Can you interpolate a function $f: \mathbb{R} \rightarrow \mathbb{R}^2$ piecewise (by two interpolations)? I am currently trying to improve on-line handwriting recognition. On-line means in this case that I have the information how the symbols are written as a list of $n$ tuples of coordinates $(x(t_i), y(t_i))$ with $i \in 1, ..., n$. I can't really influence the times I get. One idea I have is that symbol recognition might get better, if I get more points / evenly spaced points (spaced by time or probably distance).
So I need an interpolation for $(x(t), y(t))$ with $t \in [t_1, t_n]$.
I know how to calculate cubic splines for functions $\mathbb{R} \rightarrow \mathbb{R}$ and I know that they are smooth and easy to calculate.
One way to interpolate the "handwriting-function" $f:\mathbb{R}\rightarrow \mathbb{R}^2$ is to calculate two cubic splines (for $x(t)$ and $y(t)$).
I have a few questions to this:


*

*Is this a good idea / is there something (that might be) better? (This is the "soft" part of the question)

*Is the function still smooth?

*Do I loose other properties that I'm currently not aware of?

 A: One way to solve your problem is by using parametric cubic splines. As you say, this involves constructing two spline functions $x=x(t)$ and $y=y(t)$. The main problem is that, in order to do this, you have to assign a $t$ value $t=t_i$ to each of your data points $(x_i,y_i)$. How you do this will have a significant affect on the shape of your curve.
There's a bit more information in this answer, including a link that might be useful.
Some specific answers:
(1) Is it a good idea. Yes, I think so. It's not a new idea. It has been used many times before, and its problems are fairly well understood.
(2) Is the curve smooth. In a mathematical sense it is. A parametric cubic spline will have two continuous derivatives, when considered as a mapping from $\mathbb{R}$ to $\mathbb{R}^2$. But from a geometric or aesthetic point of view, it might not be "smooth". Specifically, at places where the derivative vector $(x', y')$ is zero (if any), there might be sharp corners in the curve. This doesn't happen very often in practice, though. Nasty little loops will occur if you choose the $t_i$ values badly. Again, this is a loss of aesthetic smoothness, not a loss of mathematical smoothness (differentiability).
(3) Do you lose anything. Well, in some sense, you lose your direct connection with the input data, because the $t_i$ values mentioned above are entirely fabricated.
