Background: I am currently writing a media compression algorithm. I have the ground-work ready, and now I am about to define some modes for higher level geometric representations of data.
Formulation : I want to be able to find curves which pass closely through a set of points. Let us assume every curve has a line defined for it. This line defines the end points and a local coordinate system, so that ${\bf p_2} - {\bf p_1} = {\bf e_1}$ is basis vector in a new basis in which we express the curve as a function so that : $$t\cdot {\bf e_1} + f(t)\cdot {\bf e_2}, t\in [0,1]$$ defines the curve with $\bf e_2$ chosen in some well defined manner. For example the "right hand rule" in physics.
It makes sense to make some demands on $t\to f(t)$ : for example continuity at end points: $$f(0)=f(1)=0$$
Each segment starts where the previous ends and ends where the next begins.
This leads me to believe that if $t\to f(t)$ is a polynomial, then it must be of the form $$f(t) = t(t-1)p(t)$$, where $t\to p(t)$ is another polynomial.
( The "simplest" choice of $p(t) = 0$ would correspond to the straight line. )
If this makes sense, then it would be valuable for me, as I would only need to represent and transmit information regarding $p$ and not all of $f$ which would be saving information.
Does it make sense, is there some other theoretically or practically valuable information I could derive? Particular bases of polynomials which could be more interesting to use than others?