We discussed in class how B-Splines can trade-off continuity and locality based on the degree. I've been thinking about just "how much" continuity and locality you get at each degree, for example as a cubic. It seems straightforward to determine continuity (degree 2 is a continuous function, degree 3 has a continuous derivative, degree 4 has a continuous second derivative, etc.), but I don't have an intuition for "how local" a B-spline of a certain degree is. Perhaps moving a particular vertex can only affect the spline up to d vertices away, where d is the degree?

I am confused about how the length of the interval is decided by degree?

Here's a link to an interactive app to play around with B-splines. Regarding the question on locality of a B-spline, your intuition is right that locality decreases when increasing the degree. And using the definition above, the bases B_{i,d} is non-zero over the interval t_i to t_{i+d}.

What does ai stand for? Is it the position of knots or something we need to compute?