kornia.geometry.liegroup
==========================

.. meta::
   :name: description
   :content: "The kornia.geometry.liegroup module provides mathematical tools and operations for working with Lie groups and Lie algebras, which are fundamental in many areas of robotics and computer vision. Lie groups describe smooth manifolds that satisfy group axioms, while Lie algebras represent the tangent space at the identity element of these groups. This module includes key classes for common Lie groups like `SO2`, `SO3`, `SE2`, and `SE3`, and provides functions for performing operations like the exponential and logarithmic maps, which connect Lie algebras and Lie groups."

.. 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 <https://en.wikipedia.org/wiki/Group_(mathematics)>`_

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 <https://en.wikipedia.org/wiki/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 <https://arxiv.org/pdf/1812.01537.pdf>`_

.. autoclass:: So3
   :members:
   :special-members:

.. autoclass:: Se3
   :members:
   :special-members:

.. autoclass:: So2
   :members:
   :special-members:

.. autoclass:: Se2
   :members:
   :special-members: