A remark on the Beckman/Quarles theorem Original Research Article

(BQ) In the distance space (M,d : M × M → △) let k ∈ △ and f : M → M be given such that for all a,b ∈ M: d(a,b) = k ⇒ d(f(a),f(b)) = k. Then f is an isometry. The determination of the possible f for a given distance space is called a theorem of Beckman/Quarles type. Here (BQ) is studied for the case of euclidean planes.