  In 1925, Jarnik defined a sequence of convex polygons for use in constructing
curves containing many lattice points relative to their curvatures. Properly
scaled, these polygons converge to a certain limiting curve. In this paper we
identify this limiting curve precisely, showing that it consists piecewise of
arcs of parabolas, and we discuss the analogous problem for sequences of
polygons arising from generalizations of Jarnik's construction.
