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I understand what C1 and G1 continuity are (e.g. this is not Continuity of parametric and geometric continuity).
But let's say I have a list of cubic splines that are, together, G1 continuous. Is there a canonical way of producing a re-parametrization that has C1 continuity? This may be obvious and I have failed to consider it properly.
I am not aware of a canonical way to achieve this (nor even of any article on the topic).
What you can try is to enforce the same "travel speed" $\dfrac{ds}{dt}$ where two cubic arcs meet. You know the values of $t,s$ at these points (though $s$ must be obtained by numerical integration), and you want to specify the values of the speeds, i.e. the derivatives. Considering the two endpoints of a cubic arc, that gives you four constraints that can be satisfied by Hermite interpolation in the $t$ domain.
Caution: the figure does not show the spline; it is a graph of the reparameterization function.
Assume your spline is given as a sequence of cubic polynomial segments defined through their endpoints $p_{i}^0, p_{i}^1$ and derivatives at those $d_{i}^0,d_{i}^1$ for $1\leq i \leq n$. That is, you can parametrize each piece as the following cubic polynomial curve for $t\in [0,1]$: $$u_i(t) = (2t^3 - 3t^2 +1) p_{i}^0+ (t^3 - 2t^2+t) d_{i}^0 + (-2t^3 + 3t^2) p_{i}^1 + (t^3 - t^2) d_{i}^1,$$ and you can parametrize the whole spline as: $$u(t) = u_i(t-(i-1)), \quad t \in [i-1, i].$$ $G_1$ means that the endpoints between two subsequent segments coincide $p_{i}^1=p_{i+1}^0$ (i.e. the spline is $C_0$), and that the derivatives at the inner nodes are parallel $\exists \kappa_i\ne 0, d_{i}^1 = \kappa_i d_{i+1}^0$. Finally, the points $p_{1}^0$, $p_{n}^1$, and the derivatives $d_{1}^0$, $d_{n}^1$, can be set as unrelated unless you have a looping curve. To turn this into a $C_1$ curve you need to make the derivatives equal. There are however infinitely many ways to do so, and none of those are considered canonical as far as I am aware. Let $f$ be the new tangents, some variants I can suggest are: use a linear combination of both with a fixed global parameter $\theta\in[0,1]$: $f_i^{1} = f_{i+1}^0 = \theta d_i^1 + (1-\theta)d_{i+1}^0$, e.g. you could take the average by setting $\theta = 1/2$. Another option would be to compute the derivative lengths that make the spline as close as possible to a $C_2$ spline. One way to formulate something like this is to find the spline $u$ that minimizes the squared gradient magnitude $\int_{0}^{n} \|u'(t)\|^2 \, dt$ under the constraints that $f_{i}^1=f_{i+1}^0 \propto d_i^1$. You can split the integral as a sum of the integrals over all segments, then you can take the derivative of those, take the squared magnitude, and then integrate. Taking into account the constraints this will be a function only of the lengths of the tangents, you can then take the derivatives w.r.t. those and set them equal to zero to get a system for the solution. I believe the system you will get is linear, so this shouldn't be too hard to solve.
Edit: The way I wrote this corresponds to a uniform parametrization, you can make this nonuniform with some modifications.
