Corners, cusps, and parametrizations: variations on a theorem of Epstein

According to a theorem of Epstein, in interpolation by parametric $C^2 $ cubic splines using chord length parametrization, with certain end conditions, corners cannot occur in the resulting curves. In this paper, examples are constructed to show that, under the same conditions, cusps may nevertheless occur, either at a data point or interior to a cubic piece. A simpler and more intuitive proof is also given to a slightly extended version of the theorem itself. A similar proof shows that corners will also not appear, under the same conditions, if the parametrization is given by the square roots of the chord lengths.