Question

Assign the coordinates to each point on an Affine plane of order 3.

Answer 1

If the number of points in an affine plane is finite, then if one line of the plane contains n points then:

• each line contains n points,
• each point is contained in n + 1 lines,
• there are n² points in all, and
• there is a total of n² + n lines.

The number n is called the order of the affine plane. The n² + n lines of an affine plane of order n fall into n + 1 equivalence classes of n lines apiece under the equivalence relation of parallelism.

Affine plane of order 3 : there are 9-points, 12-lines and 4- Equivalence classes of parallel lines.

Start with a finite field of order 3:

                             

Now you can draw the points as a 3×3 grid using these elements as x and y coordinates. That's the n² points you'd expect from an affine plane of order n. You can also draw lines of constant x coordinate (i.e. vertical lines), and non-vertical lines with equations y=mx+c where m and x are again elements from your field. This is n+n² lines, as expected.

It doesn't really matter how you draw the lines, as long as the points on each line come from the same equation. If the lines you draw intersect in points besides the 9 points in the plane, that simply doesn't count as an intersection. So one possible illustration would be this:

* I hope This helps you *