Monday, February 7. On Friday we ended in section 3.1.2 about representation of polynomials in terms of B-splines. The first half of today's lecture was based on Marsden's identity. We proved that B-splines are linearly independent on one knot interval (lemma 3.7) and used this to show that B-splines are also linearly independent on the total knot interval [t_{p+1}.t_{n+1}]. We then continued with sections 3.2.1, 3.2.2, and 3.2.3. Most of the time was spent on proving the differentiation formula (3.36) for B-splines. By exploiting the matrix notation, this is quite straightforward. The matrix notation also makes it clear that derivatives can be computed by algorithms completely analogous to standard evaluation algorithms in chapter 2.
Monday, January 31. We repeated algorithm 1.3 for computing the value of a point on a spline curve, and then showed how the construction could be extended to several polynomial segments that seem to join together smoothly.We then showed how the construction can be 'unwrapped' so that the resulting curve can be expressed as a linear combination of certain functions which we call B-splines. In the rest of the lecture we discussed and proved various elementary properties of B-splines as described in section 2.1.