Inflections on curves in two and three dimensions

Abstract It is shown that for parametric spline curves in two dimensions, such as B-spline curves, the number of inflections in the curve cannot exceed the number of inflections in its control polygon, provided that the control polygon does not turn too sharply. In order to consider shape properties of curves in three dimensions, we then define the inflection count to be the maximum number of inflections that the curve can appear to have when viewed from any direction. The inflection count of a curve is then classified in terms of the direction of its curvature vector. It is shown that the inflection count of a quadratic spline is equal to that of its control polygon and, with this motivation, we classify the inflection count of polygonal arcs.