{
  "id": "math/0206260",
  "version": "v1",
  "published": "2002-06-25T09:16:51.000Z",
  "updated": "2002-06-25T09:16:51.000Z",
  "title": "The Beckman-Quarles theorem for continuous mappings from R^2 to C^2",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "12 pages, 4 figures",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let \\phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 -> C^2 preserves distance d>=0 if for each x,y \\in R^2 \\phi(x,y)=d^2 implies \\phi(f(x),f(y))=d^2. We prove that if x,y \\in R^2 and |x-y|=(2\\sqrt{2}/3)^k \\cdot (\\sqrt{3})^l (k,l are non-negative integers) then there exists a finite set {x,y} \\subseteq S(x,y) \\subseteq R^2 such that each unit-distance preserving mapping from S(x,y) to C^2 preserves the distance between x and y. It implies that each continuous map from R^2 to C^2 preserving unit distance preserves all distances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-06-25T09:16:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M05"
    ],
    "keywords": [
      "beckman-quarles theorem",
      "continuous mappings",
      "preserving unit distance preserves",
      "preserves distance",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......6260T"
    }
  }
}