B-spline curve versus Bezier curve

A B-spline curve is a piecewise polynomial parametric curve defined by control points and B-spline basis functions. It is a generalisation of the Bezier curve and removes its two main weaknesses: global control and the link between the number of points and the degree.

Equation

$$\mathbf P(u)=\sum_{i=0}^{n}\mathbf B_i\,N_{i,k}(u)$$

$N_{i,k}$ are basis functions of order $k$ (degree $k-1$) given by the Cox-de Boor recursion over the knot vector $\{t_0,t_1,\dots\}$.

Terms

• Control points $\mathbf B_i$: vertices of the control polygon; the curve is attracted to them but usually does not pass through them.
• Knot vector: non-decreasing parameter values where the polynomial pieces join. Number of knots $=n+k+1$ for $n+1$ points. Uniform, open-uniform (end knots repeated $k$ times) and non-uniform vectors are used.
• Degree: polynomial degree of each segment (cubic is common); order $=$ degree $+1$.

Difference

Advantages

1. Local control: moving one point changes only $k$ segments.
2. Degree stays low (cubic) for any number of points.
3. High continuity ($C^2$ for cubic) at joints.
4. Convex hull and variation diminishing properties.
5. NURBS (rational B-splines) can represent exact conics and free-form curves in one form.