2003/07/31 by Apoloniusz Tyszka
Mathematics · #math.MG #msc:51M05
published as Abh. Math. Sem. Univ. Hamburg 74 (2004), 77-87 · LaTeX2e, 10 pages
arxiv created 2005/06/24 · arxiv updated 2009/11/30
Let F be a subfield of a commutative field extending R. Let ϕ2: F2 × F2 → F, ϕ2((x1,x2),(y1,y2))=(x1-y1)2+(x2-y2)2. We say that f:R2 → F2 preserves distance d ≥ 0 if for each x,y ∈ R2 |x-y|=d implies ϕ2(f(x),f(y))=d2. We prove that each unit-distance preserving mapping f:R2 → F2 has a form I ∘ (ρ,ρ), where ρ: R → F is a field homomorphism and I: F2 → F2 is an affine mapping with orthogonal linear part.