@ARTICLE{ANPROJ10.ABS,
    TITLE = {{C}ross-ratio in Real Vector Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\7100 La Louvi\`ere, Belgium},
    DAY = {27},
    MONTH = {February},
    YEAR = {2019},
    SUMMARY = {Using Mizar\cite{Mizar-State-2015},
     in the context of a real vector space, we introduce the concept
     of affine ratio of three aligned points (see \cite{bsl0}).\par
   It's also equivalent to the notion of ``Mesure algébrique"
   \url{https://fr.wikipedia.org/wiki/Mesure_alg\'{e}brique},
   to the opposite of the notion of Teilverh\"{a}ltnis
   \url{https://de.wikipedia.org/wiki/Teilverh\"{a}ltnis}
   or to the opposite of the ordered length ratio
   \cite{richter2011perspectives}.\par
   In the second part, we introduce the classic notion of
   ``cross-ratio" of 4 points aligned in a real vector space.\par
   Finally, we show that if the real vector space is the real line,
   the notion corresponds to the classical notion
   (\url{https://en.wikipedia.org/wiki/Cross-ratio},
  \cite{richter2011perspectives}): ``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
  $$(x_1,x_2;x_3,x_4) = \frac{x_3 - x_1}{x_3 - x_2}.
  \frac{x_4 - x_2}{x_4 - x_1}$$\par
  In the MML, the vector spaces were first defined by
  Kusak, Leo\'{n}czuk and Muzalewski in the article \cite{VECTSP_1.ABS},
  while the actual real vector space was defined by Trybulec
  \cite{RLVECT_1.abs} and the complex vector space was defined by
  Endou \cite{CLVECT_1.abs}.
  Nakasho and Shidama have developed a solution to explore the notions
  introduced by different authors
  \url{http://webmizar.cs.shinshu-u.ac.jp/mmlfe/current}
  \cite{nakasho2015documentation}.
  The definitions can be directly linked in the HTMLized version
  of the Mizar library.
  (Example: RealLinearSpace:
  \url{http://mizar.org/version/current/html/rlvect_1.html#NM2}) \par
  The study of the cross-ratio will continue within the framework
  of the Beltrami-Klein model \cite{BKMODEL1.abs,BKMODEL2.abs}.
  For a generalized cross-ratio, see Papadopoulos
  \cite{papadopoulos2012remark}.},
  MSC2010 = {15A03,51A05,03B35},
  KEYWORDS = {affine-ratio, cross-ratio, real vector space, geometry}}

 @Article{bsl0,
	title = {About a Certain Generalization of the Affine Ratio of Three Points and Unharmonic Ratio of Four Points},
	author = {Jadwiga Knop},
	journal = {Bulletin of the Section of Logic},
	volume = 32,
	number = 1-2,
	pages = {33--42},
	year = 2003
}

@article{papadopoulos2012remark,
  title={A Remark on the projective geometry of constant curvature spaces},
  author={Papadopoulos, Athanase and Yamada, Sumio},
  journal={arXiv preprint arXiv:1209.3579},
  year={2012}
}

@inproceedings{nakasho2015documentation,
  title={Documentation generator focusing on symbols for the HTML-ized Mizar library},
  author={Nakasho, Kazuhisa and Shidama, Yasunari},
  booktitle={Conferences on Intelligent Computer Mathematics},
  pages={343--347},
  year={2015},
  organization={Springer}
}
