Say I have a B-spline function (or curve) of order $k\_1$, defined over some knot vector
$\mathbf{t} = \{ t\_i\}\_1^{n\_1}$, i.e. $$f(x) = \sum\_i a^i B\_{i,k\_1}(x).$$


Do you know of a process of finding another B-spline function, say $g(u) = \sum\_j b^j B\_{j,k\_2}(u)$, of order $k\_2$ defined on some other knot vector $\mathbf{\xi} = \{ \xi\_i\}\_1^{n\_2}$ that results from some arbitrary reparameterization $x \equiv x(u)$ of $f(x)$? That is: $$g(u) = f(x(u)) \longrightarrow \sum\_j b^j B\_{j,k\_2}(u) = \sum\_i a^i B\_{i,k\_1}(x(u))$$


I know there are well-defined methods for finding the B-spline representation of $g(u)$ (i.e. exact evaluation of coefficients $b^i$ and B-spline basis $B\_{i,k\_2}(u)$) if the reparametrrization $x(u)$ is a polynomial, or another B-spline function, e.g. $x(u) = \sum\_i c^i B\_{i,k\_3}(u)$. I am interested in a general method that can handle an arbitrary reparameterization, e.g. something like
$$ x(u) = \sqrt{u^2 + 1}.$$ I am interested for an analytic method/evaluation of $b^j$ and $\mathbf{\xi} = \{ \xi\_i\}\_1^{n\_2}$ from knowledge of $\mathbf{t} = \{ t\_i\}\_1^{n\_1}$ and $a^i$, not for an approximate numerical solution. Thank you very much for your time.



General reparameterization of a B-spline
B-splines are a basis-function representation for piecwise polynomial functions. Therefore, if the reparameterization you seek cannot be represented as a piecewise polynomial it cannot, in general, be represented as a B-spline.


This can be shown with the example you gave. For the reparameterization $x(u) = \sqrt{u^2+1}$, even a simple spline function such as $f(x) = x$ cannot be represented by a polynomial.


The reparameterization will give us:
$f(u) = f(x(u)) = x(u) = \sqrt{u^2+1}$


Since, the square root function cannot, in general, be represented as a polynomial, we will not be able to represent this function as a B-spline function no matter what the parameterization will be.


As noted in the question, if the reparameterization *is* polynomial the composition of the function with the reparameterization will be polynomial and *can* be represented as a B-spline. If the reparameterization is a spline, then the composition can be divided (e.g., using [knot insertion](https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/single-insertion.html)) into intervals in which the function is polynomial, and therefore can also be represented as a B-spline. 


So, for special cases where the reparameterization can be represented as a (piecewise) polynomial function, the reparameterization is possible, but for the general case it is not. 


