Does the blending matrix change between calculating various curve segments in a uniform cubic B-splines approximation?

Question

I would like to ask about uniform (periodic) cubic B-splines (approximation, no interpolation).

$$B=1/6\begin{bmatrix}-1&3&-3&1\\3&-6&3&0\\-3&0&3&0\\1&4&1&0\end{bmatrix}$$

$B$ is the matrix of coefficients that allows calculating a single curve segment with the formula:

$$ Q_i(u)=[u^3 u^2 u^1 1]B \begin{bmatrix}P_{i-3}\\P_{i-2}\\P_{i-1}\\P_{i}\end{bmatrix} $$

When $i=3$ I calculate the first segment of the curve. My question is when $i=4$ and I want to calculate the second segment of the curve I must change the $4$ control points but does $B$ change or not?

Answer 1

No, the $B$ matrix (basis coefficient matrix) does not change from one segment to the next. It is a property of the type of spline you're using, in this case cubic B-splines. If you used Bézier splines or Hermite splines instead, you'd have a different $B$ matrix.

Answer 2

No, $B$ is constant for given type of cubic spline, e.g. B-spline, Bezier, Hermite or Catmull-Rom cubic splines have different $B$ matrix. To make B-spline continuous, you need to copy 3 control points from the previous spline segment $a$ to the control points of spline segment $b$ and add a new point as the last control point of $b$ such that:

$Pb_{i-3}=Pa_{i-2}$, $Pb_{i-2}=Pa_{i-1}$, $Pb_{i-1}=Pa_i$ and $Pb_i$=new control point.

Answer 3

If you use the same $\mathbf{B}$ matrix on each segment, then you are constructing a so-called "uniform" b-spline. This means that the parameter increment over each segment is a constant (typically $=1$). Uniform b-splines often have undesirable shapes if the control points are unevenly distributed. But, if the distances between control points don't vary too wildly, then you can use uniform b-splines and get some nice computational simplifications.