We define the geometric definition of the B-spline curve. This is clearly an extension of the geometric definition for the Bézier curve.
A Bézier Curve is a B-Spline Curve:
What would be the knots for a Bézier curve?.
The Analytic Definition of the B-spline Curve:
We can specify the blending functions in a recursive way, similar to the Bernstein polynomials for Bézier curves. Unfortunately, since B-splines are piecewise Bézier curves, these functions are somewhat messy.
Thinking in Pyramids
It is useful to use a pyramid structure to visualize the B-spline algorithms.
The DeBoor-Cox Calculation
We use the DeBoor-Cox calculation to show that the geometric and analytic definitions of B-spline curves are equivalent. This is an interesting computation, as the technique usually doesn't work in general. This time it does.
Properties of B-Spline Curves:
Lots of properties for B-spline curves. They are similar to those for Bézier curves, but more complex because we are piecing things together.