A Characterization of Similarity Maps Between Euclidean Spaces Related   to the Beckman--Quarles Theorem
  It is shown that each continuous transformation $h$ from Euclidean $m$-space ($m>1$) into Euclidean $n$-space that preserves the equality of distances (that is, fulfils the implication $|x-y|=|z-w|\Rightarrow|h(x)-h(y)|=|h(z)-h(w)|$) is a similarity map. The case of equal dimensions already follows from the Beckman--Quarles Theorem. 
Jobst Heitzig