Translation plane
Encyclopedia
In mathematics
, a translation plane is a particular kind of projective plane
, as considered as a combinatorial object.
In a projective plane,
represents a point, and 
represents a line. A central collineation
with center
and axis 
is a collineation fixing every point on 
and every line through 
. It is called an "elation" if 
is on 
, otherwise it is called a "homology". The central collineations with centre 
and axis 
form a group.
A projective plane
is called a translation plane if there exists a line 
such that the group of elations with axis 
is transitive on the affine plane Πl (the affine
derivative of Π).
is a set of q2 + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of 
into lines.
Given a spread
of 
, the André/Bruck-Bose construction1 produces a translation plane 
of order q2 as follows: Embed 
as a hyperplane of 
. Define an incidence structure 
with "points," the points of 
not on 
and "lines" the planes of 
meeting 
in a line of 
. Then 
is a translation affine plane of order q2. Let 
be the projective completion of 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a translation plane is a particular kind of projective plane
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 that do not intersect...
, as considered as a combinatorial object.
In a projective plane,
represents a point, and represents a line. A central collineationCollineation
In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. All projective linear transformations induce a collineation...
with center
and axis is a collineation fixing every point on and every line through . It is called an "elation" if is on , otherwise it is called a "homology". The central collineations with centre and axis form a group.A projective plane
is called a translation plane if there exists a line such that the group of elations with axis is transitive on the affine plane Πl (the affineAffine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
derivative of Π).
Relationship to spreads
Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction. A spread of is a set of q2 + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of into lines.Given a spread
of , the André/Bruck-Bose construction1 produces a translation plane of order q2 as follows: Embed as a hyperplane of . Define an incidence structure with "points," the points of not on and "lines" the planes of meeting in a line of . Then is a translation affine plane of order q2. Let be the projective completion of .