B-Spline Curve with Knots

This Demonstration illustrates the relation between B-spline curves and their knot vectors. Start with the control points and a knot vector , where the degree of the B-spline is . The knot vector satisfies and . The B-spline basis functions are defined as:

,

and a B-spline curve is defined as:

.

For nonperiodic B-splines, the first knots are equal to 0 and the last knots are equal to 1. If duplication happens at the other knots, the curve becomes times differentiable. So, by overlapping the knots, you can generate a curve with sharp turns or even discontinuities.

When the number of control points is , the basis functions are reduced to Bernstein polynomial, thus the curve becomes a Bézier curve.