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 $F$ be a field, $F^3 = F × F × F$ the set of all tuples of three elements over $F$ and $x ∈ F^3$ with $x = (X,Y, Z)$. Then there is exactly one line $L_x$ in $F^3$ that intersects both $(0, 0, 0)$ and $x$, given by the set $L_x = {(k · X, k ·Y, k · Z) | k ∈ F}$. A point in the projective plane over $F$ can then be defined as such a line if we exclude the intersection of that line with $(0, 0, 0)$. This leads to the following definition of a point in projective geometry:

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

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

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

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

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

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

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