I have a cubic Bezier curve defined by 4 control points ($A(x_a,y_a)$,$B(x_b,y_b)$,$C(x_c,y_c)$ and $D(x_d,y_d)$). I am searching for condition, which will say will resulting curve have monotonous slope or not. To clarify what I need see figure below.
So, red curve has 'monotonous' slope, blue - not. Black curve looks like a border between these two situation. I try to play with this curves, and I have the following idea:
The border condition is: $BC \cdot AD =0$,
Is it correct? In case I'am right could you help me to prove it. Or what should it be?
It will be perfect, if solution may be generalised to 3D case.
Update:
Thanks @Hagen von Eitzen. Now I understand myself much better. What I really need is: I have given angles between AB and AD, and between BC and AD. So, I need to know some border conditions, when resulting curve changed qualitatively: start to go "backwards" or inflection point appears, what else may happens.
It's somewhat complicated. Two papers on the subject are:
Stone, M.C. and DeRose, T.D.,
A Geometric Characterization of Parametric Cubic Curves,
ACM Transactions on Graphics, Vol. 8, pp. 147–163, 1989.
Wang, C.Y.,
Shape Classification of the Parametric Cubic Curve and Parametric B-spline Cubic Curve,
Computer Aided Design, Vol. 113, pp. 199–206, 1981.
Some examples of possible shapes are shown here:
The one at the top right has a cusp (a sharp corner), and the one at the bottom right has two inflexion points.