A term in combinatorics. Over any finite field F one can construct a two-dimensional affine geometry. If the field is of size n, the affine plane consists of a set P of points which are ordered pairs of elements of the field (thus there are n^2 points), and a set L of lines which are themselves sets of points. The points on a given line consist of all pairs (x,y) which are solutions of a linear equation ( y=mx+b , or x=k, with m,b and k in F). The lines can be further divided up into "parallel classes" based on the value of m in
their defining equation, (the lines of the form x=k are said to be in the "infinite slope" parallel class). By adding an additional "point at infinity" for each of the parallel classes and an additional "line at infinity" which connects the points at infinity, one can construct a projective plane from an affine plane.

--back to combinatorics--

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