Geometry
In geometry, similarities of a Euclidean space preserve circles and spheres. Conversely, Abouabdillah's theorem states that every injective or surjective transformation of a Euclidean space that preserves circles or spheres is a similarity.
More precisely:
Theorem. Let be a Euclidean affine space of dimension at least 2. Then:
1. Every surjective mapping that transforms any four concyclic points into four concyclic points is a similarity.
2. Every injective mapping that transforms any circle into a circle is a similarity.
Read more about this topic: Abouabdillah's Theorem
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