Base class for C1 Lie group types. More...

#include <c1.hpp>

Inheritance diagram for C1Base< _Derived >:

Public Member Functions
Scalar	angle () const
	Rotation angle.

Scalar	scaling () const
	Scaling.

SO2< Scalar >	so2 () const
	Rotation.

std::complex< Scalar >	c1 () const
	Complex number representation.

template<typename EigenDerived >
Eigen::Matrix< Scalar, 2, 1 >	operator* (const Eigen::MatrixBase< EigenDerived > &v) const
	Rotation and scaling action on 2D vector.

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

template<typename _Derived>
class C1Base< _Derived >

Base class for C1 Lie group types.

Memory layout

Group: \( \mathbf{x} = [a, b] \)
Tangent: \( \mathbf{a} = [s, \omega_z] \)

Constraints

Group: \( a^2 + b^2 > 0 \)
Tangent: \( -\pi < \omega_z \leq \pi \)

Lie group matrix form

\[ \mathbf{X} = \begin{bmatrix} b & -a \\ a & b \end{bmatrix} \in \mathbb{R}^{2 \times 2} \]

Lie algebra matrix form

\[ \mathbf{a}^\wedge = \begin{bmatrix} s & -\omega_z \\ \omega_z & s \\ \end{bmatrix} \in \mathbb{R}^{2 \times 2} \]

Definition at line 61 of file c1.hpp.

Member Function Documentation

◆ angle()

template<typename _Derived >

Scalar C1Base< _Derived >::angle ( ) const

inline

Rotation angle.

Definition at line 74 of file c1.hpp.

◆ c1()

template<typename _Derived >

std::complex< Scalar > C1Base< _Derived >::c1 ( ) const

inline

Complex number representation.

Definition at line 104 of file c1.hpp.

◆ operator*()

template<typename _Derived >

template<typename EigenDerived >

Eigen::Matrix< Scalar, 2, 1 > C1Base< _Derived >::operator* ( const Eigen::MatrixBase< EigenDerived > & v ) const

inline

Rotation and scaling action on 2D vector.

Definition at line 114 of file c1.hpp.

◆ scaling()

template<typename _Derived >

Scalar C1Base< _Derived >::scaling ( ) const

inline

Scaling.

Definition at line 84 of file c1.hpp.

◆ so2()

template<typename _Derived >

SO2< Scalar > C1Base< _Derived >::so2 ( ) const

inline

Rotation.

Definition at line 96 of file c1.hpp.

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS

template<typename _Derived >

C1Base< _Derived >::SMOOTH_INHERIT_TYPEDEFS

Definition at line 69 of file c1.hpp.

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

include/smooth/c1.hpp

Public Member Functions

Public Attributes

Additional Inherited Members

Detailed Description

Memory layout

Constraints

Lie group matrix form

Lie algebra matrix form

Member Function Documentation

◆ angle()

◆ c1()

◆ operator*()

◆ scaling()

◆ so2()

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS