The line of AG(2,9) joining two points
The 81 black and blue points here are the points of the finite affine plane AG(2,9). The blue diamond-shaped points are the points of AG(2,3) embedded inside AG(2,9). Move the red points marked A and B to points in the plane, and the points on the line joining A and B will be marked with pink circles. Some questions you might investigate: What happens if A and B are both on the blue AG(2,3) points? What happens if they are points within the same cluster of 9 points? What happens if they are matching points in two different clusters?