Notation for a transformation of a parametric curve

Question

Although I'm conceptually familiar in the area of parametric curves, it's quite new to me notation-wise. I ask this simple question here because searching the web for "notation for curve transformation" yields only pre-calculus level graph transformations.

Suppose I have some parametric equation $\gamma:\mathbb{R}\rightarrow\mathbb{R}^2$, and I'd like to "transform" it. I.e; scaling, rotation, translation, or a more complicated maneuver.

I imagine, like all things in math, a given transformation can be represented by a symbol such as $T$; and that $T$ may take inputs that affect the resulting transformation. These inputs could simply be numbers or more abstract objects like other curves.

What would the notation be to represent a curve $\gamma$ transformed by $T$?

Would it be as simple as

$$T(\gamma,x_1,x_2,\dots)$$

where $x_1,x_2,\dots$ are other inputs that affect the translation?

Answer 1

Think of the curve and its transformation as mappings from one space to another. In your case, the curve is a mapping of the real line $\mathbb R$ to some subset of the plane $\mathbb R^2$. Every real number corresponding to a value of the parameter in the domain is mapped to an ordered pair $(x,y) \in \mathbb R^2$. We can express such a parametrization componentwise; e.g., $$\gamma : \mathbb R \mapsto \mathbb R^2, \\ \gamma(t) = (x(t), y(t)).$$

A transformation $T$ of the plane to itself can be expressed as some function $$T : \mathbb R^2 \to \mathbb R^2 \\ T(x,y) = (u(x,y), v(x,y)),$$ again as some componentwise operation. Then the transformation of $\gamma$ under $T$ is simply the composition of mappings and is a mapping from $\mathbb R$ to $\mathbb R^2$: $$T(\gamma(t)) = (u(x(t), y(t)), v(x(t), y(t))).$$

In a sense, the mappings $\gamma$ and $T$ differ only in the dimensions of the spaces involved. We can more generally talk about arbitrary mappings from $\mathbb R^m \to \mathbb R^n$, and compositions of such mappings.