Affine plane

In geometry, an affine plane is a two-dimensional affine space. Typical examples of affine planes are

• Euclidean planes, which are affine planes over the reals, equipped with a metric, the Euclidean distance
• Vector spaces of dimension two, in which the zero vector is not considered as different from the other elements
• For every field or division ring F, the set F² of the pairs of elements of F
• The result of removing any single line (and all the points on this line) from any projective plane

All the affine planes defined over a field are isomorphic. More precisely, the choice of an affine coordinate system (or, in the real case, a Cartesian coordinate system) for an affine plane P over a field F induces an isomorphism of affine planes between P and F².

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.

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.

In the applications of mathematics, there are often situations where an affine plane without the Euclidean metric is used instead of the Euclidean plane. For example, in a graph, which can be drawn on paper, and in which the position of a particle is plotted against time, the Euclidean metric is not adequate for its interpretation, since the distances between its points or the measures of the angles between its lines have, in general, no physical importance.^[1][2]

General References

• Artin, Emil (1987), "II. Affine and Projective Geometry", Geometric Algebra, Interscience Publishers, ISBN 0-470-03432-7 
• Blumenthal, Leonard M. (1980) [1961], "IV. Coordinates in an Affine Plane", A Modern View of Geometry, Dover, ISBN 0-486-63962-2 
• Gruenberg, K.W.; Weir, A.J. (1977), "II. Affine and Projective Geometry", Linear Geometry (2nd ed.), Springer-Verlag, ISBN 0-387-90227-9 
• Snapper, Ernst; Troyer, Robert J. (1989) [1971], Metric Affine Geometry, Dover, ISBN 0-486-66108-3 
• Yale, Paul B. (1968), "Chapter 5 Affine Spaces", Geometry and Symmetry, Holden-Day 

References