A Characterization of Similarity Maps Between Euclidean Spaces Related to the Beckman--Quarles Theorem

2002/03/22 by Jobst Heitzig, Heitzig, Jobst
Decision Sciences · Mathematics · #Fuzzy and Soft Set Theory #math.GN #math.MG #msc:15A04 #msc:51F20 #msc:51F25

paper · pdf · doi:10.48550/arxiv.math/0203233

5 pages

arxiv created 2002/03/22 · arxiv updated 2009/11/30

Abstract

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|⇒|h(x)-h(y)|=|h(z)-h(w)|) is a similarity map. The case of equal dimensions already follows from the Beckman--Quarles Theorem.

Citations

• On isometries of Euclidean spaces

Related

• The Beckman-Quarles theorem for continuous mappings from Rⁿ to Cⁿ
• Qualitative Euclidean embedding of Disjoint Sets of Points
• The Beckman-Quarles theorem for continuous mappings from R² to C²
• Beckman-Quarles type theorems for mappings from ℝⁿ to ℂⁿ
• Homotopy similarity of maps. Strong similarity