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.
Definitions
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

maps lines to lines, and
maps parallel lines to parallel lines.

Proof
(1) Let L be the line through U with direction vector d ≠ 0.
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.

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. Definitions 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 maps lines to lines, and maps parallel lines to parallel lines. Proof (1) Let L be the line through U with direction vector d ≠ 0. 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.