The Klein View of Geometry

The similarity group

Let E denote the euclidean plane.

Definition 1

A similarity is a transformation of E which maps lines to lines, and preserves the size of angles.

The set of all similarities is the similarity group S(2).

Since each isometry of E preserves lines and angles, an isometry is a similarity.
Thus E(2) is a subgroup of S(2).

But there are similarities which are not isometries.
An obvious example is s_k, the dilation about the origin by a factor k (>0).
In terms of complex coordinates, s_k(z) = kz.

The Scaling Theorem
A similarity scales all distances by a fixed positive factor.

Corollary
Each similarity can be written as the composite of an isometry and a dilation about O.

Proof of Theorem and Corollary

Note that the scaling factor associated with a similarity is uniquely defined.
We can thus recover E(2) from S(2) as follows:

E(2) is the subgroup of S(2) consisting of similarities with scaling factor 1.

Since similarities preserve, the size of angles, they may be direct or indirect (or possibly neither).
Once we observe that each dilation is direct, the Corollary shows that each is direct or indirect.

We can now describe the elements of S(2).

Theorem S2
If t is a direct similarity, then t(z) = az + b, where a, b are in C, and |a| non-zero.
If t is an indirect similarity, then t(z) = az* + b, where a, b are in C, and |a| non-zero.

Proof of Theorem S2

As for the euclidean group, we often want to use coordinates.

Theorem S3
E(2) = {t(x) = kAx + b, where A is a real orthogonal 2×2 matrix, b a real vector and k > 0}.
The direct isometries are those with det(A)=1.

This follows from Theorem S2 above, and Theorem E2 for euclidean geometry

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