SO3TangentVector.SO3TangentVector

The left tangent space of SO(3) in some rotation R can be described by

\[ T_R(SO(3)) = \{ s\cdot R | s = -s^T \} \]

where \(R\) denotes the rotation matrix and \(s\) are skew symmetric matrices which look like

\[ \left(\begin{matrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{matrix}\right).\]

Hence we describe an element of of the tangent space T_R(SO(3)) by the vector \((a,b,c)^T\) which in fact is an vector3d and the corresponding rotation.

Note that \( \{ R\cdot t | t = -t^T \} \) is another possible representation of the tangent space. It is called right tangent space.

We denote whether an SO3TangentVector v is described w.r.t. the left tangent space or right tangent space by the property v.tangentSpace. Moreover we can change the representation of the tangentSpace by using the methods right(v) and left(v).

Description

The symmetries of a tangent vector are the symmetries of |oriRef| -
there is no separate symmetry argument. A field or function that owns
the pair therefore states it on the reference it hands in, e.g.
|SO3TangentVector(v,orientation(rot,SO3F.CS,SO3F.SS),tS)|, rather than
passing it alongside. That leaves exactly one place a symmetry can come
from, so there is nothing to arbitrate.

|left| means oriRef multiplies from the right (the default), |right|
that it multiplies from the left.

Syntax

SO3TV = SO3TangentVector(v,oriRef)
SO3TV = SO3TangentVector(v,oriRef,SO3TangentSpace.rightVector)
SO3TV = SO3TangentVector(v,oriRef,'right')

Input

Output

See also

vector3d.vector3d SO3VectorField.SO3VectorField SO3VectorFieldHarmonic.eval SO3Fun.grad SO3FunHarmonic.grad