“Moulton plane”的意思、由来-开放百科全书

词条

Moulton plane

释义

Formal definition
Application
Notes
References

In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues's theorem does not hold. It is named after the American astronomer Forest Ray Moulton. The points of the Moulton plane are simply the points in the real plane R² and the lines are the regular lines as well with the exception that for lines with a negative slope, the slope doubles when they pass the y-axis.

Formal definition

The Moulton plane is an incidence structure , where denotes the set of points, the set of lines and the incidence relation "lies on":

is just a formal symbol for an element . It is used to describe vertical lines, which you may think of as lines with an infinitely large slope.

The incidence relation is defined as follows:

For and we have

Application

The Moulton plane is an affine plane in which Desargues' theorem does not hold.^[1] The associated projective plane is consequently non-desarguesian as well. This means that there are projective planes not isomorphic to for any (skew) field F. Here is the projective plane determined by a 3-dimensional vector space over the (skew) field F.

Notes

1. ^{{harvnb|Beutelspacher|Rosenbaum|1998|page=[https://books.google.com/books?id=I4OqBcaKAJ0C&pg=PA77 77]}}

References

{{citation|last1=Beutelspacher|first1=Albrecht|last2=Rosenbaum|first2=Ute|author1-link=Albrecht Beutelspacher|title=Projective Geometry : From Foundations to Applications|year= 1998|publisher=Cambridge University Press|isbn=978-0-521-48364-3|pages=[https://books.google.com/books?id=I4OqBcaKAJ0C&pg=PA76 76–78]}}
{{Citation | last1=Moulton | first1=Forest Ray | title=A Simple Non-Desarguesian Plane Geometry | jstor=1986419 | publisher=American Mathematical Society | location=Providence, R.I. | year=1902 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=3 | issue=2 | pages=192–195 | doi=10.2307/1986419}}
Richard S. Millman, George D. Parker: Geometry: A Metric Approach with Models. Springer 1991, {{isbn|9780387974125}}, pp. [https://books.google.com/books?id=KpQ49uySA-EC&pg=PA97 97-104]

1 : Incidence geometry

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。