Shapeworks Studio  2.1
Shape analysis software suite
•All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Friends Modules Pages
List of all members | Public Types | Public Member Functions | Static Public Member Functions | Protected Attributes | Friends
Eigen::AngleAxis< _Scalar > Class Template Reference
Geometry_Module

Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. More...

#include <AngleAxis.h>

+ Inheritance diagram for Eigen::AngleAxis< _Scalar >:
+ Collaboration diagram for Eigen::AngleAxis< _Scalar >:

Public Types

enum  { Dim = 3 }
 
enum  { Dim = 3 }
 
typedef _Scalar Scalar
 
typedef Matrix< Scalar, 3, 3 > Matrix3
 
typedef Matrix< Scalar, 3, 1 > Vector3
 
typedef Quaternion< Scalar > QuaternionType
 
typedef _Scalar Scalar
 
typedef Matrix< Scalar, 3, 3 > Matrix3
 
typedef Matrix< Scalar, 3, 1 > Vector3
 
typedef Quaternion< Scalar > QuaternionType
 
- Public Types inherited from Eigen::RotationBase< AngleAxis< _Scalar >, 3 >
enum  
 
enum  
 
typedef ei_traits< AngleAxis< _Scalar > >::Scalar Scalar
 
typedef internal::traits< AngleAxis< _Scalar > >::Scalar Scalar
 
typedef Matrix< Scalar, Dim, Dim > RotationMatrixType
 
typedef Matrix< Scalar, Dim, Dim > RotationMatrixType
 
typedef Matrix< Scalar, Dim, 1 > VectorType
 

Public Member Functions

 AngleAxis ()
 
template<typename Derived >
 AngleAxis (Scalar angle, const MatrixBase< Derived > &axis)
 
 AngleAxis (const QuaternionType &q)
 
template<typename Derived >
 AngleAxis (const MatrixBase< Derived > &m)
 
Scalar angle () const
 
Scalar & angle ()
 
const Vector3 & axis () const
 
Vector3 & axis ()
 
QuaternionType operator* (const AngleAxis &other) const
 
QuaternionType operator* (const QuaternionType &other) const
 
Matrix3 operator* (const Matrix3 &other) const
 
Vector3 operator* (const Vector3 &other) const
 
AngleAxis inverse () const
 
AngleAxis & operator= (const QuaternionType &q)
 
template<typename Derived >
AngleAxis & operator= (const MatrixBase< Derived > &m)
 
template<typename Derived >
AngleAxis & fromRotationMatrix (const MatrixBase< Derived > &m)
 
Matrix3 toRotationMatrix (void) const
 
template<typename NewScalarType >
internal::cast_return_type< AngleAxis, AngleAxis< NewScalarType > >::type cast () const
 
template<typename OtherScalarType >
 AngleAxis (const AngleAxis< OtherScalarType > &other)
 
bool isApprox (const AngleAxis &other, typename NumTraits< Scalar >::Real prec=precision< Scalar >()) const
 
 AngleAxis ()
 
template<typename Derived >
 AngleAxis (const Scalar &angle, const MatrixBase< Derived > &axis)
 
template<typename QuatDerived >
 AngleAxis (const QuaternionBase< QuatDerived > &q)
 
template<typename Derived >
 AngleAxis (const MatrixBase< Derived > &m)
 
Scalar angle () const
 
Scalar & angle ()
 
const Vector3 & axis () const
 
Vector3 & axis ()
 
QuaternionType operator* (const AngleAxis &other) const
 
QuaternionType operator* (const QuaternionType &other) const
 
AngleAxis inverse () const
 
template<class QuatDerived >
AngleAxis & operator= (const QuaternionBase< QuatDerived > &q)
 
template<typename Derived >
AngleAxis & operator= (const MatrixBase< Derived > &m)
 
template<typename Derived >
AngleAxis & fromRotationMatrix (const MatrixBase< Derived > &m)
 
Matrix3 toRotationMatrix (void) const
 
template<typename NewScalarType >
internal::cast_return_type< AngleAxis, AngleAxis< NewScalarType > >::type cast () const
 
template<typename OtherScalarType >
 AngleAxis (const AngleAxis< OtherScalarType > &other)
 
bool isApprox (const AngleAxis &other, const typename NumTraits< Scalar >::Real &prec=NumTraits< Scalar >::dummy_precision()) const
 
template<typename Derived >
AngleAxis< Scalar > & operator= (const MatrixBase< Derived > &mat)
 
template<typename QuatDerived >
AngleAxis< Scalar > & operator= (const QuaternionBase< QuatDerived > &q)
 
template<typename Derived >
AngleAxis< Scalar > & fromRotationMatrix (const MatrixBase< Derived > &mat)
 Sets *this from a 3x3 rotation matrix.
 
- Public Member Functions inherited from Eigen::RotationBase< AngleAxis< _Scalar >, 3 >
const AngleAxis< _Scalar > & derived () const
 
AngleAxis< _Scalar > & derived ()
 
const AngleAxis< _Scalar > & derived () const
 
AngleAxis< _Scalar > & derived ()
 
RotationMatrixType toRotationMatrix () const
 
RotationMatrixType toRotationMatrix () const
 
AngleAxis< _Scalar > inverse () const
 
AngleAxis< _Scalar > inverse () const
 
Transform< Scalar, Dim > operator* (const Translation< Scalar, Dim > &t) const
 
RotationMatrixType operator* (const Scaling< Scalar, Dim > &s) const
 
Transform< Scalar, Dim > operator* (const Transform< Scalar, Dim > &t) const
 
Transform< Scalar, Dim, Isometry > operator* (const Translation< Scalar, Dim > &t) const
 
RotationMatrixType operator* (const UniformScaling< Scalar > &s) const
 
EIGEN_STRONG_INLINE internal::rotation_base_generic_product_selector< AngleAxis< _Scalar >, OtherDerived, OtherDerived::IsVectorAtCompileTime >::ReturnType operator* (const EigenBase< OtherDerived > &e) const
 
Transform< Scalar, Dim, Mode > operator* (const Transform< Scalar, Dim, Mode, Options > &t) const
 
RotationMatrixType matrix () const
 
VectorType _transformVector (const OtherVectorType &v) const
 

Static Public Member Functions

static const AngleAxis Identity ()
 

Protected Attributes

Vector3 m_axis
 
Scalar m_angle
 

Friends

QuaternionType operator* (const QuaternionType &a, const AngleAxis &b)
 
Matrix3 operator* (const Matrix3 &a, const AngleAxis &b)
 
QuaternionType operator* (const QuaternionType &a, const AngleAxis &b)
 

Detailed Description

template<typename _Scalar>
class Eigen::AngleAxis< _Scalar >

Represents a 3D rotation as a rotation angle around an arbitrary 3D axis.

Parameters
_Scalarthe scalar type, i.e., the type of the coefficients.

The following two typedefs are provided for convenience:

AngleAxisForEuler How to define a rotation from Euler-angles

Combined with MatrixBase::Unit{X,Y,Z}, AngleAxis can be used to easily mimic Euler-angles. Here is an example:

Output: 

Note
This class is not aimed to be used to store a rotation transformation, but rather to make easier the creation of other rotation (Quaternion, rotation Matrix) and transformation objects.


See also
class Quaternion, class Transform, MatrixBase::UnitX()


Parameters

  

    
_Scalarthe scalar type, i.e., the type of the coefficients.

  

  




Warning
When setting up an AngleAxis object, the axis vector must be normalized.


The following two typedefs are provided for convenience: 


Combined with MatrixBase::Unit{X,Y,Z}, AngleAxis can be used to easily mimic Euler-angles. Here is an example: 
 Output: 
Note
This class is not aimed to be used to store a rotation transformation, but rather to make easier the creation of other rotation (Quaternion, rotation Matrix) and transformation objects.


See also
class Quaternion, class Transform, MatrixBase::UnitX() 



Definition at line 235 of file ForwardDeclarations.h.


Member Typedef Documentation








template<typename _Scalar> 

      

        

          typedef _Scalar Eigen::AngleAxis< _Scalar >::Scalar
        

      




the scalar type of the coefficients 



Definition at line 56 of file AngleAxis.h.













template<typename _Scalar> 

      

        

          typedef _Scalar Eigen::AngleAxis< _Scalar >::Scalar
        

      




the scalar type of the coefficients 



Definition at line 59 of file AngleAxis.h.







Constructor & Destructor Documentation








template<typename _Scalar> 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          )
          
        

      

  		
  
inline  
  






Default constructor without initialization. 



Definition at line 69 of file AngleAxis.h.


   69 {}












template<typename _Scalar> 



template<typename Derived > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          Scalar 
          angle, 
        

        

          
          
          const MatrixBase< Derived > & 
          axis 
        

        

          
          )
          
        

      

  		
  
inline  
  






Constructs and initialize the angle-axis rotation from an angle in radian and an axis which must be normalized. 



Definition at line 73 of file AngleAxis.h.


   73 : m_axis(axis), m_angle(angle) {}












template<typename _Scalar> 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const QuaternionType & 
          q)
          
        

      

  		
  
inline  
  






Constructs and initialize the angle-axis rotation from a quaternion q. 



Definition at line 75 of file AngleAxis.h.


   75 { *this = q; }












template<typename _Scalar> 



template<typename Derived > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const MatrixBase< Derived > & 
          m)
          
        

      

  		
  
inlineexplicit  
  






Constructs and initialize the angle-axis rotation from a 3x3 rotation matrix. 



Definition at line 78 of file AngleAxis.h.


   78 { *this = m; }












template<typename _Scalar> 



template<typename OtherScalarType > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const AngleAxis< OtherScalarType > & 
          other)
          
        

      

  		
  
inlineexplicit  
  






Copy constructor with scalar type conversion 



Definition at line 133 of file AngleAxis.h.


  134   {
  135     m_axis = other.axis().template cast<Scalar>();
  136     m_angle = Scalar(other.angle());
  137   }
Eigen::AngleAxis::Scalar
_Scalar Scalar
Definition: AngleAxis.h:56














template<typename _Scalar> 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          )
          
        

      

  		
  
inline  
  






Default constructor without initialization. 



Definition at line 72 of file AngleAxis.h.


   72 {}












template<typename _Scalar> 



template<typename Derived > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const Scalar & 
          angle, 
        

        

          
          
          const MatrixBase< Derived > & 
          axis 
        

        

          
          )
          
        

      

  		
  
inline  
  






Constructs and initialize the angle-axis rotation from an angle in radian and an axis which must be normalized.


Warning
If the axis vector is not normalized, then the angle-axis object represents an invalid rotation. 



Definition at line 79 of file AngleAxis.h.


   79 : m_axis(axis), m_angle(angle) {}












template<typename _Scalar> 



template<typename QuatDerived > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const QuaternionBase< QuatDerived > & 
          q)
          
        

      

  		
  
inlineexplicit  
  






Constructs and initialize the angle-axis rotation from a quaternion q. 



Definition at line 81 of file AngleAxis.h.


   81 { *this = q; }












template<typename _Scalar> 



template<typename Derived > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const MatrixBase< Derived > & 
          m)
          
        

      

  		
  
inlineexplicit  
  






Constructs and initialize the angle-axis rotation from a 3x3 rotation matrix. 



Definition at line 84 of file AngleAxis.h.


   84 { *this = m; }












template<typename _Scalar> 



template<typename OtherScalarType > 



  

  
      

        

          Eigen::AngleAxis< _Scalar >::AngleAxis 
          (
          const AngleAxis< OtherScalarType > & 
          other)
          
        

      

  		
  
inlineexplicit  
  






Copy constructor with scalar type conversion 



Definition at line 128 of file AngleAxis.h.


  129   {
  130     m_axis = other.axis().template cast<Scalar>();
  131     m_angle = Scalar(other.angle());
  132   }
Eigen::AngleAxis::Scalar
_Scalar Scalar
Definition: AngleAxis.h:56








Member Function Documentation








template<typename _Scalar> 



template<typename NewScalarType > 



  

  
      

        

          internal::cast_return_type<AngleAxis,AngleAxis<NewScalarType> >::type Eigen::AngleAxis< _Scalar >::cast 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
*this with scalar type casted to NewScalarType 


Note that if NewScalarType is equal to the current scalar type of *this then this function smartly returns a const reference to *this. 



Definition at line 123 of file AngleAxis.h.


  124   { return typename internal::cast_return_type<AngleAxis,AngleAxis<NewScalarType> >::type(*this); }












template<typename _Scalar> 



template<typename NewScalarType > 



  

  
      

        

          internal::cast_return_type<AngleAxis,AngleAxis<NewScalarType> >::type Eigen::AngleAxis< _Scalar >::cast 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
*this with scalar type casted to NewScalarType 


Note that if NewScalarType is equal to the current scalar type of *this then this function smartly returns a const reference to *this. 



Definition at line 128 of file AngleAxis.h.


  129   { return typename internal::cast_return_type<AngleAxis,AngleAxis<NewScalarType> >::type(*this); }












template<typename _Scalar> 



  

  
      

        

          AngleAxis Eigen::AngleAxis< _Scalar >::inverse 
          (
          void 
          )
           const
        

      

  		
  
inline  
  






Returns
the inverse rotation, i.e., an angle-axis with opposite rotation angle 



Definition at line 105 of file AngleAxis.h.


  106   { return AngleAxis(-m_angle, m_axis); }














template<typename _Scalar> 



  

  
      

        

          AngleAxis Eigen::AngleAxis< _Scalar >::inverse 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the inverse rotation, i.e., an angle-axis with opposite rotation angle 



Definition at line 111 of file AngleAxis.h.


  112   { return AngleAxis(-m_angle, m_axis); }














template<typename _Scalar> 



  

  
      

        

          bool Eigen::AngleAxis< _Scalar >::isApprox 
          (
          const AngleAxis< _Scalar > & 
          other, 
        

        

          
          
          const typename NumTraits< Scalar >::Real & 
          prec = NumTraits<Scalar>::dummy_precision() 
        

        

          
          )
           const
        

      

  		
  
inline  
  






Returns
true if *this is approximately equal to other, within the precision determined by prec.


See also
MatrixBase::isApprox() 



Definition at line 140 of file AngleAxis.h.


  141   { return m_axis.isApprox(other.m_axis, prec) && internal::isApprox(m_angle,other.m_angle, prec); }












template<typename _Scalar> 



  

  
      

        

          bool Eigen::AngleAxis< _Scalar >::isApprox 
          (
          const AngleAxis< _Scalar > & 
          other, 
        

        

          
          
          typename NumTraits< Scalar >::Real 
          prec = precision<Scalar>() 
        

        

          
          )
           const
        

      

  		
  
inline  
  






Returns
true if *this is approximately equal to other, within the precision determined by prec.


See also
MatrixBase::isApprox() 



Definition at line 143 of file AngleAxis.h.


  144   { return m_axis.isApprox(other.m_axis, prec) && ei_isApprox(m_angle,other.m_angle, prec); }












template<typename _Scalar> 



  

  
      

        

          QuaternionType Eigen::AngleAxis< _Scalar >::operator* 
          (
          const AngleAxis< _Scalar > & 
          other)
           const
        

      

  		
  
inline  
  






Concatenates two rotations 



Definition at line 87 of file AngleAxis.h.


   88   { return QuaternionType(*this) * QuaternionType(other); }












template<typename _Scalar> 



  

  
      

        

          QuaternionType Eigen::AngleAxis< _Scalar >::operator* 
          (
          const QuaternionType & 
          other)
           const
        

      

  		
  
inline  
  






Concatenates two rotations 



Definition at line 91 of file AngleAxis.h.


   92   { return QuaternionType(*this) * other; }












template<typename _Scalar> 



  

  
      

        

          QuaternionType Eigen::AngleAxis< _Scalar >::operator* 
          (
          const AngleAxis< _Scalar > & 
          other)
           const
        

      

  		
  
inline  
  






Concatenates two rotations 



Definition at line 93 of file AngleAxis.h.


   94   { return QuaternionType(*this) * QuaternionType(other); }












template<typename _Scalar> 



  

  
      

        

          QuaternionType Eigen::AngleAxis< _Scalar >::operator* 
          (
          const QuaternionType & 
          other)
           const
        

      

  		
  
inline  
  






Concatenates two rotations 



Definition at line 97 of file AngleAxis.h.


   98   { return QuaternionType(*this) * other; }












template<typename _Scalar> 



  

  
      

        

          Matrix3 Eigen::AngleAxis< _Scalar >::operator* 
          (
          const Matrix3 & 
          other)
           const
        

      

  		
  
inline  
  






Concatenates two rotations 



Definition at line 99 of file AngleAxis.h.


  100   { return toRotationMatrix() * other; }
Eigen::AngleAxis::toRotationMatrix
Matrix3 toRotationMatrix(void) const 
Definition: AngleAxis.h:189














template<typename _Scalar> 



  

  
      

        

          Vector3 Eigen::AngleAxis< _Scalar >::operator* 
          (
          const Vector3 & 
          other)
           const
        

      

  		
  
inline  
  






Applies rotation to vector 



Definition at line 107 of file AngleAxis.h.


  108   { return toRotationMatrix() * other; }
Eigen::AngleAxis::toRotationMatrix
Matrix3 toRotationMatrix(void) const 
Definition: AngleAxis.h:189














template<typename Scalar > 

      

        

          AngleAxis< Scalar > & Eigen::AngleAxis< Scalar >::operator= 
          (
          const QuaternionType & 
          q)
          
        

      




Set *this from a quaternion. The axis is normalized. 



Definition at line 158 of file AngleAxis.h.


  159 {
  160   Scalar n2 = q.vec().squaredNorm();
  161   if (n2 < precision<Scalar>()*precision<Scalar>())
  162   {
  163     m_angle = 0;
  164     m_axis << 1, 0, 0;
  165   }
  166   else
  167   {
  168     m_angle = 2*std::acos(q.w());
  169     m_axis = q.vec() / ei_sqrt(n2);
  170   }
  171   return *this;
  172 }
Eigen::AngleAxis::Scalar
_Scalar Scalar
Definition: AngleAxis.h:56














template<typename _Scalar> 



template<typename QuatDerived > 

      

        

          AngleAxis<Scalar>& Eigen::AngleAxis< _Scalar >::operator= 
          (
          const QuaternionBase< QuatDerived > & 
          q)
          
        

      




Set *this from a unit quaternion. The axis is normalized.


Warning
As any other method dealing with quaternion, if the input quaternion is not normalized then the result is undefined. 



Definition at line 159 of file AngleAxis.h.


  160 {
  161   using std::acos;
  162   using std::min;
  163   using std::max;
  164   using std::sqrt;
  165   Scalar n2 = q.vec().squaredNorm();
  166   if (n2 < NumTraits<Scalar>::dummy_precision()*NumTraits<Scalar>::dummy_precision())
  167   {
  168     m_angle = 0;
  169     m_axis << 1, 0, 0;
  170   }
  171   else
  172   {
  173     m_angle = Scalar(2)*acos((min)((max)(Scalar(-1),q.w()),Scalar(1)));
  174     m_axis = q.vec() / sqrt(n2);
  175   }
  176   return *this;
  177 }
Eigen::AngleAxis::Scalar
_Scalar Scalar
Definition: AngleAxis.h:56














template<typename Scalar > 



template<typename Derived > 

      

        

          AngleAxis< Scalar > & Eigen::AngleAxis< Scalar >::operator= 
          (
          const MatrixBase< Derived > & 
          mat)
          
        

      




Set *this from a 3x3 rotation matrix mat. 



Definition at line 178 of file AngleAxis.h.


  179 {
  180   // Since a direct conversion would not be really faster,
  181   // let's use the robust Quaternion implementation:
  182   return *this = QuaternionType(mat);
  183 }












template<typename Scalar > 

      

        

          AngleAxis< Scalar >::Matrix3 Eigen::AngleAxis< Scalar >::toRotationMatrix 
          (
          void 
          )
           const
        

      




Constructs and 
Returns
an equivalent 3x3 rotation matrix. 



Definition at line 189 of file AngleAxis.h.


  190 {
  191   Matrix3 res;
  192   Vector3 sin_axis  = ei_sin(m_angle) * m_axis;
  193   Scalar c = ei_cos(m_angle);
  194   Vector3 cos1_axis = (Scalar(1)-c) * m_axis;
  195 
  196   Scalar tmp;
  197   tmp = cos1_axis.x() * m_axis.y();
  198   res.coeffRef(0,1) = tmp - sin_axis.z();
  199   res.coeffRef(1,0) = tmp + sin_axis.z();
  200 
  201   tmp = cos1_axis.x() * m_axis.z();
  202   res.coeffRef(0,2) = tmp + sin_axis.y();
  203   res.coeffRef(2,0) = tmp - sin_axis.y();
  204 
  205   tmp = cos1_axis.y() * m_axis.z();
  206   res.coeffRef(1,2) = tmp - sin_axis.x();
  207   res.coeffRef(2,1) = tmp + sin_axis.x();
  208 
  209   res.diagonal() = (cos1_axis.cwise() * m_axis).cwise() + c;
  210 
  211   return res;
  212 }
Eigen::AngleAxis::Scalar
_Scalar Scalar
Definition: AngleAxis.h:56








Friends And Related Function Documentation








template<typename _Scalar> 



  

  
      

        

          QuaternionType operator* 
          (
          const QuaternionType & 
          a, 
        

        

          
          
          const AngleAxis< _Scalar > & 
          b 
        

        

          
          )
          
        

      

  		
  
friend  
  






Concatenates two rotations 



Definition at line 95 of file AngleAxis.h.


   96   { return a * QuaternionType(b); }












template<typename _Scalar> 



  

  
      

        

          QuaternionType operator* 
          (
          const QuaternionType & 
          a, 
        

        

          
          
          const AngleAxis< _Scalar > & 
          b 
        

        

          
          )
          
        

      

  		
  
friend  
  






Concatenates two rotations 



Definition at line 101 of file AngleAxis.h.


  102   { return a * QuaternionType(b); }












template<typename _Scalar> 



  

  
      

        

          Matrix3 operator* 
          (
          const Matrix3 & 
          a, 
        

        

          
          
          const AngleAxis< _Scalar > & 
          b 
        

        

          
          )
          
        

      

  		
  
friend  
  






Concatenates two rotations 



Definition at line 103 of file AngleAxis.h.


  104   { return a * b.toRotationMatrix(); }






The documentation for this class was generated from the following files:





Generated on Mon Aug 15 2016 13:47:40 for Shapeworks Studio by  
doxygen
 1.8.11