Cross-Ratio in Real Vector Space

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 algebrique" 1, to the opposite of the notion of Teilverhaltnis(2) 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 notion(3) [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x(1), x(2), x(3), x(4) is given by: (x(1), x(2); x(3), x(4)) = x(3)-x(1)/x(3)-x(2) . x(4)-x(2)/x(4)-x(1) In 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 authors(4) [7]. The definitions can be directly linked in the HTMLized version of the Mizar library(5). 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].