// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2015 Tal Hadad <tal_hd@hotmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_EULERSYSTEM_H
#define EIGEN_EULERSYSTEM_H
namespace Eigen
{
// Forward declerations
template <typename _Scalar, class _System>
class EulerAngles;
namespace internal
{
// TODO: Check if already exists on the rest API
template <int Num, bool IsPositive = (Num > 0)>
struct Abs
{
enum { value = Num };
};
template <int Num>
struct Abs<Num, false>
{
enum { value = -Num };
};
template <int Axis>
struct IsValidAxis
{
enum { value = Axis != 0 && Abs<Axis>::value <= 3 };
};
}
#define EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT(COND,MSG) typedef char static_assertion_##MSG[(COND)?1:-1]
/** \brief Representation of a fixed signed rotation axis for EulerSystem.
*
* \ingroup EulerAngles_Module
*
* Values here represent:
* - The axis of the rotation: X, Y or Z.
* - The sign (i.e. direction of the rotation along the axis): positive(+) or negative(-)
*
* Therefore, this could express all the axes {+X,+Y,+Z,-X,-Y,-Z}
*
* For positive axis, use +EULER_{axis}, and for negative axis use -EULER_{axis}.
*/
enum EulerAxis
{
EULER_X = 1, /*!< the X axis */
EULER_Y = 2, /*!< the Y axis */
EULER_Z = 3 /*!< the Z axis */
};
/** \class EulerSystem
*
* \ingroup EulerAngles_Module
*
* \brief Represents a fixed Euler rotation system.
*
* This meta-class goal is to represent the Euler system in compilation time, for EulerAngles.
*
* You can use this class to get two things:
* - Build an Euler system, and then pass it as a template parameter to EulerAngles.
* - Query some compile time data about an Euler system. (e.g. Whether it's tait bryan)
*
* Euler rotation is a set of three rotation on fixed axes. (see \ref EulerAngles)
* This meta-class store constantly those signed axes. (see \ref EulerAxis)
*
* ### Types of Euler systems ###
*
* All and only valid 3 dimension Euler rotation over standard
* signed axes{+X,+Y,+Z,-X,-Y,-Z} are supported:
* - all axes X, Y, Z in each valid order (see below what order is valid)
* - rotation over the axis is supported both over the positive and negative directions.
* - both tait bryan and proper/classic Euler angles (i.e. the opposite).
*
* Since EulerSystem support both positive and negative directions,
* you may call this rotation distinction in other names:
* - _right handed_ or _left handed_
* - _counterclockwise_ or _clockwise_
*
* Notice all axed combination are valid, and would trigger a static assertion.
* Same unsigned axes can't be neighbors, e.g. {X,X,Y} is invalid.
* This yield two and only two classes:
* - _tait bryan_ - all unsigned axes are distinct, e.g. {X,Y,Z}
* - _proper/classic Euler angles_ - The first and the third unsigned axes is equal,
* and the second is different, e.g. {X,Y,X}
*
* ### Intrinsic vs extrinsic Euler systems ###
*
* Only intrinsic Euler systems are supported for simplicity.
* If you want to use extrinsic Euler systems,
* just use the equal intrinsic opposite order for axes and angles.
* I.e axes (A,B,C) becomes (C,B,A), and angles (a,b,c) becomes (c,b,a).
*
* ### Convenient user typedefs ###
*
* Convenient typedefs for EulerSystem exist (only for positive axes Euler systems),
* in a form of EulerSystem{A}{B}{C}, e.g. \ref EulerSystemXYZ.
*
* ### Additional reading ###
*
* More information about Euler angles: https://en.wikipedia.org/wiki/Euler_angles
*
* \tparam _AlphaAxis the first fixed EulerAxis
*
* \tparam _AlphaAxis the second fixed EulerAxis
*
* \tparam _AlphaAxis the third fixed EulerAxis
*/
template <int _AlphaAxis, int _BetaAxis, int _GammaAxis>
class EulerSystem
{
public:
// It's defined this way and not as enum, because I think
// that enum is not guerantee to support negative numbers
/** The first rotation axis */
static const int AlphaAxis = _AlphaAxis;
/** The second rotation axis */
static const int BetaAxis = _BetaAxis;
/** The third rotation axis */
static const int GammaAxis = _GammaAxis;
enum
{
AlphaAxisAbs = internal::Abs<AlphaAxis>::value, /*!< the first rotation axis unsigned */
BetaAxisAbs = internal::Abs<BetaAxis>::value, /*!< the second rotation axis unsigned */
GammaAxisAbs = internal::Abs<GammaAxis>::value, /*!< the third rotation axis unsigned */
IsAlphaOpposite = (AlphaAxis < 0) ? 1 : 0, /*!< weather alpha axis is negative */
IsBetaOpposite = (BetaAxis < 0) ? 1 : 0, /*!< weather beta axis is negative */
IsGammaOpposite = (GammaAxis < 0) ? 1 : 0, /*!< weather gamma axis is negative */
IsOdd = ((AlphaAxisAbs)%3 == (BetaAxisAbs - 1)%3) ? 0 : 1, /*!< weather the Euler system is odd */
IsEven = IsOdd ? 0 : 1, /*!< weather the Euler system is even */
IsTaitBryan = ((unsigned)AlphaAxisAbs != (unsigned)GammaAxisAbs) ? 1 : 0 /*!< weather the Euler system is tait bryan */
};
private:
EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT(internal::IsValidAxis<AlphaAxis>::value,
ALPHA_AXIS_IS_INVALID);
EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT(internal::IsValidAxis<BetaAxis>::value,
BETA_AXIS_IS_INVALID);
EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT(internal::IsValidAxis<GammaAxis>::value,
GAMMA_AXIS_IS_INVALID);
EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT((unsigned)AlphaAxisAbs != (unsigned)BetaAxisAbs,
ALPHA_AXIS_CANT_BE_EQUAL_TO_BETA_AXIS);
EIGEN_EULER_ANGLES_CLASS_STATIC_ASSERT((unsigned)BetaAxisAbs != (unsigned)GammaAxisAbs,
BETA_AXIS_CANT_BE_EQUAL_TO_GAMMA_AXIS);
enum
{
// I, J, K are the pivot indexes permutation for the rotation matrix, that match this Euler system.
// They are used in this class converters.
// They are always different from each other, and their possible values are: 0, 1, or 2.
I = AlphaAxisAbs - 1,
J = (AlphaAxisAbs - 1 + 1 + IsOdd)%3,
K = (AlphaAxisAbs - 1 + 2 - IsOdd)%3
};
// TODO: Get @mat parameter in form that avoids double evaluation.
template <typename Derived>
static void CalcEulerAngles_imp(Matrix<typename MatrixBase<Derived>::Scalar, 3, 1>& res, const MatrixBase<Derived>& mat, internal::true_type /*isTaitBryan*/)
{
using std::atan2;
using std::sin;
using std::cos;
typedef typename Derived::Scalar Scalar;
typedef Matrix<Scalar,2,1> Vector2;
res[0] = atan2(mat(J,K), mat(K,K));
Scalar c2 = Vector2(mat(I,I), mat(I,J)).norm();
if((IsOdd && res[0]<Scalar(0)) || ((!IsOdd) && res[0]>Scalar(0))) {
if(res[0] > Scalar(0)) {
res[0] -= Scalar(EIGEN_PI);
}
else {
res[0] += Scalar(EIGEN_PI);
}
res[1] = atan2(-mat(I,K), -c2);
}
else
res[1] = atan2(-mat(I,K), c2);
Scalar s1 = sin(res[0]);
Scalar c1 = cos(res[0]);
res[2] = atan2(s1*mat(K,I)-c1*mat(J,I), c1*mat(J,J) - s1 * mat(K,J));
}
template <typename Derived>
static void CalcEulerAngles_imp(Matrix<typename MatrixBase<Derived>::Scalar,3,1>& res, const MatrixBase<Derived>& mat, internal::false_type /*isTaitBryan*/)
{
using std::atan2;
using std::sin;
using std::cos;
typedef typename Derived::Scalar Scalar;
typedef Matrix<Scalar,2,1> Vector2;
res[0] = atan2(mat(J,I), mat(K,I));
if((IsOdd && res[0]<Scalar(0)) || ((!IsOdd) && res[0]>Scalar(0)))
{
if(res[0] > Scalar(0)) {
res[0] -= Scalar(EIGEN_PI);
}
else {
res[0] += Scalar(EIGEN_PI);
}
Scalar s2 = Vector2(mat(J,I), mat(K,I)).norm();
res[1] = -atan2(s2, mat(I,I));
}
else
{
Scalar s2 = Vector2(mat(J,I), mat(K,I)).norm();
res[1] = atan2(s2, mat(I,I));
}
// With a=(0,1,0), we have i=0; j=1; k=2, and after computing the first two angles,
// we can compute their respective rotation, and apply its inverse to M. Since the result must
// be a rotation around x, we have:
//
// c2 s1.s2 c1.s2 1 0 0
// 0 c1 -s1 * M = 0 c3 s3
// -s2 s1.c2 c1.c2 0 -s3 c3
//
// Thus: m11.c1 - m21.s1 = c3 & m12.c1 - m22.s1 = s3
Scalar s1 = sin(res[0]);
Scalar c1 = cos(res[0]);
res[2] = atan2(c1*mat(J,K)-s1*mat(K,K), c1*mat(J,J) - s1 * mat(K,J));
}
template<typename Scalar>
static void CalcEulerAngles(
EulerAngles<Scalar, EulerSystem>& res,
const typename EulerAngles<Scalar, EulerSystem>::Matrix3& mat)
{
CalcEulerAngles(res, mat, false, false, false);
}
template<
bool PositiveRangeAlpha,
bool PositiveRangeBeta,
bool PositiveRangeGamma,
typename Scalar>
static void CalcEulerAngles(
EulerAngles<Scalar, EulerSystem>& res,
const typename EulerAngles<Scalar, EulerSystem>::Matrix3& mat)
{
CalcEulerAngles(res, mat, PositiveRangeAlpha, PositiveRangeBeta, PositiveRangeGamma);
}
template<typename Scalar>
static void CalcEulerAngles(
EulerAngles<Scalar, EulerSystem>& res,
const typename EulerAngles<Scalar, EulerSystem>::Matrix3& mat,
bool PositiveRangeAlpha,
bool PositiveRangeBeta,
bool PositiveRangeGamma)
{
CalcEulerAngles_imp(
res.angles(), mat,
typename internal::conditional<IsTaitBryan, internal::true_type, internal::false_type>::type());
if (IsAlphaOpposite == IsOdd)
res.alpha() = -res.alpha();
if (IsBetaOpposite == IsOdd)
res.beta() = -res.beta();
if (IsGammaOpposite == IsOdd)
res.gamma() = -res.gamma();
// Saturate results to the requested range
if (PositiveRangeAlpha && (res.alpha() < 0))
res.alpha() += Scalar(2 * EIGEN_PI);
if (PositiveRangeBeta && (res.beta() < 0))
res.beta() += Scalar(2 * EIGEN_PI);
if (PositiveRangeGamma && (res.gamma() < 0))
res.gamma() += Scalar(2 * EIGEN_PI);
}
template <typename _Scalar, class _System>
friend class Eigen::EulerAngles;
};
#define EIGEN_EULER_SYSTEM_TYPEDEF(A, B, C) \
/** \ingroup EulerAngles_Module */ \
typedef EulerSystem<EULER_##A, EULER_##B, EULER_##C> EulerSystem##A##B##C;
EIGEN_EULER_SYSTEM_TYPEDEF(X,Y,Z)
EIGEN_EULER_SYSTEM_TYPEDEF(X,Y,X)
EIGEN_EULER_SYSTEM_TYPEDEF(X,Z,Y)
EIGEN_EULER_SYSTEM_TYPEDEF(X,Z,X)
EIGEN_EULER_SYSTEM_TYPEDEF(Y,Z,X)
EIGEN_EULER_SYSTEM_TYPEDEF(Y,Z,Y)
EIGEN_EULER_SYSTEM_TYPEDEF(Y,X,Z)
EIGEN_EULER_SYSTEM_TYPEDEF(Y,X,Y)
EIGEN_EULER_SYSTEM_TYPEDEF(Z,X,Y)
EIGEN_EULER_SYSTEM_TYPEDEF(Z,X,Z)
EIGEN_EULER_SYSTEM_TYPEDEF(Z,Y,X)
EIGEN_EULER_SYSTEM_TYPEDEF(Z,Y,Z)
}
#endif // EIGEN_EULERSYSTEM_H