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 be a field, the set of all tuples of three elements over and with . Then there is exactly one line in that intersects both and , given by the set . A point in the projective plane over can then be defined as such a line if we exclude the intersection of that line with . This leads to the following definition of a point in projective geometry:
Points in projective geometry are therefore lines in where the intersection with is excluded. Given a field , the projective plane of that field is then defined as the set of all points excluding the point :
It follows from this that points are not simply described by fixed coordinates , but by sets of coordinates, where two different coordinates and describe the same point if and only if there is some non-zero field element such that . Points are called projective coordinates.
Projective coordinates of the form are descriptions of so-called affine points. Projective coordinates of the form are descriptions of so-called points at infinity. In particular, the projective coordinate describes the so-called line at infinity.
A projective plane over a finite field contains number of elements.
Visual explanation of projective coordinates: https://www.youtube.com/watch?v=XXzhqStLG-4
Last updated