The Chebyshev T-polynomials have at least two natural definitions, either via the characterizing property $\cos(n\theta) = T_n(\cos(\theta))$, which I will call the geometric definition, or via a recurrence relation $T_{n+1} = 2x\ T_n - T_{n-1}$. My question concerns the relationship between these two definitions, and specifically asks if other families of polynomials defined by similar recurrence relations have a natural geometric interpretation, similarly to how $T_n$ expresses the relation between x-coordinates of particular points on the circle.

Starting from the geometric definition of $T_n$, it is straightforward to derive the recurrent definition. Is there a natural way of going the other direction? One thought I have had is as follows. If we treat the $T_n$ as elements of the coordinate ring of the circle, then $T_n$ expresses the relationship between the two parameterizations of the x-coordinates given by $\theta \mapsto \cos(\theta)$ and $\theta \mapsto \cos(n\theta)$. Can we do the same sort of thing with other families of polynomials defined by similar recurrence relations (say, for simplicity, second-order linear polynomial recurrences with coefficients of degree $\leq 1$)?

One potential obstacle I have encountered is that $\cosh(n\theta) = T_n(\cosh(\theta))$ as well, so that the hyperbola $X^2 - Y^2 = 1$ is just as natural a choice as the circle for a geometric object associated to $T_n$.

Here's my main question: given a particular family of polynomials $P_n$ related by a polynomial recurrence of an appropriately "nice" type, can one associate one or more algebraic curves (or other geometric objects) so that $P_n$ expresses some relationship between various points on the curve?