parallel projections and the affine group

We introduce a family of transformations of the plane which
turn out to generate the affine group A(2). The definition is
purely geometric. It can be motivated by considering the
relationship between all the shadows of an object.

If Π and Π' are planes in R3, and v is a vector not parallel
to either, then parallel projection from Π to Π' along v
is the map from Π to Π' mapping the point P on Π to P',
the point where the line through P parallel to v cuts Π'.

The definition of a parallel projection t from Π to Π' is purely
geometrical. To obtain an algebraic description, we suppose
that Π and Π' are each equipped with (rectangular) x- and
y-axes. Then the transformation has a familiar form.

Theorem A1
A parallel projection is an element of A(2).

proof of theorem A1

Although every parallel projection is an affine transformation,
not every affine transformation is a parallel projection. To see
this, we can use the following

A parallel projection t preserves the lengths of segments
of at least one line.

If the planes Π and Π' are parallel, then it is easy to see that
the map t preserves the length of every segment.
Otherwise Π and Π' meet in a line L. The points of L are fixed
pointwise by t, so the length of every segment of L is fixed.

Now note that the dilation about O with scale factor 2 doubles
the length of every segment. This is the affine transformation
taking x to 2Ix+0. Thus, we have an affine transformation
which is not a parallel projection.

However, we can obtain any affine transformation by
composing parallel projections:

Theorem A2
The affine group A(2) is generated by parallel projections.

proof of theorem A2

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