Summary
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:
$$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$
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 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].
Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics