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 2012-04-20