Geometry module

Detailed Description

This module provides support for:

• fixed-size homogeneous transformations
• translation, scaling, 2D and 3D rotations
• quaternions
• cross products (MatrixBase::cross, MatrixBase::cross3)
• orthognal vector generation (MatrixBase::unitOrthogonal)
• some linear components: parametrized-lines and hyperplanes
• axis aligned bounding boxes
• least-square transformation fitting

#include <Eigen/Geometry>

Topics
	Global aligned box typedefs

Classes
class	Eigen::AlignedBox< Scalar_, AmbientDim_ >
	An axis aligned box. More...
class	Eigen::AngleAxis< Scalar_ >
	Represents a 3D rotation as a rotation angle around an arbitrary 3D axis. More...
class	Eigen::Homogeneous< MatrixType, Direction_ >
	Expression of one (or a set of) homogeneous vector(s) More...
class	Eigen::Hyperplane< Scalar_, AmbientDim_, Options_ >
	A hyperplane. More...
class	Eigen::Map< const Quaternion< Scalar_ >, Options_ >
	Quaternion expression mapping a constant memory buffer. More...
class	Eigen::Map< Quaternion< Scalar_ >, Options_ >
	Expression of a quaternion from a memory buffer. More...
class	Eigen::ParametrizedLine< Scalar_, AmbientDim_, Options_ >
	A parametrized line. More...
class	Eigen::Quaternion< Scalar_, Options_ >
	The quaternion class used to represent 3D orientations and rotations. More...
class	Eigen::QuaternionBase< Derived >
	Base class for quaternion expressions. More...
class	Eigen::Rotation2D< Scalar_ >
	Represents a rotation/orientation in a 2 dimensional space. More...
class	Eigen::Transform< Scalar_, Dim_, Mode_, Options_ >
	Represents an homogeneous transformation in a N dimensional space. More...
class	Eigen::Translation< Scalar_, Dim_ >
	Represents a translation transformation. More...
class	Eigen::UniformScaling< Scalar_ >
	Represents a generic uniform scaling transformation. More...

Typedefs
typedef AngleAxis< double >	Eigen::AngleAxisd
typedef AngleAxis< float >	Eigen::AngleAxisf
typedef Quaternion< double >	Eigen::Quaterniond
typedef Quaternion< float >	Eigen::Quaternionf
typedef Map< Quaternion< double >, Aligned >	Eigen::QuaternionMapAlignedd
typedef Map< Quaternion< float >, Aligned >	Eigen::QuaternionMapAlignedf
typedef Map< Quaternion< double >, 0 >	Eigen::QuaternionMapd
typedef Map< Quaternion< float >, 0 >	Eigen::QuaternionMapf
typedef Rotation2D< double >	Eigen::Rotation2Dd
typedef Rotation2D< float >	Eigen::Rotation2Df

Functions
Matrix< Scalar, 3, 1 >	Eigen::MatrixBase< Derived >::canonicalEulerAngles (Index a0, Index a1, Index a2) const
template<typename OtherDerived >
std::conditional_t< SizeAtCompileTime==2, Scalar, PlainObject >	Eigen::MatrixBase< Derived >::cross (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
const VectorwiseOp< ExpressionType, Direction >::CrossReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::cross (const MatrixBase< OtherDerived > &other) const
template<typename OtherDerived >
MatrixBase< Derived >::PlainObject	Eigen::MatrixBase< Derived >::cross3 (const MatrixBase< OtherDerived > &other) const
EIGEN_DEPRECATED Matrix< Scalar, 3, 1 >	Eigen::MatrixBase< Derived >::eulerAngles (Index a0, Index a1, Index a2) const
const HNormalizedReturnType	Eigen::MatrixBase< Derived >::hnormalized () const
	homogeneous normalization
const HNormalizedReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized () const
	column or row-wise homogeneous normalization
HomogeneousReturnType	Eigen::MatrixBase< Derived >::homogeneous () const
HomogeneousReturnType	Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous () const
template<typename Derived , typename OtherDerived >
internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type	Eigen::umeyama (const MatrixBase< Derived > &src, const MatrixBase< OtherDerived > &dst, bool with_scaling=true)
	Returns the transformation between two point sets.
PlainObject	Eigen::MatrixBase< Derived >::unitOrthogonal (void) const

Typedef Documentation

AngleAxisd

typedef AngleAxis<double> Eigen::AngleAxisd

double precision angle-axis type

AngleAxisf

typedef AngleAxis<float> Eigen::AngleAxisf

single precision angle-axis type

Quaterniond

typedef Quaternion<double> Eigen::Quaterniond

double precision quaternion type

Quaternionf

typedef Quaternion<float> Eigen::Quaternionf

single precision quaternion type

QuaternionMapAlignedd

typedef Map<Quaternion<double>, Aligned> Eigen::QuaternionMapAlignedd

Map a 16-byte aligned array of double precision scalars as a quaternion

QuaternionMapAlignedf

typedef Map<Quaternion<float>, Aligned> Eigen::QuaternionMapAlignedf

Map a 16-byte aligned array of single precision scalars as a quaternion

QuaternionMapd

typedef Map<Quaternion<double>, 0> Eigen::QuaternionMapd

Map an unaligned array of double precision scalars as a quaternion

QuaternionMapf

typedef Map<Quaternion<float>, 0> Eigen::QuaternionMapf

Map an unaligned array of single precision scalars as a quaternion

Rotation2Dd

typedef Rotation2D<double> Eigen::Rotation2Dd

double precision 2D rotation type

Rotation2Df

typedef Rotation2D<float> Eigen::Rotation2Df

single precision 2D rotation type

Function Documentation

canonicalEulerAngles()

template<typename Derived >

Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > Eigen::MatrixBase< Derived >::canonicalEulerAngles ( Index a0, Index a1, Index a2 ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

the canonical Euler-angles of the rotation matrix *this using the convention defined by the triplet (a0,a1,a2)

Each of the three parameters a0,a1,a2 represents the respective rotation axis as an integer in {0,1,2}. For instance, in:

Vector3f ea = mat.eulerAngles(2, 0, 2); 

"2" represents the z axis and "0" the x axis, etc. The returned angles are such that we have the following equality:

mat == AngleAxisf(ea[0], Vector3f::UnitZ())
     * AngleAxisf(ea[1], Vector3f::UnitX())
     * AngleAxisf(ea[2], Vector3f::UnitZ()); 

This corresponds to the right-multiply conventions (with right hand side frames).

For Tait-Bryan angle configurations (a0 != a2), the returned angles are in the ranges [-pi:pi]x[-pi/2:pi/2]x[-pi:pi]. For proper Euler angle configurations (a0 == a2), the returned angles are in the ranges [-pi:pi]x[0:pi]x[-pi:pi].

The approach used is also described here: https://d3cw3dd2w32x2b.cloudfront.net/wp-content/uploads/2012/07/euler-angles.pdf

See also

class AngleAxis

cross() [1/2]

template<typename Derived >

template<typename OtherDerived >

std::conditional_t< SizeAtCompileTime==2, Scalar, PlainObject > Eigen::MatrixBase< Derived >::cross ( const MatrixBase< OtherDerived > & other ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

the cross product of *this and other. This is either a scalar for size-2 vectors or a size-3 vector for size-3 vectors.

This method is implemented for two different cases: between vectors of fixed size 2 and between vectors of fixed size 3.

For vectors of size 3, the output is simply the traditional cross product.

For vectors of size 2, the output is a scalar. Given vectors \( v = \begin{bmatrix} v_1 & v_2 \end{bmatrix} \) and \( w = \begin{bmatrix} w_1 & w_2 \end{bmatrix} \), the result is simply \( v\times w = \overline{v_1 w_2 - v_2 w_1} = \text{conj}\left|\begin{smallmatrix} v_1 & w_1 \\ v_2 & w_2 \end{smallmatrix}\right| \); or, to put it differently, it is the third coordinate of the cross product of \( \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} \) and \( \begin{bmatrix} w_1 & w_2 & w_3 \end{bmatrix} \). For real-valued inputs, the result can be interpreted as the signed area of a parallelogram spanned by the two vectors.

Note

With complex numbers, the cross product is implemented as \( (\mathbf{a}+i\mathbf{b}) \times (\mathbf{c}+i\mathbf{d}) = (\mathbf{a} \times \mathbf{c} - \mathbf{b} \times \mathbf{d}) - i(\mathbf{a} \times \mathbf{d} + \mathbf{b} \times \mathbf{c})\)

See also

MatrixBase::cross3()

cross() [2/2]

template<typename ExpressionType , int Direction>

template<typename OtherDerived >

const VectorwiseOp< ExpressionType, Direction >::CrossReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::cross

(

const MatrixBase< OtherDerived > &

other

)

const

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

a matrix expression of the cross product of each column or row of the referenced expression with the other vector.

The referenced matrix must have one dimension equal to 3. The result matrix has the same dimensions than the referenced one.

See also

MatrixBase::cross()

cross3()

template<typename Derived >

template<typename OtherDerived >

MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::cross3 ( const MatrixBase< OtherDerived > & other ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

the cross product of *this and other using only the x, y, and z coefficients

The size of *this and other must be four. This function is especially useful when using 4D vectors instead of 3D ones to get advantage of SSE/AltiVec vectorization.

See also

MatrixBase::cross()

eulerAngles()

template<typename Derived >

EIGEN_DEPRECATED Matrix< typename MatrixBase< Derived >::Scalar, 3, 1 > Eigen::MatrixBase< Derived >::eulerAngles ( Index a0, Index a1, Index a2 ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

the Euler-angles of the rotation matrix *this using the convention defined by the triplet (a0,a1,a2)

NB: The returned angles are in non-canonical ranges [0:pi]x[-pi:pi]x[-pi:pi]. For canonical Tait-Bryan/proper Euler ranges, use canonicalEulerAngles.

See also

MatrixBase::canonicalEulerAngles

class AngleAxis

hnormalized() [1/2]

template<typename Derived >

const MatrixBase< Derived >::HNormalizedReturnType Eigen::MatrixBase< Derived >::hnormalized ( ) const

inline

homogeneous normalization

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

a vector expression of the N-1 first coefficients of *this divided by that last coefficient.

This can be used to convert homogeneous coordinates to affine coordinates.

It is essentially a shortcut for:

this->head(this->size()-1)/this->coeff(this->size()-1);

Example:

Vector4d v = Vector4d::Random();
Projective3d P(Matrix4d::Random());
cout << "v                   = " << v.transpose() << "]^T" << endl;
cout << "v.hnormalized()     = " << v.hnormalized().transpose() << "]^T" << endl;
cout << "P*v                 = " << (P * v).transpose() << "]^T" << endl;
cout << "(P*v).hnormalized() = " << (P * v).hnormalized().transpose() << "]^T" << endl;

Output:

See also

VectorwiseOp::hnormalized()

hnormalized() [2/2]

template<typename ExpressionType , int Direction>

const VectorwiseOp< ExpressionType, Direction >::HNormalizedReturnType Eigen::VectorwiseOp< ExpressionType, Direction >::hnormalized ( ) const

inline

column or row-wise homogeneous normalization

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

an expression of the first N-1 coefficients of each column (or row) of *this divided by the last coefficient of each column (or row).

This can be used to convert homogeneous coordinates to affine coordinates.

It is conceptually equivalent to calling MatrixBase::hnormalized() to each column (or row) of *this.

Example:

Matrix4Xd M = Matrix4Xd::Random(4, 5);
Projective3d P(Matrix4d::Random());
cout << "The matrix M is:" << endl << M << endl << endl;
cout << "M.colwise().hnormalized():" << endl << M.colwise().hnormalized() << endl << endl;
cout << "P*M:" << endl << P * M << endl << endl;
cout << "(P*M).colwise().hnormalized():" << endl << (P * M).colwise().hnormalized() << endl << endl;

Output:

See also

MatrixBase::hnormalized()

homogeneous() [1/2]

template<typename Derived >

MatrixBase< Derived >::HomogeneousReturnType Eigen::MatrixBase< Derived >::homogeneous ( ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

a vector expression that is one longer than the vector argument, with the value 1 symbolically appended as the last coefficient.

This can be used to convert affine coordinates to homogeneous coordinates.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

Example:

Vector3d v = Vector3d::Random(), w;
Projective3d P(Matrix4d::Random());
cout << "v                                   = [" << v.transpose() << "]^T" << endl;
cout << "h.homogeneous()                     = [" << v.homogeneous().transpose() << "]^T" << endl;
cout << "(P * v.homogeneous())               = [" << (P * v.homogeneous()).transpose() << "]^T" << endl;
cout << "(P * v.homogeneous()).hnormalized() = [" << (P * v.homogeneous()).eval().hnormalized().transpose() << "]^T"
     << endl;

Output:

See also

VectorwiseOp::homogeneous(), class Homogeneous

homogeneous() [2/2]

template<typename ExpressionType , int Direction>

Homogeneous< ExpressionType, Direction > Eigen::VectorwiseOp< ExpressionType, Direction >::homogeneous ( ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

an expression where the value 1 is symbolically appended as the final coefficient to each column (or row) of the matrix.

This can be used to convert affine coordinates to homogeneous coordinates.

Example:

Matrix3Xd M = Matrix3Xd::Random(3, 5);
Projective3d P(Matrix4d::Random());
cout << "The matrix M is:" << endl << M << endl << endl;
cout << "M.colwise().homogeneous():" << endl << M.colwise().homogeneous() << endl << endl;
cout << "P * M.colwise().homogeneous():" << endl << P * M.colwise().homogeneous() << endl << endl;
cout << "P * M.colwise().homogeneous().hnormalized(): " << endl
     << (P * M.colwise().homogeneous()).colwise().hnormalized() << endl
     << endl;

Output:

See also

MatrixBase::homogeneous(), class Homogeneous

umeyama()

template<typename Derived , typename OtherDerived >

internal::umeyama_transform_matrix_type< Derived, OtherDerived >::type Eigen::umeyama	(	const MatrixBase< Derived > &	src,
		const MatrixBase< OtherDerived > &	dst,
		bool	with_scaling = true )

Returns the transformation between two point sets.

This is defined in the Geometry module.

#include <Eigen/Geometry> 

The algorithm is based on: "Least-squares estimation of transformation parameters between two point patterns", Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573

It estimates parameters \( c, \mathbf{R}, \) and \( \mathbf{t} \) such that

\begin{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2 \end{align*}

is minimized.

The algorithm is based on the analysis of the covariance matrix \( \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} \) of the input point sets \( \mathbf{x} \) and \( \mathbf{y} \) where \(d\) is corresponding to the dimension (which is typically small). The analysis is involving the SVD having a complexity of \(O(d^3)\) though the actual computational effort lies in the covariance matrix computation which has an asymptotic lower bound of \(O(dm)\) when the input point sets have dimension \(d \times m\).

Currently the method is working only for floating point matrices.

Parameters

src	Source points \( \mathbf{x} = \left( x_1, \hdots, x_n \right) \).
dst	Destination points \( \mathbf{y} = \left( y_1, \hdots, y_n \right) \).
with_scaling	Sets \( c=1 \) when false is passed.

Returns

The homogeneous transformation

\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}

minimizing the residual above. This transformation is always returned as an Eigen::Matrix.

unitOrthogonal()

template<typename Derived >

MatrixBase< Derived >::PlainObject Eigen::MatrixBase< Derived >::unitOrthogonal ( void ) const

inline

This is defined in the Geometry module.

#include <Eigen/Geometry> 

Returns

a unit vector which is orthogonal to *this

The size of *this must be at least 2. If the size is exactly 2, then the returned vector is a counter clock wise rotation of *this, i.e., (-y,x).normalized().

See also

cross()