  Given a set $S$ of elements in a number field $k$, we discuss the existence
of planar algebraic curves over $k$ which possess rational points whose
$x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is
either $4,5$, or $6$, we exhibit infinite families of (twisted) Edwards curves
and (general) Huff curves for which the elements of $S$ are realized as the
$x$-coordinates of rational points on these curves. This generalizes earlier
work on progressions of certain types on some algebraic curves.
