MA 323 Geometric Modelling Course Notes: Day 08 Cubic Curves and Cubic Hermite Interpolation

We can generalize some of the constructions in the previous section to higher order curves. One problem with using parabolas as the basic curve element for modelling is that a parabola is a convex curve, convexity being defined as the curve lying on one one side of its tangent lines. This means that in order to model a curve that is not convex like S-curve one must use piecewise parabolic curves. By increasing the order of the polynomial curves that are used one can create a simple model of non-convex curve with one curve. The simplest polynomial curve that is not generically convex is a cubic curve. In fact, one of the reasons that cubic curves are heavily used in geometric modelling is because cubic curves are the lowest degree polynomial curves that are not generically convex.