InfoIn geometry, an affine plane is a two-dimensional affine space.more...Typical examples of affine planes are All the affine planes defined over a field are isomorphic.

The affine subspaces are model surfaces -- they are the simplest surfaces in R 3, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.Every smooth surface S has a unique affine plane tangent to it at each point.The family of all such planes in R 3, one attached to each point of S, is called the congruence of tangent planes.…

Over the complex projective plane there are only two types of degenerate conics - two different lines, which necessarily intersect in one point, or one double line.Over the real affine plane the situation is more complicated.Reducible conics - those whose equation factors - consist of two lines in the plane.…

In the more general situation, where the affine planes are not defined over a field, they will in general not be isomorphic.Two affine planes arising from the same non-Desarguesian projective plane by the removal of different lines may not be isomorphic.…

If we call both circles and such completed lines cycles, we get an incidence structure in which every three points determine exactly one cycle.In an affine plane the parallel relation between lines is essential.In the geometry of cycles, this relation is generalized to the touching relation.…

The classical example is based on the geometry of lines and circles in the real affine plane.

If the number of points in an affine plane is finite, then if one line of the plane contains n points then: The number n is called the order of the affine plane.All known finite affine planes have orders that are prime or prime power integers.The smallest affine plane (of order 2) is obtained by removing a line and the three points on that line from the Fano plane.…

An affine plane of order q can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane.Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.…

A line from a vertex to the opposite vertex would then be a line parallel to one of the five tangent lines.Brianchon's theorem stated only for the affine plane would be uninformative about such a situation.The projective dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane.…

The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept.In the affine plane, a line extends in two opposite directions.In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity.…

In this situation, although the planes being rolled over the surface are tangent planes in a naive sense, the notion of a tangent space is really an infinitesimal notion, whereas the planes, as affine subspaces of R 3, are infinite in extent.However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point.The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point.…

There are two main kinds of finite plane geometry: affine and projective.In an affine plane, the normal sense of parallel lines applies.In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist.…

The Euclidean plane and the Moulton plane are examples of infinite affine planes.A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed.The order of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes).…

For a given a\, the closer e\ is to 1, the smaller is the semi-minor axis.Conics may be defined over other fields, and may also be classified in the projective plane rather than in the affine plane.…

The simplest affine plane contains only four points; it is called the affine plane of order 2.Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines.…

Brianchon's theorem is true in both the affine plane and the real projective plane.However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane.Consider, for example, five tangent lines to a parabola.…

There are two ways to formally define affine planes, which are equivalent for affine planes over a field.The first one consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively.In incidence geometry, an affine plane is defined as an abstract system of points and lines satisfying a system of axioms.…

The polar reciprocal and projective dual of this theorem give Pascal's theorem.Brianchon's theorem is true in both the affine plane and the real projective plane.However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane.…

In order to provide a context for such geometry as well as those where Desargues theorem is valid, the concept of a ternary ring has been developed.Rudimentary affine planes are constructed from ordered pairs taken from a ternary ring.A plane is said to have the "minor affine Desargues property" when two triangles in parallel perspective, having two parallel sides, must also have the third sides parallel.…

An affine plane of order n exists if and only if a projective plane of order n exists (the definitions of order in these cases is not the same).Thus, there is no affine plane of order 6 or order 10.The Bruck-Ryser-Chowla theorem provides further limitations on the order of a projective plane, and thus, the order of an affine plane.…

For example, the three-dimensional Euclidean space is not a countable union of its affine planes.The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints.…