Multiple Transforms
We will often want to specify complicated transformations by stringing together
sequences of simple manipulations.
For instance, if you want to translate
points and then rotate them about the origin.
Suppose that the translation is accomplished by the matrix operator T,
and the rotation is achieved using the matrix, R.
Given what we know now it is a simple matter to construct this series of operations.
Each step in the process can be considered as a transformation of coordinates.
Alternatively, we could have considered the same sequence of operations as follows:
Where each step is considered as a change of basis frames.
These are alternate interpretations of the same transformations. They mean entirely different
things, however they result in the same set of transformed points. The first sequence is considered
as a transformation about a global frame. The second sequence is considered as a change in
local frames. Frequently, we will mix together these ideas in a single transformation.
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