# 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'. Proof 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 αa+βb+γc, with α+β+γ=1. A similar result holds for a',b',c'. For such a point, Ap = αa'+βb'+γc', 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.