual,
@ref Quaternion, @ref Matrix4
*/
template<class T> class DualQuaternion: public Dual<Quaternion<T>> {
public:
typedef T Type; /**< @brief Underlying data type */
/**
* @brief Rotation dual quaternion
* @param angle Rotation angle (counterclockwise)
* @param normalizedAxis Normalized rotation axis
*
* Expects that the rotation axis is normalized. @f[
* \hat q = [\boldsymbol a \cdot sin \frac \theta 2, cos \frac \theta 2] + \epsilon [\boldsymbol 0, 0]
* @f]
* @see @ref rotation() const, @ref Quaternion::rotation(),
* @ref Matrix4::rotation(), @ref DualComplex::rotation(),
* @ref Vector3::xAxis(), @ref Vector3::yAxis(),
* @ref Vector3::zAxis(), @ref Vector::isNormalized()
*/
static DualQuaternion<T> rotation(Rad<T> angle, const Vector3<T>& normalizedAxis) {
return {Quaternion<T>::rotation(angle, normalizedAxis), {{}, T(0)}};
}
/** @todo Rotation about axis with arbitrary origin, screw motion */
/**
* @brief Translation dual quaternion
* @param vector Translation vector
*
* @f[
* \hat q = [\boldsymbol 0, 1] + \epsilon [\frac{\boldsymbol v}{2}, 0]
* @f]
* @see @ref translation() const,
* @ref Matrix4::translation(const Vector3<T>&),
* @ref DualComplex::translation(), @ref Vector3::xAxis(),
* @ref Vector3::yAxis(), @ref Vector3::zAxis()
*/
static DualQuaternion<T> translation(const Vector3<T>& vector) {
return {{}, {vector/T(2), T(0)}};
}
/**
* @brief Create dual quaternion from transformation matrix
*
* Expects that the matrix represents rigid transformation.
* @see @ref toMatrix(), @ref Quaternion::fromMatrix(),
* @ref Matrix4::isRigidTransformation()
*/
static DualQuaternion<T> fromMatrix(const Matrix4<T>& matrix) {
CORRADE_ASSERT(matrix.isRigidTransformation(),
"Math::DualQuaternion::fromMatrix(): the matrix doesn't represent rigid transformation", {});
Quaternion<T> q = Implementation::quaternionFromMatrix(matrix.rotationScaling());
return {q, Quaternion<T>(matrix.translation()/2)*q};
}
/**
* @brief Default constructor
*
* Creates unit dual quaternion. @f[
* \hat q = [\boldsymbol 0, 1] + \epsilon [\boldsymbol 0, 0]
* @f]
*/
constexpr /*implicit*/ DualQuaternion(IdentityInitT = IdentityInit)
/** @todoc remove workaround when doxygen is sane */
#ifndef DOXYGEN_GENERATING_OUTPUT
: Dual<Quaternion<T>>({}, {{}, T(0)})
#endif
{}
/** @brief Construct zero-initialized dual quaternion */
constexpr explicit DualQuaternion(ZeroInitT)
/** @todoc remove workaround when doxygen is sane */
#ifndef DOXYGEN_GENERATING_OUTPUT
/* MSVC 2015 can't handle {} here */
: Dual<Quaternion<T>>(Quaternion<T>{ZeroInit}, Quaternion<T>{ZeroInit})
#endif
{}
/** @brief Construct without initializing the contents */
explicit DualQuaternion(NoInitT)
/** @todoc remove workaround when doxygen is sane */
#ifndef DOXYGEN_GENERATING_OUTPUT
/* MSVC 2015 can't handle {} here */
: Dual<Quaternion<T>>(NoInit)
#endif
{}
/**
* @brief Construct dual quaternion from real and dual part
*
* @f[
* \hat q = q_0 + \epsilon q_\epsilon
* @f]
*/
constexpr /*implicit*/ DualQuaternion(const Quaternion<T>& real, const Quaternion<T>& dual = Quaternion<T>({}, T(0))): Dual<Quaternion<T>>(real, dual) {}
/**
* @brief Construct dual quaternion from dual vector and scalar parts
*
* @f[
* \hat q = [\hat{\boldsymbol v}, \hat s] = [\boldsymbol v_0, s_0] + \epsilon [\boldsymbol v_\epsilon, s_\epsilon]
* @f]
*/
constexpr /*implicit*/ DualQuaternion(const Dual<Vector3<T>>& vector, const Dual<T>& scalar)
#ifndef DOXYGEN_GENERATING_OUTPUT
/* MSVC 2015 can't handle {} here */
: Dual<Quaternion<T>>({vector.real(), scalar.real()}, {vector.dual(), scalar.dual()})
#endif
{}
/**
* @brief Construct dual quaternion from vector
*
* To be used in transformations later. @f[
* \hat q = [\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]
* @f]
* @see @ref transformPointNormalized()
*/
#ifdef DOXYGEN_GENERATING_OUTPUT
constexpr explicit DualQuaternion(const Vector3<T>& vector);
#else
constexpr explicit DualQuaternion(const Vector3<T>& vector): Dual<Quaternion<T>>({}, {vector, T(0)}) {}
#endif
/**
* @brief Construct dual quaternion from another of different type
*
* Performs only default casting on the values, no rounding or anything
* else.
*/
template<class U> constexpr explicit DualQuaternion(const DualQuaternion<U>& other): Dual<Quaternion<T>>(other) {}
/** @brief Construct dual quaternion from external representation */
template<class U, class V = decltype(Implementation::DualQuaternionConverter<T, U>::from(std::declval<U>()))>
#ifndef CORRADE_MSVC2015_COMPATIBILITY
/* Can't use delegating constructors with constexpr -- https://connect.microsoft.com/VisualStudio/feedback/details/1579279/c-constexpr-does-not-work-with-delegating-constructors */
constexpr
#endif
explicit DualQuaternion(const U& other): DualQuaternion{Implementation::DualQuaternionConverter<T, U>::from(other)} {}
/** @brief Copy constructor */
constexpr DualQuaternion(const Dual<Quaternion<T>>& other): Dual<Quaternion<T>>(other) {}
/** @brief Convert dual quaternion to external representation */
template<class U, class V = decltype(Implementation::DualQuaternionConverter<T, U>::to(std::declval<DualQuaternion<T>>()))> constexpr explicit operator U() const {
return Implementation::DualQuaternionConverter<T, U>::to(*this);
}
/**
* @brief Whether the dual quaternion is normalized
*
* Dual quaternion is normalized if it has unit length: @f[
* |\hat q|^2 = |\hat q| = 1 + \epsilon 0
* @f]
* @see @ref lengthSquared(), @ref normalized()
* @todoc Improve the equation as in Quaternion::isNormalized()
*/
bool isNormalized() const {
/* Comparing dual part classically, as comparing sqrt() of it would
lead to overly strict precision */
Dual<T> a = lengthSquared();
return Implementation::isNormalizedSquared(a.real()) &&
TypeTraits<T>::equals(a.dual(), T(0));
}
/**
* @brief Rotation part of unit dual quaternion
*
* @see @ref Quaternion::angle(), @ref Quaternion::axis()
*/
constexpr Quaternion<T> rotation() const {
return Dual<Quaternion<T>>::real();
}
/**
* @brief Translation part of unit dual quaternion
*
* @f[
* \boldsymbol a = 2 (q_\epsilon q_0^*)_V
* @f]
* @see @ref translation(const Vector3<T>&)
*/
Vector3<T> translation() const {
return (Dual<Quaternion<T>>::dual()*Dual<Quaternion<T>>::real().conjugated()).vector()*T(2);
}
/**
* @brief Convert dual quaternion to transformation matrix
*
* @see @ref fromMatrix(), @ref Quaternion::toMatrix()
*/
Matrix4<T> toMatrix() const {
return Matrix4<T>::from(Dual<Quaternion<T>>::real().toMatrix(), translation());
}
/**
* @brief Quaternion-conjugated dual quaternion
*
* @f[
* \hat q^* = q_0^* + q_\epsilon^*
* @f]
* @see @ref dualConjugated(), @ref conjugated(),
* @ref Quaternion::conjugated()
*/
DualQuaternion<T> quaternionConjugated() const {
return {Dual<Quaternion<T>>::real().conjugated(), Dual<Quaternion<T>>::dual().conjugated()};
}
/**
* @brief Dual-conjugated dual quaternion
*
* @f[
* \overline{\hat q} = q_0 - \epsilon q_\epsilon
* @f]
* @see @ref quaternionConjugated(), @ref conjugated(),
* @ref Dual::conjugated()
*/
DualQuaternion<T> dualConjugated() const {
return Dual<Quaternion<T>>::conjugated();
}
/**
* @brief Conjugated dual quaternion
*
* Both quaternion and dual conjugation. @f[
* \overline{\hat q^*} = q_0^* - \epsilon q_\epsilon^* = q_0^* + \epsilon [\boldsymbol q_{V \epsilon}, -q_{S \epsilon}]
* @f]
* @see @ref quaternionConjugated(), @ref dualConjugated(),
* @ref Quaternion::conjugated(), @ref Dual::conjugated()
*/
DualQuaternion<T> conjugated() const {
return {Dual<Quaternion<T>>::real().conjugated(), {Dual<Quaternion<T>>::dual().vector(), -Dual<Quaternion<T>>::dual().scalar()}};
}
/**
* @brief Dual quaternion length squared
*
* Should be used instead of @ref length() for comparing dual
* quaternion length with other values, because it doesn't compute the
* square root. @f[
* |\hat q|^2 = \sqrt{\hat q^* \hat q}^2 = q_0 \cdot q_0 + \epsilon 2 (q_0 \cdot q_\epsilon)
* @f]
*/
Dual<T> lengthSquared() const {
return {Dual<Quaternion<T>>::real().dot(), T(2)*dot(Dual<Quaternion<T>>::real(), Dual<Quaternion<T>>::dual())};
}
/**
* @brief Dual quaternion length
*
* See @ref lengthSquared() which is faster for comparing length with other
* values. @f[
* |\hat q| = \sqrt{\hat q^* \hat q} = |q_0| + \epsilon \frac{q_0 \cdot q_\epsilon}{|q_0|}
* @f]
*/
Dual<T> length() const {
return Math::sqrt(lengthSquared());
}
/**
* @brief Normalized dual quaternion (of unit length)
*
* @see @ref isNormalized()
*/
DualQuaternion<T> normalized() const {
return (*this)/length();
}
/**
* @brief Inverted dual quaternion
*
* See @ref invertedNormalized() which is faster for normalized dual
* quaternions. @f[
* \hat q^{-1} = \frac{\hat q^*}{|\hat q|^2}
* @f]
*/
DualQuaternion<T> inverted() const {
return quaternionConjugated()/lengthSquared();
}
/**
* @brief Inverted normalized dual quaternion
*
* Equivalent to @ref quaternionConjugated(). Expects that the
* quaternion is normalized. @f[
* \hat q^{-1} = \frac{\hat q^*}{|\hat q|^2} = \hat q^*
* @f]
* @see @ref isNormalized(), @ref inverted()
*/
DualQuaternion<T> invertedNormalized() const {
CORRADE_ASSERT(isNormalized(),
"Math::DualQuaternion::invertedNormalized(): dual quaternion must be normalized", {});
return quaternionConjugated();
}
/**
* @brief Rotate and translate point with dual quaternion
*
* See @ref transformPointNormalized(), which is faster for normalized
* dual quaternions. @f[
* v' = \hat q v \overline{\hat q^{-1}} = \hat q ([\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]) \overline{\hat q^{-1}}
* @f]
* @see @ref DualQuaternion(const Vector3<T>&), @ref dual(),
* @ref Matrix4::transformPoint(),
* @ref Quaternion::transformVector(),
* @ref DualComplex::transformPoint()
*/
Vector3<T> transformPoint(const Vector3<T>& vector) const {
return ((*this)*DualQuaternion<T>(vector)*inverted().dualConjugated()).dual().vector();
}
/**
* @brief Rotate and translate point with normalized dual quaternion
*
* Faster alternative to @ref transformPoint(), expects that the dual
* quaternion is normalized. @f[
* v' = \hat q v \overline{\hat q^{-1}} = \hat q v \overline{\hat q^*} = \hat q ([\boldsymbol 0, 1] + \epsilon [\boldsymbol v, 0]) \overline{\hat q^*}
* @f]
* @see @ref isNormalized(), @ref DualQuaternion(const Vector3<T>&),
* @ref dual(), @ref Matrix4::transformPoint(),
* @ref Quaternion::transformVectorNormalized(),
* @ref DualComplex::transformPoint()
*/
Vector3<T> transformPointNormalized(const Vector3<T>& vector) const {
CORRADE_ASSERT(isNormalized(),
"Math::DualQuaternion::transformPointNormalized(): dual quaternion must be normalized", {});
return ((*this)*DualQuaternion<T>(vector)*conjugated()).dual().vector();
}
MAGNUM_DUAL_SUBCLASS_IMPLEMENTATION(DualQuaternion, Quaternion, T)
MAGNUM_DUAL_SUBCLASS_MULTIPLICATION_IMPLEMENTATION(DualQuaternion, Quaternion)
};
MAGNUM_DUAL_OPERATOR_IMPLEMENTATION(DualQuaternion, Quaternion, T)
/** @debugoperator{Magnum::Math::DualQuaternion} */
template<class T> Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug& debug, const DualQuaternion<T>& value) {
return debug << "DualQuaternion({{" << Corrade::Utility::Debug::nospace
<< value.real().vector().x() << Corrade::Utility::Debug::nospace << ","
<< value.real().vector().y() << Corrade::Utility::Debug::nospace << ","
<< value.real().vector().z() << Corrade::Utility::Debug::nospace << "},"
<< value.real().scalar() << Corrade::Utility::Debug::nospace << "}, {{"
<< Corrade::Utility::Debug::nospace
<< value.dual().vector().x() << Corrade::Utility::Debug::nospace << ","
<< value.dual().vector().y() << Corrade::Utility::Debug::nospace << ","
<< value.dual().vector().z() << Corrade::Utility::Debug::nospace << "},"
<< value.dual().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 DualQuaternion<Float>&);
#ifndef MAGNUM_TARGET_GLES
extern template MAGNUM_EXPORT Corrade::Utility::Debug& operator<<(Corrade::Utility::Debug&, const DualQuaternion<Double>&);
#endif
#endif
}}
#endif