The Klein View of Geometry

the affine group

In Theorem E2, we showed that each element t of E(2) can be written in matrix form as
t(x) = Ax + b, where A is an orthogonal 2x2 matrix, and bεR2. In Theorem S2, we showed
that each element of S(2) has a similar from, with A replaced by a scalar multiple kA.

Both of these groups are clearly subgroups of the group obtained by allowing A to be any
invertible 2x2 matrix.

The affine group A(2) consists of all transformations of R2 of the form
t(x) = Ax + b, with A a 2x2 invertible matrix, and bεR2.
The elements of A(2) are called affine transformations.

Affine geometry is the geometry with set R2 and group A(2).

Since E(2) is a subgroup of A(2), this geometry is related to euclidean geometry. Indeed,
many classical theorems, such as those of Ceva and Menelaus, are really affine theorems.

For the moment, we will concentrate on the fundamental theorem, developing only such
results as are required for this purpose.

Note that, throughout our proofs, we will use the convention that the position
vector of a point P is represented by p, the corresponding lower case letter.

Theorem A0
An affine transformation t

  1. maps lines to lines, and
  2. maps parallel lines to parallel lines.

(1) Let L be the line through U with direction vector d0.
Then each point P on L has vector p = u + kd, for some k.
Suppose that t is affine, so that t(x) = Ax + b. Then
t(p) = A(u + kd) + b = (Au + b) + kAd, so that t(L) is the
line through the point (Au + b) with direction vector Ad.

(2) If M is a line parallel to L, then we may take d as a direction
vector for M. As in (1), t(M) is a line with direction vector Ad,
so is parallel to t(L).

Thus, collinearity is an affine property. It follows that a list of non-collinear
points can map only to another list of non-collinear points.

The Fundamental Theorem of Affine Geometry
If L =(A,B,C) and L' = (A',B',C') are lists of non-collinear points of R2,
then there is a unique element of A(2) mapping L to L'.

Much as in inversive geometry, this follows easily from a special case with
the standard points O=(0,0), X=(1,0) and Y=(0,1) in one of the lists.

The (O,X,Y) Theorem
If L =(A,B,C) is a list of non-collinear points of R2, then
there is a unique element of A(2) mapping (O,X,Y) to L.

proofs of the theorems

This result allows us to find geometrically defined generators for A(2).

The study of the properties of affine geometry does not require any
knowledge of the generators, so can be begun from here.