Template Class SO2Base

Inheritance Relationships

Derived Type

Class Documentation

template<class Derived>
class SO2Base

SO2 base type - implements SO2 class but is storage agnostic.

SO(2) is the group of rotations in 2d. As a matrix group, it is the set of matrices which are orthogonal such that R * R' = I (with R' being the transpose of R) and have a positive determinant. In particular, the determinant is 1. Let theta be the rotation angle, the rotation matrix can be written in close form: 

 | cos(theta) -sin(theta) |
 | sin(theta)  cos(theta) |
As a matter of fact, the first column of those matrices is isomorph to the set of unit complex numbers U(1). Thus, internally, SO2 is represented as complex number with length 1.

SO(2) is a compact and commutative group. First it is compact since the set of rotation matrices is a closed and bounded set. Second it is commutative since R(alpha) * R(beta) = R(beta) * R(alpha), simply because alpha + beta = beta + alpha with alpha and beta being rotation angles (about the same axis).

Class invariant: The 2-norm of unit_complex must be close to 1. Technically speaking, it must hold that:

|unit_complex().squaredNorm() - 1| <= Constants::epsilon().

Subclassed by Sophus::SO2< Scalar, Options >

Public Types

using Scalar = typename Eigen::internal::traits<Derived>::Scalar
using ComplexT = typename Eigen::internal::traits<Derived>::ComplexType
using ComplexTemporaryType = Sophus::Vector2<Scalar, Options>
using Transformation = Matrix<Scalar, N, N>
using Point = Vector2<Scalar>
using HomogeneousPoint = Vector3<Scalar>
using Line = ParametrizedLine2<Scalar>
using Hyperplane = Hyperplane2<Scalar>
using Tangent = Scalar
using Adjoint = Scalar
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 SO2 operations.

template<typename OtherDerived>
using SO2Product = SO2<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.

It simply 1, since SO(2) is a commutative group.

template<class NewScalarType> inline SOPHUS_FUNC SO2< NewScalarType > cast () const

Returns copy of instance casted to NewScalarType.

inline SOPHUS_FUNC Scalar * data ()

This provides unsafe read/write access to internal data. SO(2) is represented by a unit complex number (two parameters). When using direct write access, the user needs to take care of that the complex number stays normalized.

inline SOPHUS_FUNC Scalar const  * data () const

Const version of data() above.

inline SOPHUS_FUNC SO2< Scalar > inverse () const

Returns group inverse.

inline SOPHUS_FUNC Scalar log () const

Logarithmic map

Computes the logarithm, the inverse of the group exponential which maps element of the group (rotation matrices) to elements of the tangent space (rotation angles).

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

inline SOPHUS_FUNC void normalize ()

It re-normalizes unit_complex to unit length.

Note: Because of the class invariant, there is typically no need to call this function directly.

inline SOPHUS_FUNC Transformation matrix () const

Returns 2x2 matrix representation of the instance.

For SO(2), the matrix representation is an orthogonal matrix R with det(R)=1, thus the so-called “rotation matrix”.

template<class OtherDerived> inline SOPHUS_FUNC SO2Base< Derived > & operator= (SO2Base< OtherDerived > const &other)

Assignment-like operator from OtherDerived.

template<typename OtherDerived> inline SOPHUS_FUNC SO2Product< OtherDerived > operator* (SO2Base< 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 a 2 dimensional point p by the SO2 element bar_R_foo (= rotation matrix): p_bar = bar_R_foo * p_foo.

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.

This function rotates a homogeneous 2 dimensional point p by the SO2 element bar_R_foo (= rotation matrix): p_bar = bar_R_foo * p_foo.

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

Group action on lines.

This function rotates a parametrized line l(t) = o + t * d by the SO2 element:

Both direction d and origin o are rotated as a 2 dimensional point

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 SO2 element:

Normal vector n is rotated Offset d is left unchanged

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 SO2Base< Derived > operator*= (SO2Base< 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 Matrix< Scalar, num_parameters, DoF > Dx_this_mul_exp_x_at_0 () const

Returns derivative of this * SO2::exp(x) wrt. x at x=0.

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

Returns internal parameters of SO(2).

It returns (c[0], c[1]), with c being the unit complex number.

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 void setComplex (Point const &complex)

Takes in complex number / tuple and normalizes it.

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

inline SOPHUS_FUNC ComplexT const  & unit_complex () const

Accessor of unit quaternion.

Public Static Attributes

static constexpr int Options = Eigen::internal::traits<Derived>::Options
static constexpr int DoF = 1

Degrees of freedom of manifold, number of dimensions in tangent space (one since we only have in-plane rotations).

static constexpr int num_parameters = 2

Number of internal parameters used (complex numbers are a tuples).

static constexpr int N = 2

Group transformations are 2x2 matrices.

static constexpr int Dim = 2

Points are 3-dimensional.