the affine group revisited

Since A(2) is isomorphic to a subgroup of P(2), it is possible to
obtain the Fundamental Theorem for affine geometry from that
for projective geometry.

The key is the observation that while a projective transformation,
as a map of an extended plane, may map the centroid of a triangle
to any point not on the image triangle, an affine transformation of
the plane maps the centroid to the centroid of the image triangle.

The Fundamental Theorem of Affine Geometry
If L = (A,B,C) and L' = (A',B',C') are lists of non-collinear points,
then there is a unique element of A(2) mapping L to L'.

We consider the affine group A(2) as acting on the plane E.
To relate it to projective geometry by placing E on the plane
(i.e. p-line) Π : z=1.

For a point P(x,y) on E, we write p for the corresponding point
(x,y,1) on Π, and [p] for the p-point [x,y,1].

Let S, S' denote the centroids of triangles ABC, A'B'C' respectively.
Then we have s = (1/3)(a + b + c), and s' = (1/3)(a' + b' + c').
As A,B,C are non-collinear, a,b,c are linearly independent, and
it is easy to see that any three of a,b,c,s are also independent.
It follows that no three of [a],[b],[c],[s] are collinear. A similar
argument shows that no three of [a'],[b'],[c'],[s'] are collinear.

By the Fundamental Theorem of Projective Geometry, there is a
projective transformation t mapping the first four p-points to the
second four. Suppose that A is a matrix for t. Then we have
Aa = ka', Ab = lb', Ac = mc', As = ns', where k,l,m,n are non-zero
constants. (See the proof of the theorem for projective geometry.)
But we also have formulae for s and s'. These yield
As = (1/3)(ka' + lb' + mc'), and so we have
(1/3)(ka' + lb' + mc') = (1/3)(na' + nb' + nc').
Then, since a',b',c' are independent, we must have k = l = m = n.
Now, we may use any (non-zero) multiple of A as a matrix for t, so
we may as well assume that k,l,m,n are all equal to 1. We then have
Aa = a', Ab = b', Ac = c', As = s'.

Now, a,b,c are non-collinear points on Π, any point p on Π has the
form αabc, with α+β+γ=1. A similar result holds for a',b',c'.
For such a point, Ap = αa'b'c', and so Ap also lies on Π. Thus
the transformation t lies in the subgroup A(2), as described earlier.

Restricting our attention to Π, we have an affine transformation
which maps L to L'.

The uniqueness of the transformation follows from the uniqueness of t.

return to the affine and projective page