Projective Planes

Projective planes are particular geometric constructs defined over a given field. In a sense, projective planes extend the concept of the ordinary Euclidean plane by including points at infinity.

To understand the idea of constructing of projective planes, note that, in an ordinary Eu- clidean plane, two lines either intersect in a single point or are parallel. In the latter case, both lines are either the same, that is, they intersect in all points, or do not intersect at all. A projec- tive plane can then be thought of as an ordinary plane, but equipped with an additional point at infinity such that two different lines always intersect in a single point. Parallel lines intersect at infinity.

Such an inclusion of infinity points makes projective planes particularly useful in the description of elliptic curves, as the description of such a curve in an ordinary plane needs an additional symbol for the point at infinity to give the set of points on the curve the structure of a group. Translating the curve into projective geometry includes this point at infinity more naturally into the set of all points on a projective plane.

To be more precise, let FF be a field, F3=F×F×FF^3 = F × F × F the set of all tuples of three elements over FF and xF3x ∈ F^3 with x=(X,Y,Z)x = (X,Y, Z). Then there is exactly one line LxL_x in F3F^3 that intersects both (0,0,0)(0, 0, 0) and xx, given by the set Lx=(kX,kY,kZ)kFL_x = {(k · X, k ·Y, k · Z) | k ∈ F}. A point in the projective plane over FF can then be defined as such a line if we exclude the intersection of that line with (0,0,0)(0, 0, 0). This leads to the following definition of a point in projective geometry:

[X:Y:Z]={(kX,kY,kZ)kF}[X : Y : Z] = \{(k · X, k ·Y, k · Z) | k ∈ F^∗\}

Points in projective geometry are therefore lines in F3F^3 where the intersection with (0,0,0)(0, 0, 0) is excluded. Given a field FF, the projective plane of that field is then defined as the set of all points excluding the point [0:0:0][0 : 0 : 0]:

FP2={[X:Y:Z](X,Y,Z)F3 with (X,Y,Z)(0,0,0)}FP^2 = \{[X : Y : Z] | (X,Y, Z) ∈ F^3\text{ with }(X,Y, Z) \neq (0, 0, 0)\}

It follows from this that points [X:Y:Z]FP2[X : Y : Z] ∈ FP^2 are not simply described by fixed coordinates (X,Y,Z)(X, Y, Z), but by sets of coordinates, where two different coordinates (X1,Y1,Z1)(X_1,Y_1, Z_1) and (X2,Y2,Z2)(X_2,Y_2, Z_2) describe the same point if and only if there is some non-zero field element kFk ∈ F^∗ such that (X1,Y1,Z1)=(kX2,kY2,kZ2)(X_1,Y_1, Z_1) = (k · X_2, k ·Y_2, k · Z_2). Points [X:Y:Z][X : Y : Z] are called projective coordinates.

Projective coordinates of the form [X:Y:1][X : Y : 1] are descriptions of so-called affine points. Projective coordinates of the form [X:Y:0][X : Y : 0] are descriptions of so-called points at infinity. In particular, the projective coordinate [1:0:0][1 : 0 : 0] describes the so-called line at infinity.

A projective plane over a finite field FpmF_{p^m} contains p2m+pm+1p^{2m} + p^m + 1 number of elements.

Visual explanation of projective coordinates: https://www.youtube.com/watch?v=XXzhqStLG-4

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