kornia.geometry.liegroup ========================== .. currentmodule:: kornia.geometry.liegroup The Lie group encompasses the concepts of `group` and `smooth manifold` in a unique body. A group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. See more: `Group `_ A Lie group :math:`G` is a smooth manifold whose elements satisfy the group axioms.You can visualize the idea of manifold like a curved, smooth (hyper)-surface, with no edges or spikes, embedded in a space of higher dimension. See more: `Manifold `_ In robotics, we say that our state vector evolves on this surface, that is, the manifold describes or is defined by the constraints imposed on the state. lie algebra =========== .. image:: data/lie.png If :math:`M` is the manifold that represents a lie group, the tangent space at the identity is called the Lie algebra of :math:`M`. The Lie algebra :math:`m` is a vector space. As such, its elements can be identified with vectors in :math:`R^d`, whose dimension :math:`d` is the number of degrees of freedom of :math:`M`. For example :math:`d` would be 3 in the case of lie group :math:`SO3` lie group and lie algebra ========================== Every Lie group has an associated Lie algebra. We relate the Lie group with its Lie algebra through the following facts #. The Lie algebra :math:`m` is a vector space. As such, its elements can be identified with vectors in :math:`R^d`, whose dimension :math:`d` is the number of degrees of freedom of :math:`M`. #. The exponential map, `exp` : :math:`m` → :math:`M`, exactly converts elements of the Lie algebra into elements of the group. The `log` map is the inverse operation. .. image:: data/lie_ops.png Reference: `Micro lie theory `_ .. autoclass:: So3 :members: :special-members: .. autoclass:: Se3 :members: :special-members: .. autoclass:: So2 :members: :special-members: .. autoclass:: Se2 :members: :special-members: