## affine plane

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• Info In geometry, an affine plane is a two-dimensional affine space. more...
• Every smooth surface S has a unique affine plane tangent to it at each point.
• Over the real affine plane the situation is more complicated.
• In the more general situation, where the affine planes are not defined over a field, they will in general not be isomorphic.
• In an affine plane the parallel relation between lines is essential.
• The classical example is based on the geometry of lines and circles in the real affine plane.
• All known finite affine planes have orders that are prime or prime power integers.
• Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.
• Brianchon's theorem stated only for the affine plane would be uninformative about such a situation.
• In the affine plane, a line extends in two opposite directions.
• 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.
• In an affine plane, the normal sense of parallel lines applies.
• A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed.
• 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.
• However, its statement in the affine plane is in a sense less informative and more complicated than that in the projective plane.
• The first one consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively.
• Brianchon's theorem is true in both the affine plane and the real projective plane.
• Rudimentary affine planes are constructed from ordered pairs taken from a ternary ring.
• Thus, there is no affine plane of order 6 or order 10.
• For example, the three-dimensional Euclidean space is not a countable union of its affine planes.