On Systems of Functions Satisfying Hodograph Properties

We say that a system of functions is hodograph diminishing if the tangent directions of the curve are contained in the cone generated by the sides of the control polygon. We characterize such systems and conclude that Bézier curves, B-spline curves and, more generally, curves generated by totally positive bases are hodograph diminishing. A characterization is also found for rational curves. Hodograph diminution can be used to identify when two curves have at most one intersection. We also consider conditions for bounding the number of intersections by two. §