Pyramid Algorithms

7.8 Uniform B-Splines

The simplest, most common knot vectors have evenly spaced knots—for example, knot vectors where the knots are located at the integers {…, −2,−1,0,1,2,…}. When the knots are simple knots (knots of multiplicity one) located at the integers, the knot sequence is called a uniform knot sequence and the associated B-splines are called uniform B-splines. These uniform B-splines have some especially nice properties, which we are now going to investigate.

For uniform B-splines it follows easily from the de Boor algorithm that all the B-splines of the same degree are translates of a single B-spline so that

(7.20)

(see Exercise 1