Managing the shape of planar splines by their control polygons (Q1801864)

The following shape-preservation theorem is proved for uniform planar quadratic \((C^ 1)\) and cubic \((C^ 2)\) \(B\)-splines and also for beta2- splines: ``Every spline-curve segment has essentially the same shape characterization as the corresponding 4-points control polygon.'' The possible control polygons defined separately on each set of four consecutive control points are of the following types: convex or hyperconvex (concave or hyperconcave), linear, inflected or looped polygons.