Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials (Bezier curves), if it makes a difference.
It is well known that if two polynomials of degree $n$ are equal at $n+1$ points, then they are equal everywhere.
Is it possible to use that fact to derive a similar statement to the following effect: if $P, Q$ differ by no more than $\delta$ (by "differ" I mean $L_2$ norm of the difference, but $L_1$ or $L_\infty$ would also be OK), at some set of $n+1$ time values in $[0, 1]$, then there is no time value for which they differ by more than $f(n, \delta)$?
A further question would be whether it is possible to generalize such a statement to images of the curves. There are many Bezier curves that have different control points, but their images contain the same points. (For example, all the possible curves that form a straight line between endpoints.) I would be interested in a statement similar to: if for a set of $n+1$ time values in $P$, the closest point in the image of $Q$ is no further than $\delta$, then the images of $P$ and $Q$ are nowhere more distant than some $f(n, \delta)$.
Probably not, take $P_1(x)=k x$ and $P_2(x)=-k x$. Since they have the same value at $x=0$ and are continuous, for any $\delta>0$ we can find $2$ (distinct) points such that the difference between the functions is less than $\delta$ (just pick points close enough to zero). The difference between the two functions at $x=1$ is $2k$. Now we can make $k$ arbitrarily large to reach a contradiction.