In
incidence geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidenc ...
, the Moulton plane is an example of an
affine plane
In geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
* Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...
in which
Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
does not hold. It is named after the American astronomer
Forest Ray Moulton
Forest Ray Moulton (April 29, 1872 – December 7, 1952) was an American astronomer.
Biography
He was born in Le Roy, Michigan, and was educated at Albion College. After graduating in 1894 (Bachelor of Arts, A.B.), he performed his graduate s ...
. 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
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, the slope doubles when they pass the ''y''-axis.
Formal definition
The Moulton plane is an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
, 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.
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
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
determined by a 3-dimensional vector space over the (skew) field ''F''.
Notes
References
*
*
*Richard S. Millman, George D. Parker: ''Geometry: A Metric Approach with Models''. Springer 1991, , pp
97-104
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Incidence geometry