[hwjkpvhd] Generating points in general position

Start with a coarse square grid within a finite shape.  Pick some of those grid points, perhaps randomly, so that all the selected points are in general position, for some given definition of general position.  Switch to a finer grid, perhaps when it is no longer possible to add a coarse grid point to the general position set.  Keep picking points and refining the grid.

You get an infinite sequence of points in general position, each point having rational coordinates.  In contrast, the traditional way to generate points in general position is to randomly sample infinite precision real numbers, and then their infinite precision coordinates need to be recorded.  Both of these tasks are practically difficult if not impossible.

Can one select points on the grids not-so-randomly in order to decrease the computation effort needed to test whether each new point is in general position with respect to all the previous points?  But we would still like to select a number of points on each grid level not too much less than the density of random, not, say, 1 point per grid level.