Affine plane (incidence geometry)
In geometry, an affine plane is a system of points and lines that satisfy the following axioms:[1]
- Any two distinct points lie on a unique line.
- Each line has at least two points.
- Given a point and line there is a unique line which contains the point and is parallel to the line.
- There exist three non-collinear points (points not on a single line).
In an affine plane, two lines are called parallel if they are equal or disjoint.
Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry.
The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and division rings), there are also many non-Desarguesian planes, not derived from coordinates in a division ring, satisfying these axioms. The Moulton plane is an example of one of these.
An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a line at infinity, each of whose points is that point at infinity where an equivalence class of parallel lines meets. If the projective plane is non-Desarguesian, the removal of different lines could result in non-isomorphic affine planes.
Finite affine plane
If the number of points in an affine plane is finite, then if one line of the plane contains n points then:
- all lines contain n points,
- every point is contained in n + 1 lines,
- there are n2 points in all, and
- there are a total of n2 + n lines.
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 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.
Affine spaces
Affine spaces can be defined in an analogous manner to the construction of affine planes from projective planes. It is also possible to provide a system of axioms for the higher-dimensional affine spaces which does not refer to the corresponding projective space.[2]
Notes
- ↑ Hartshorne 2000, pg. 71
- ↑ Lenz 1961, pg. 138, but see also Cameron 1991, chapter 3
References
- Cameron, Peter J. (1991), Projective and Polar Spaces, QMW Maths Notes 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019
- Casse, Rey (2006), Projective Geometry: An Introduction, Oxford: Oxford University Press, ISBN 0-19-929886-6
- Dembowski, Peter (1968), Finite Geometries, Berlin: Springer Verlag
- R. Hartshorne (2000). Geometry: Euclid and Beyond. Springer. ISBN 0387986502.
- D. Hughes and F. Piper (1973). Projective Planes. Springer-Verlag. ISBN 0-387-90044-6.
- Kárteszi, F. (1976), Introduction to Finite Geometries, Amsterdam: North-Holland, ISBN 0-7204-2832-7
- Lenz, H. (1961), Grundlagen der Elementarmathematik, Berlin: Deutscher Verlag d. Wiss.
- Lindner, Charles C. and Christopher A. Rodger (eds.) Design Theory, CRC-Press; 1 edition (October 31, 1997). ISBN 0-8493-3986-3.
- Lüneburg, Heinz (1980), Translation Planes, Berlin: Springer Verlag, ISBN 0-387-09614-0
- Moulton, Forest Ray (1902), "A Simple Non-Desarguesian Plane Geometry", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 3 (2): 192–195, ISSN 0002-9947, JSTOR 1986419
- Stevenson, Frederick W. (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 0-7167-0443-9