A rotation is a special case of a linear transformation, which is
generally expressed by an
matrix,
, assuming the
transformations are performed over
. Consider transforming a
point
in a 2D robot,
, as
The scaling, shearing, and rotation matrices may be multiplied
together to yield a general transformation matrix that explicitly
parameterizes each effect. It is also possible to extend the from
to
to obtain a homogeneous
transformation matrix that includes translation. Also, the concepts
extend in a straightforward way to
and beyond. This enables
the additional effects of scaling and shearing to be incorporated
directly into the concepts from Sections
3.2-3.4.
Steven M LaValle 2020-08-14