Addition of two B-spline curves

Question

Suppose I have two B-splines, both with the same degree, $p$, and uniformly distributed knots, but with different numbers of knots and control points.

Is it possible to sum the two splines to obtain a new B-spline?

Answer 1

As suggested in the comments, you can just add knots until the two curves have the same knot sequence. They will then have the same number of control points, and "adding" the curves is just adding corresponding control points.

The final knot sequence of each curve will be the union of the two original knot sequences. You say the original knot sequences were uniform. So, if we rescale both of them to the interval $[0,1]$, they will have the form $$ \left\{\frac{i}{m} : 0 \le i \le m \right\} \quad \text{and} \quad \left\{\frac{j}{n} : 0 \le j \le n \right\} $$ These sets are not likely to have many members in common, so, when you form the union, you will probably end up with a knot sequence with roughly $m+n$ entries.

If you know that you're going to be doing this sort of thing a lot, and you're worried about "knot explosion", one idea is to choose the original $m$ and $n$ to be powers of $2$.