We have observed that, regarded as groups of transformations of R^{2}, we have the inclusions E(2) ≤ S(2) ≤ A(2). Also, S(2) is isomorphic to the subgroup S(2) = {t ε I(2) : t(∞) = ∞} of I(2).
In a similar way, we show that A(2) is isomorphic to a geometrically


Suppose that t is the projective transformation with the matrix A given on the right. Then t maps the plane Π : z=0 to itself if and only if, for all x,y, t maps [x,y,0] to a ppoint with the third coordinate equal to zero i.e. gx + hy + i0 = 0. Since this holds for all x,y, we must have g = h = 0. As A is nonsingular, i ≠ 0. Finally, since t may also be given by λA, we may assume i = 1.
Then a simple calculation shows that A maps (x,y,1) to the point
Conversely, an affine transformation of the above form leads to a
Definition The above calculation shows that A(2) is isomorphic to A(2).
We can interpret our proof as choosing a pline L in RP^{2}, which
The relation between the groups leads to a relation between

