Differentiability conditions and bounds on singular points

A least upper bound for the number of singular points on normal arcs and curves of cyclic order four

“…In the analysis of such ~¢4's it will be necessary to include (2, 2) points ( [5], 2.1) as possible singular points as well as (1,3) and (3,1) points; (2, 2)-singular points are not possible if Condition I is satisfied ( [5], 2.2). The definition of such singular points on A4 is included for the convenience of the reader.…”

“…In [5] it is shown that a much weaker differentiability criterion--namely, Condition I ( [3], 1.5)--is sufficient to obtain four as the least upper bound for such arcs. Assuming no differentiability conditions, the best bound obtained so far is eleven ( [4] and [1], 4…”