Talk:Perspectivity

This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Mid‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Mid This article has been rated as Mid-priority on the project's priority scale. Arts This article is within the scope of WikiProject Arts, a collaborative effort to improve the coverage of Arts on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.ArtsWikipedia:WikiProject ArtsTemplate:WikiProject ArtsWikiProject Arts

The article says

Given two lines ℓ {\displaystyle \ell } and m {\displaystyle m} in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of ℓ {\displaystyle \ell } and the range of m {\displaystyle 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 ⩞ Y . {\displaystyle X\doublebarwedge Y.} In this notation, to show that the center of perspectivity is P, write X   ⩞ P   Y . {\displaystyle X\ {\overset {P}{\doublebarwedge }}\ Y.} Using the language of functions, a central perspectivity with center P is a function f P : [ ℓ ] ↦ [ m ] {\displaystyle f_{P}\colon [\ell ]\mapsto [m]} (where the square brackets indicate the projective range of the line) defined by f P ( X ) = Y  whenever  P ∈ X Y {\displaystyle f_{P}(X)=Y{\text{ whenever }}P\in XY} . This map is an involution, that is, f P ( f P ( X ) ) = X  for all  X ∈ [ ℓ ] {\displaystyle 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Perhaps I am out of date, but it puzzles me that it talks about f P ( f P ( X ) ) {\displaystyle f_{P}(f_{P}(X))} . It defines f P ( X ) = Y {\displaystyle f_{P}(X)=Y} . But how is f P ( Y ) {\displaystyle f_{P}(Y)} defined?Chjoaygame (talk) 06:36, 24 May 2019 (UTC)[reply]

If someone is out of date, this is certainly not you, but the author of the article that uses a terminology that dates from the time (before the 20th century) where a line was not the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 ℓ {\displaystyle \ell } and the range of m {\displaystyle m} " is an old-fashioned wording for "the bijective mapping between the line ℓ {\displaystyle \ell } and the line m {\displaystyle m} ".

About your question, the given definition of f P {\displaystyle f_{P}} is incorrect: it should be "a central perspectivity with center P is a function f P , m : [ ℓ ] ↦ [ m ] {\displaystyle f_{P,m}\colon [\ell ]\mapsto [m]} defined by f P ( X ) = Y {\displaystyle f_{P}(X)=Y} whenever Y ∈ [ m ] {\displaystyle Y\in [m]} and P ∈ X Y {\displaystyle P\in XY} ". With this more accurate definition, one has f P , m ( f P , ℓ ( X ) ) = X {\displaystyle f_{P,m}(f_{P,\ell }(X))=X} for all X ∈ [ ℓ ] . {\displaystyle 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)[reply]

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