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**The projective plane**

A cool way to think about a projective plane is that it is something that has points and lines which come endowed with an incidence correspondence (i.e., we know what it means for a point to be on a line) subject to the following axioms:

- Given any two distinct points there exists a unique line passing through both of them.
- Given any two distinct lines there exists a unique point which lies on both of them.

It turns out that given a field K we can find a particular projective plane, namely **P**^2(K)
whose points correspond to 1-dimensional sub vector spaces of K^3. We will usually think of a
point in **P**^2(K) as a triple (a : b : c) of elements of K, not all zero. Thus (a : b : c)
defines the same point as (2a : 2b : 2c) provided that 2 isn't zero in K (which actually
sometimes happens, right?). Anyway, a line of **P**^2(K) is typically given by an equation

- L : Ax_0 + Bx_1 + Cx_2 = 0

where A, B, C are elements of K, not all zero. We say that (a : b : c) *lies on* L if and
only if

- Aa + Bb + Cc = 0

This of course means that L depends on (A, B, C) only up to a scalar. Hence we obtain the pleasing feature of this particular projective plane that the set of lines and set of points have the same cardinality.

**Exercise 1:**
Prove that an axiomatic projective plane has the same number of points as lines

**Exercise 2:**
Prove that the example **P**^2(K) as defined above is indeed a projective plane

**Exercise 3:**
If K is a finite field of order q how many points does **P**^2(K) have?

**Exercise 4:**
If K = Z/pZ write a script finding the intersection point of two lines

**Exercise 5:**
If K = Z/pZ write a script finding the line passing through two given points

**Exercise 6:**
Find other ways to think about lines in **P**^2(K) by parametrizing lines.

Continue reading about the projective_line. Back to the start page.