Linear space (geometry)
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A linear space is a basic structure in incidence geometry. Linear geometry can be described as having an angle of 180° with thee atoms making the arrangement. The second atom of the three is the vertex.
Linear spaces can be seen as a generalization of 2 − (v,k,1)designs, where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line. The term linear space was coined by Libois in 1964, though many results about linear spaces are much older.
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[edit] Definition
Let L = (P, G, I) be an incidence structure, for which the elements of P are called points and the elements of G are called lines. L is a linear space if the following three axioms hold:
- (L1) two points are incident with exactly one line.
- (L2) every line incident to at least two points.
- (L3) L contains at least two lines.
[edit] Examples
The regular Euclidean plane with its points and lines constitutes a linear space, moreover all affine and projective spaces are linear spaces as well.
The following table lists all nontrivial linear spaces of five points with the usual convention, that the lines being incident with only two points are not drawn. The trivial case is simply a line through five points.
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| 10 lines | 8 lines | 6 lines | 5 lines |
A linear space of n points containg a line being incident with n − 1 points is called a near pencil.
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| near pencil with 10 points |
[edit] See also
[edit] References
- A. Beutelspacher: Einführung in die endliche Geometrie II. Page 159, Bibliographisches Institut 1983, ISBN 3-411-01648-5
- J.H. van Lint, R.M. Wilson: A Course in Combinatorics. Page 188, Cambridge University Press 1992,ISBN 0-521-42260-4
- L.M. Batten, A. Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
