Template Class SE2Base

Class Documentation

template<class Derived>
class SE2Base

SE2 base type - implements SE2 class but is storage agnostic.

SE(2) is the group of rotations and translation in 2d. It is the semi-direct product of SO(2) and the 2d Euclidean vector space. The class is represented using a composition of SO2Group for rotation and a 2-vector for translation.

SE(2) is neither compact, nor a commutative group.

See SO2Group for more details of the rotation representation in 2d.

Public Types

using Scalar = typename Eigen::internal::traits<Derived>::Scalar
using TranslationType = typename Eigen::internal::traits<Derived>::TranslationType
using SO2Type = typename Eigen::internal::traits<Derived>::SO2Type
using Transformation = Matrix<Scalar, N, N>
using Point = Vector2<Scalar>
using HomogeneousPoint = Vector3<Scalar>
using Line = ParametrizedLine2<Scalar>
using Hyperplane = Hyperplane2<Scalar>
using Tangent = Vector<Scalar, DoF>
using Adjoint = Matrix<Scalar, DoF, DoF>
template<typename OtherDerived>
using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<Scalar, typename OtherDerived::Scalar>::ReturnType

For binary operations the return type is determined with the ScalarBinaryOpTraits feature of Eigen. This allows mixing concrete and Map types, as well as other compatible scalar types such as Ceres::Jet and double scalars with SE2 operations.

template<typename OtherDerived>
using SE2Product = SE2<ReturnScalar<OtherDerived>>
template<typename PointDerived>
using PointProduct = Vector2<ReturnScalar<PointDerived>>
template<typename HPointDerived>
using HomogeneousPointProduct = Vector3<ReturnScalar<HPointDerived>>

Public Functions

inline SOPHUS_FUNC Adjoint Adj () const

Adjoint transformation

This function return the adjoint transformation Ad of the group element A such that for all x it holds that hat(Ad_A * x) = A * hat(x) A^{-1}. See hat-operator below.

template<class NewScalarType> inline SOPHUS_FUNC SE2< NewScalarType > cast () const

Returns copy of instance casted to NewScalarType.

inline SOPHUS_FUNC Matrix< Scalar, num_parameters, DoF > Dx_this_mul_exp_x_at_0 () const

Returns derivative of this * exp(x) wrt x at x=0.

inline SOPHUS_FUNC Matrix< Scalar, DoF, num_parameters > Dx_log_this_inv_by_x_at_this () const

Returns derivative of log(this^{-1} * x) by x at x=this.

inline SOPHUS_FUNC SE2< Scalar > inverse () const

Returns group inverse.

inline SOPHUS_FUNC Tangent log () const

Logarithmic map

Computes the logarithm, the inverse of the group exponential which maps element of the group (rigid body transformations) to elements of the tangent space (twist).

To be specific, this function computes vee(logmat(.)) with logmat(.) being the matrix logarithm and vee(.) the vee-operator of SE(2).

inline SOPHUS_FUNC void normalize ()

Normalize SO2 element

It re-normalizes the SO2 element.

inline SOPHUS_FUNC Transformation matrix () const

Returns 3x3 matrix representation of the instance.

It has the following form:

| R t | | o 1 |

where R is a 2x2 rotation matrix, t a translation 2-vector and o a 2-column vector of zeros.

inline SOPHUS_FUNC Matrix< Scalar, 2, 3 > matrix2x3 () const

Returns the significant first two rows of the matrix above.

template<class OtherDerived> inline SOPHUS_FUNC SE2Base< Derived > & operator= (SE2Base< OtherDerived > const &other)

Assignment-like operator from OtherDerived.

template<typename OtherDerived> inline SOPHUS_FUNC SE2Product< OtherDerived > operator* (SE2Base< OtherDerived > const &other) const

Group multiplication, which is rotation concatenation.

template<typename PointDerived, typename = typename std::enable_if<                IsFixedSizeVector<PointDerived, 2>::value>::type> inline SOPHUS_FUNC PointProduct< PointDerived > operator* (Eigen::MatrixBase< PointDerived > const &p) const

Group action on 2-points.

This function rotates and translates a two dimensional point p by the SE(2) element bar_T_foo = (bar_R_foo, t_bar) (= rigid body transformation):

p_bar = bar_R_foo * p_foo + t_bar.

template<typename HPointDerived, typename = typename std::enable_if<                IsFixedSizeVector<HPointDerived, 3>::value>::type> inline SOPHUS_FUNC HomogeneousPointProduct< HPointDerived > operator* (Eigen::MatrixBase< HPointDerived > const &p) const

Group action on homogeneous 2-points. See above for more details.

inline SOPHUS_FUNC Line operator* (Line const &l) const

Group action on lines.

This function rotates and translates a parametrized line l(t) = o + t * d by the SE(2) element:

Origin o is rotated and translated using SE(2) action Direction d is rotated using SO(2) action

inline SOPHUS_FUNC Hyperplane operator* (Hyperplane const &p) const

Group action on hyper-planes.

This function rotates a hyper-plane n.x + d = 0 by the SE2 element:

Normal vector n is rotated Offset d is adjusted for translation

Note that in 2d-case hyper-planes are just another parametrization of lines

template<typename OtherDerived, typename = typename std::enable_if<                std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type> inline SOPHUS_FUNC SE2Base< Derived > & operator*= (SE2Base< OtherDerived > const &other)

In-place group multiplication. This method is only valid if the return type of the multiplication is compatible with this SO2’s Scalar type.

inline SOPHUS_FUNC Sophus::Vector< Scalar, num_parameters > params () const

Returns internal parameters of SE(2).

It returns (c[0], c[1], t[0], t[1]), with c being the unit complex number, t the translation 3-vector.

inline SOPHUS_FUNC Matrix< Scalar, 2, 2 > rotationMatrix () const

Returns rotation matrix.

inline SOPHUS_FUNC void setComplex (Sophus::Vector2< Scalar > const &complex)

Takes in complex number, and normalizes it.

Precondition: The complex number must not be close to zero.

inline SOPHUS_FUNC void setRotationMatrix (Matrix< Scalar, 2, 2 > const &R)

Sets so3 using rotation_matrix.

Precondition: R must be orthogonal and det(R)=1.

inline SOPHUS_FUNC SO2Type & so2 ()

Mutator of SO3 group.

inline SOPHUS_FUNC SO2Type const  & so2 () const

Accessor of SO3 group.

inline SOPHUS_FUNC TranslationType & translation ()

Mutator of translation vector.

inline SOPHUS_FUNC TranslationType const  & translation () const

Accessor of translation vector

inline SOPHUS_FUNC Eigen::internal::traits< Derived >::SO2Type::ComplexT const  & unit_complex () const

Accessor of unit complex number.

Public Static Attributes

static constexpr int DoF = 3

Degrees of freedom of manifold, number of dimensions in tangent space (two for translation, three for rotation).

static constexpr int num_parameters = 4

Number of internal parameters used (tuple for complex, two for translation).

static constexpr int N = 3

Group transformations are 3x3 matrices.

static constexpr int Dim = 2

Points are 2-dimensional.