# Notation for a transformation of a parametric curve

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?

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.
• Aha, it's all just mappings! Thanks for the insight. What is the difference between $\mapsto$ in the curve, and $\rightarrow$ in the transformation? Aug 17, 2020 at 5:29