Course MT3818 Topics in Geometry

 Previous page (Affine theorems) Contents Next page (Projective spaces)

Similarity geometry

Similarity geometry lies between Euclidean geometry (with group I(Rn)) and Affine geometry (group A(Rn)).

Definition

A similarity transformation or similitude is an affine map which preserves angles.

Theorem
A similarity transformation can be written TaλL with L ∈ O(n) and λ > 0.

Proof
Since f is an affine transformation it can be written as TaL with L linear. Since f preserves angles, so does L and such a map must stretch all vectors by the same amount and must be a non-zero scalar multiple of an orthogonal transformation.

Any two figures related by a similarity transformation are called similar.
For example, any two squares are similar; any two circles are similar; two triangles are similar if their corresponding angles are equal.

Many of the theorems of so-called Euclidean geometry are in fact theorems of Similarity geometry.
For example, the well-known theorem of Pythagoras can be proved by "similar triangle" methods

Pythagoras's theorem
If a triangle ABC has a right angle at A then AB2 + AC2 = BC2

Proof
Drop a perpendicular to P as shown.
Then the triangles ABC, PBA and PAC are similar and so have their sides in proportion.
Hence AB/PB = BC/BA and AC/PC = BC/AC and so AB2 = BP.BC and AC2 = BC.PC
Thus AB2 + AC2 = (BP + PC).BC = BC2

 Previous page (Affine theorems) Contents Next page (Projective spaces)

JOC March 2003