Talk:Perspectivity

Question
The article says


 * Given two lines $$\ell$$ and $$m$$ in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of $$\ell$$ and the range of $$m$$ determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P). A special symbol has been used to show that points X and Y are related by a perspectivity; $$X \doublebarwedge Y .$$ In this notation, to show that the center of perspectivity is P, write $$X \ \overset {P}{\doublebarwedge} \ Y.$$ Using the language of functions, a central perspectivity with center P is a function $$f_P \colon [\ell] \mapsto [m]$$ (where the square brackets indicate the projective range of the line) defined by $$f_P (X) = Y \text{ whenever } P \in XY$$. This map is an involution, that is, $$f_P (f_P (X)) = X \text{ for all }X \in [\ell]$$.

Perhaps I am out of date, but it puzzles me that it talks about $$f_P (f_P (X))$$. It defines $$f_P (X) = Y$$. But how is $$f_P (Y)$$ defined?Chjoaygame (talk) 06:36, 24 May 2019 (UTC)
 * If someone is out of date, this is certainly not you, but the author of the article that uses a terminology that dates from the time (before the 20th century) where a line was not the set of its points (at that time, infinite sets were not accepted by mathematicians). Therefore "the bijective mapping between the points of the range of $$\ell$$ and the range of $$m$$" is an old-fashioned wording for "the bijective mapping between the line $$\ell$$ and the line $$m$$".
 * About your question, the given definition of $$f_P$$ is incorrect: it should be "a central perspectivity with center P is a function $$f_{P, m} \colon [\ell] \mapsto [m]$$ defined by $$f_P (X) = Y$$ whenever $$Y \in [m]$$ and $$P \in XY$$". With this more accurate definition, one has $$f_{P, m} (f_{P, \ell} (X)) = X$$ for all $$X \in [\ell].$$ This is not an involution, since the domain and the codomain of an involution must be equal. Therefore, I'll remove the sentence.
 * By the way the article has other fundamental issues:
 * The lead is about perspectivities in the 3D space, and the article body is about perspectivities in the plane.
 * It confuses two different concepts of perspectivities, which are (in the plane case) a mapping (projection) from the plane but a point to a line, and the restriction of this mapping to a line, which is bijective transformation between two lines.
 * The lead is about the first concept, while the body is about the second concept. D.Lazard (talk) 08:59, 24 May 2019 (UTC)


 * Thank you, Editor D.Lazard.Chjoaygame (talk) 09:21, 24 May 2019 (UTC)