Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.

Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.

Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.

Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.

Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.

Returns the ith infinitesimal generators of SE(3). The infinitesimal generators of SE(3) are: ``` | 0 0 0 1 | G_0 = | 0 0 0 0 | | 0 0 0

Returns the ith infinitesimal generators of SE(2). The infinitesimal generators of SE(2) are: ``` | 0 0 1 | G_0 = | 0 0 0 | | 0 0 0 | | 0

Returns the ith infinitesimal generators of SO(3). The infinitesimal generators of SO(3) are: ``` | 0 0 0 | G_0 = | 0 0 -1 | | 0 1 0 | | 0

Returns the ith infinitesimal generators of Sim(2). The infinitesimal generators of Sim(2) are: ``` | 0 0 1 | G_0 = | 0 0 0 | | 0 0 0 | |

Returns the ith infinitesimal generators of Sim(3). The infinitesimal generators of Sim(3) are: ``` | 0 0 0 1 | G_0 = | 0 0 0 0 | | 0 0 0

Returns the infinitesimal generators of SO(2). The infinitesimal generators of SO(2) is: | 0 1 | | -1 0 |

hat-operator It takes in the 6-vector representation (= twist) and returns the corresponding matrix representation of Lie algebra element. Formally,

hat-operator It takes in the 3-vector representation (= twist) and returns the corresponding matrix representation of Lie algebra element. Formally,

hat-operator It takes in the 3-vector representation ``omega`` (= rotation vector) and returns the corresponding matrix representation of Lie algebra

hat-operator It takes in the 4-vector representation and returns the corresponding matrix representation of Lie algebra element. Formally, the hat()

hat-operator It takes in the 7-vector representation and returns the corresponding matrix representation of Lie algebra element. Formally, the hat()

hat-operator It takes in the scalar representation ``theta`` (= rotation angle) and returns the corresponding matrix representation of Lie algebra el

Returns 3x3 matrix representation of the instance. For SO(3), the matrix representation is an orthogonal matrix ``R`` with ``det(R)=1``, thus the so-

Returns rotation matrix.

Returns rotation matrix.

quaternion multiplication

We use the following convention [real, imag]

We use the following convention [q0, q1, q2, q3, t0, t1, t2]

We use the following convention [q0, q1, q2, q3, t0, t1, t2]

We use the following convention [vec0, vec1, vec2, real]

Dual quaternion consists of a real quaternion, and an infinitesimal quaternion

internally represented by a unit complex number z and a translation 2-vector

internally represented by a unit quaternion q and a translation 3-vector

Quaternion consists of a real scalar, and an imaginary 3-vector

internally represented by a unit quaternion q

internally represented by a unit complex number z

complex multiplication

dual quaternion multiplication

left-multiplication either rotation concatenation or point-transform

left-multiplication either rotation concatenation or point-transform

quaternion multiplication

left-multiplication either rotation concatenation or point-transform

left-multiplication either rotation concatenation or point-transform