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 Tytuł: Cross-Ratio in Real Vector Space Autorzy: Coghetto, Roland Słowa kluczowe: affine ratiocross-ratioreal vector spacegeometry Data wydania: 2019 Data dodania: 21-maj-2019 Wydawca: DeGruyter Open Źródło: Formalized Mathematics, Volume 27, Issue 1, Pages 47-60 Abstrakt: Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]).It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9].In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space.Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]:The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by:(x1,x2;x3,x4)=x3-x1x3-x2.x4-x2x4-x1In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5.The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8]. Afiliacja: Rue de la Brasserie 5, 7100 La Louvière, Belgium URI: http://hdl.handle.net/11320/7839 DOI: 10.2478/forma-2019-0005 ISSN: 1426-2630 e-ISSN: 1898-9934 metadata.dc.identifier.orcid: 0000-0002-4901-0766 Typ Dokumentu: Article Występuje w kolekcji(ach): Formalized Mathematics, 2019, Volume 27, Issue 1

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