similarity geometry
Photocopy a drawing at any enlargement, then slide, spin, or flip the copy: the result looks like the original, just possibly bigger or smaller. Nothing about its shape has changed, only its overall scale and placement. Similarity geometry is the geometry that treats two figures as the same exactly when one is a scaled, possibly reflected, repositioned copy of the other — it is the geometry of shape, blind to absolute size.
In the Erlangen scheme, similarity geometry is the geometry whose transformation group is the similarity group: all maps built from a rigid motion (rotation, reflection, translation) followed by a uniform scaling by some positive factor. Such a map multiplies every length by the same factor k, so it does not preserve length, but because the scaling is uniform it preserves the ratio of any two lengths and it preserves every angle. The invariants of similarity geometry are therefore shape itself, angles, ratios of lengths, ratios of areas, and parallelism and collinearity inherited from below; what is lost compared with Euclidean geometry is absolute length and absolute area. Two triangles are 'the same' in similarity geometry precisely when they are similar in the school sense — equal angles, proportional sides — which is why the AA, SAS, and SSS similarity criteria are the natural congruence tests at this level.
Similarity geometry sits one level above Euclidean geometry and one below affine geometry in the hierarchy: it forgets the unit of length that Euclidean geometry keeps, but still remembers angle, which affine geometry throws away. It is the right setting for any statement that involves proportion and shape but not an absolute scale — most of the theory of similar triangles, the Pythagorean theorem read as a statement about ratios, fractal self-similarity. A natural caution: 'similar' allows a reflection, so a left-handed and a right-handed version of a shape count as similar even though no rotation alone superimposes them.
Two maps of the same country, one at scale 1:50000 and one at 1:100000, are related by a similarity: every distance on the second is half the corresponding distance on the first, every angle is identical, and every shape is the same. In similarity geometry the two maps are 'equal', because what differs between them — the scale — is exactly what this geometry refuses to see.
Similarity geometry sees shape, angle, and ratio, but is blind to absolute scale.
Similar is not the same as congruent: congruence (Euclidean) demands equal size as well as equal shape, while similarity allows any uniform rescaling and even a reflection, so all squares are similar but not all are congruent.