Affine Transformations
As with vectors, we can operate on points using
matrices. However, we will need to use 4 by 4 matrices
since our basis set has four components. However, we
will initially limit ourselves to transforms that
preserve the integrity of our points and vectors.
Literally, those transforms that produce a point or
vector when given one of the same.
This subset of 4 by 4 matrices has the property that a
point will be obtained from any input point, and a
vector will be obtained from an input vector,
independent of the point or vector's coordinates.
This subset of matrices is called, you guessed it, the
affine subset.
Our rules for interpreting left and right association
that we developed when transforming vectors still
apply here. We can transform affine frames and we can
transform the coordinates of points. The next time we
meet we will discuss, and give names to, various
sub-subsets of these transformations. But doing so is
merely mechanics, the magic is all here.
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