The Moulton plane. Lines sloping down and to the right are bent where they cross the yaxis.
In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues' 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 R^{2} 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 yaxis.
Contents

Formal definition 1

Application 2

Notes 3

References 4
Formal definition
The Moulton plane is an incidence structure \mathfrak M=\langle P, G,\textrm I\rangle, where P denotes the set of points, G the set of lines and \textrm I the incidence relation "lies on":

P:=\mathbb R^2 \,

G:=(\mathbb R \cup \{\infty\}) \times \mathbb R,
\infty is just a formal symbol for an element \not\in\mathbb R. 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 p = (x, y) ∈ P and g = (m, b) ∈ G we have

p\,\textrm I\,g\iff\begin{cases} x=b&\text{if }m=\infty\\ y=\frac{1}{2}mx+b&\text{if }m\leq 0, x\leq 0\\ y=mx+b&\text{if }m\geq 0 \text{ or } x\geq 0. \end{cases}
Application
The Moulton plane is an affine plane in which Desargues' theorem does not hold.^{[1]} The associated projective plane is consequently nondesarguesian as well. This means that there are projective planes not isomorphic to PG(2,F) for any (skew) field F. Here PG(2,F) is the projective plane P(F^3) determined by a 3dimensional vector space over the (skew) field F.
Notes

^ Beutelspacher & Rosenbaum 1998, p. 77
References


Moulton, Forest Ray (1902), "A Simple NonDesarguesian Plane Geometry",

Richard S. Millman, George D. Parker: Geometry: A Metric Approach with Models. Springer 1991, ISBN 9780387974125, pp. 97104
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