Base class for SO2 Lie group types. More...
#include <so2.hpp>
Inheritance diagram for SO2Base< _Derived >:
Public Member Functions | |
| Scalar | angle () const |
| Angle representation, in \([-\pi,\pi]\) range. | |
| Scalar | angle_cw () const |
| Angle representation in clockwise direction, in \([-2\pi,0]\) range. | |
| Scalar | angle_ccw () const |
| Angle representation in clockwise direction, in \([0,2\pi]\) range. | |
| Eigen::Vector2< Scalar > | unit_complex () const |
| Unit complex number (U(1)) representation. | |
| std::complex< Scalar > | u1 () const |
| Unit complex number (U(1)) representation. | |
| template<typename EigenDerived > | |
| Eigen::Vector2< Scalar > | operator* (const Eigen::MatrixBase< EigenDerived > &v) const |
| Rotation action on 2D vector. | |
| template<typename EigenDerived > | |
| Eigen::Matrix< Scalar, 2, 1 > | dr_action (const Eigen::MatrixBase< EigenDerived > &v) const |
| Jacobian of rotation action w.r.t. group. | |
| SO3< Scalar > | lift_so3 () const |
| Lift to SO3. | |
 Public Member Functions inherited from LieGroupBase< _Derived > | |
| auto & | coeffs () const |
| Access underlying storages. | |
| const auto & | coeffs () const |
| Const access underlying storages. | |
| auto * | data () const |
| Access raw pointer. | |
| const auto * | data () const |
| Const access raw pointer. | |
| _Derived & | operator= (const LieGroupBase< OtherDerived > &o) noexcept |
| Eigen::Index | dof () const noexcept |
| Dynamic size (degrees of freedom). | |
| void | setIdentity () noexcept |
| Set to group identity element. | |
| void | setRandom () noexcept |
| Set to a random element. | |
| Matrix | matrix () const noexcept |
| Return as matrix Lie group element in \( \mathbb{R}^{\mathtt{dim} \times \mathtt{dim}}
\). | |
| bool | isApprox (const LieGroupBase< OtherDerived > &o, const Scalar &eps=Eigen::NumTraits< Scalar >::dummy_precision()) const noexcept |
Check if (approximately) equal to other element o. | |
| CastT< NewScalar > | cast () const noexcept |
| Cast to different scalar type. | |
| PlainObject | operator* (const LieGroupBase< OtherDerived > &o) const noexcept |
| Group binary composition operation. | |
| _Derived & | operator*= (const LieGroupBase< OtherDerived > &o) noexcept |
| Inplace group binary composition operation. | |
| PlainObject | inverse () const noexcept |
| Group inverse operation. | |
| Tangent | log () const noexcept |
| Lie group logarithm. | |
| TangentMap | Ad () const noexcept |
| Lie group adjoint. | |
| PlainObject | operator+ (const Eigen::MatrixBase< TangentDerived > &a) const noexcept |
| Right-plus. | |
| _Derived & | operator+= (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Inplace right-plus: \( \mathbf{x} \leftarrow \mathbf{x} \circ \exp(\mathbf{a}) \). | |
| Tangent | operator- (const LieGroupBase< OtherDerived > &xo) const noexcept |
| Right-minus. | |
Public Attributes | |
| SMOOTH_INHERIT_TYPEDEFS | |
Additional Inherited Members | |
| Public Types inherited from LieGroupBase< _Derived > | |
| using | Scalar = typename traits::Scalar |
| Scalar type. | |
| using | Matrix = Eigen::Matrix< Scalar, Dim, Dim > |
| Lie group matrix type. | |
| using | Tangent = Eigen::Matrix< Scalar, Dof, 1 > |
| Lie group parameterized tangent type. | |
| using | TangentMap = Eigen::Matrix< Scalar, Dof, Dof > |
| Matrix representing map between tangent elements. | |
| using | Hessian = Eigen::Matrix< Scalar, Dof, Dof *Dof > |
| Plain return type with different scalar. | |
| using | CastT = typename traits::template PlainObject< NewScalar > |
| Plain return type with different scalar. | |
| using | PlainObject = CastT< Scalar > |
| Plain return type. | |
| Static Public Member Functions inherited from LieGroupBase< _Derived > | |
| static PlainObject | Identity () noexcept |
| Construct the identity element. | |
| static PlainObject | Random () noexcept |
| Construct a random element. | |
| static PlainObject | exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Lie group exponential map. | |
| static Matrix | hat (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Lie algebra hat map. | |
| static Tangent | vee (const Eigen::MatrixBase< MatrixDerived > &A) noexcept |
| Lie alebra vee map. | |
| static TangentMap | ad (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Lie algebra adjoint. | |
| static Tangent | lie_bracket (const Eigen::MatrixBase< TangentDerived1 > &a, const Eigen::MatrixBase< TangentDerived2 > &b) noexcept |
| Lie algebra bracket. | |
| static TangentMap | dr_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Right jacobian of the exponential map. | |
| static TangentMap | dr_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Inverse of right jacobian of the exponential map. | |
| static TangentMap | dl_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Left jacobian of the exponential map. | |
| static TangentMap | dl_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Inverse of left jacobian of the exponential map. | |
| static Hessian | d2r_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Right Hessian of the exponential map. | |
| static Hessian | d2r_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Right Hessian of the log map. | |
| static Hessian | d2l_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Left Hessian of the exponential map. | |
| static Hessian | d2l_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept |
| Left Hessian of the log map. | |
| Static Public Attributes inherited from LieGroupBase< _Derived > | |
| static constexpr int | RepSize |
| Number of scalars in internal representation. | |
| static constexpr int | Dof |
| Degrees of freedom of manifold (equal to tangent space dimension). | |
| static constexpr int | Dim |
| Side of Lie group matrix representation. | |
| static constexpr bool | IsCommutative |
| Commutativity of group. A commutative group has a zero Lie bracket. | |
| Protected Types inherited from LieGroupBase< _Derived > | |
| using | traits = liebase_info< _Derived > |
| CRTP traits. | |
| using | Impl = typename traits::Impl |
| Group-specific Lie group implementation. | |
| Static Protected Attributes inherited from LieGroupBase< _Derived > | |
| static constexpr bool | is_mutable |
| True if underlying storage supports modification. | |
Detailed Description
Base class for SO2 Lie group types.
Internally represented as \(\mathbb{U}(1)\) (unit complex numbers).
Memory layout
- Group: \( \mathbf{x} = [q_z, q_w] \)
- Tangent: \( \mathbf{a} = [\omega_z] \)
Constraints
- Group: \(q_z^2 + q_w^2 = 1 \)
- Tangent: \( -\pi < \omega_z \leq \pi \)
Lie group matrix form
\[ \mathbf{X} = \begin{bmatrix} q_w & -q_z \\ q_z & q_w \end{bmatrix} \in \mathbb{R}^{2 \times 2} \]
Lie algebra matrix form
\[ \mathbf{a}^\wedge = \begin{bmatrix} 0 & -\omega_z \\ \omega_z & 0 \\ \end{bmatrix} \in \mathbb{R}^{2 \times 2} \]
Member Function Documentation
◆ angle()
◆ angle_ccw()
◆ angle_cw()
◆ dr_action()
template<typename EigenDerived >
|
inline |
◆ lift_so3()
◆ operator*()
template<typename EigenDerived >
|
inline |
◆ u1()
◆ unit_complex()
Member Data Documentation
◆ SMOOTH_INHERIT_TYPEDEFS
The documentation for this class was generated from the following file:
- include/smooth/so2.hpp
