as follows. Points of
are the points of V, and the lines of
are defined to be the subsets of the form t+U where t is an
element of V and
All affine translation planes are of this form, up to
isomorphism.
Related pagesMoufang planesSpreadsets Spreads Lenz classification |
is called a translation plane if there exists a line l
such that the group of
elations with axis l
is transitive on the affine plane
.
Each such line l is called a translation axis.
Usually, the translation axis l is unique. In fact, a
projective plane admits
two distinct translation axes if and only if the plane is
a Moufang plane.
The plane P(K,T) over a ternary field (K,T) is a translation plane with respect to the line at infinity if and only if (K,T) is a quasifield.
An affine plane is called an affine translation plane if it's projective closure is a translation plane with respect to the line at infinity.
Given a vector space V over a skewfield K, and a
spread S, then we may
define an affine translation plane
as follows. Points of
are the points of V, and the lines of
are defined to be the subsets of the form t+U where t is an
element of V and
All affine translation planes are of this form, up to
isomorphism.
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