Here is a representation of some rational points on the projective plane. More precisely, the first cursor provides a value $B$ for the bound on the height

\[H ( [ x : y : z ]) = \mathrm{max} ( | x | , | y | , | z |)\]

where \((x,y,z)\in \mathbf{Z}^3_\mathrm{prim}\) and we randomly and uniformly pick a certain number of points of height at most $B$. You can make this number of points vary with the third cursor.

The second cursor controls a certain value $B_\mathrm{small} \leqslant B$ and the script plots in red the points of height at most $B_\mathrm{small}$.

This script is inspired by E.Peyre’s beautiful illustrations.

You will find here a similar figure for the 3-dimensional projective space.