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Perhaps I am out of date, but it puzzles me that it talks about $f_{P}(f_{P}(X))$.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If someone is out of date, this is certainly not you, but the author of the article that uses a terminology that dates from the time (before the 20th century) where a line was not the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)