AG(2,9) and AG(2,3)
These 81 points are the affine field plane of order 9. This plane is an extension of the affine field plane of order 3, which are the nine points shown as blue diamonds.
The coordinates of these 81 points are pairs of numbers from the field of order 9. That is, the coordinates are numbers of the form a+bi, where a and b are chosen from -1, 0, 1 and addition and multiplication is modulo 3.
You can enter coefficients for a line equation using these numbers and the applet will mark all 9 points on the line with a pink circle.