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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Beckman–Quarles theorem ： ウィキペディア英語版 Beckman–Quarles theorem In geometry, the Beckman–Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all distances. Equivalently, every automorphism of the unit distance graph of the plane must be an isometry of the plane. Beckman and Quarles published this result in 1953;〔.〕 it was later rediscovered by other authors.〔.〕〔.〕 ==Formal statement== Formally, the result is as follows. Let be a function or multivalued function from a -dimensional Euclidean space to itself, and suppose that, for every pair of points and that are at unit distance from each other, every pair of images and are also at unit distance from each other. Then must be an isometry: it is a one-to-one function that preserves distances between all pairs of points.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Beckman–Quarles theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.