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 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-x1$$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$

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].