There is a well known kind of parametric surface known as sweeping surfaces, where one curve is swept across another, forming a surface in 3d.
i.e. one obtains a formulation of the form:
$f(u,v) = R(u)C(v)$
Where $R$ is a rotation matrix defined by the frenet frame of the first curve and $C$ is the parametrization of the second curve.
Assume you don't have just a curve, but a more topologically complex skeleton, for now, just a y joint where two curves intersect.
Is there a way to produce a smooth parametrization (explicitly, i.e. an actual computation) that also "sweeps" a curve but splits it into 2 at the joint?
So for example in the case of a Y joint in a topological skeleton (a graph).
We wuld get something like this.
Note that the graph does not need to be planar. A topological joint can look like a sea urchin with arbitrarily many edges in 3D connected to the same point in a sphere like fashion.