Curve

A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing"

Definitions

Level curve

Given a $$f:\mathbb{R}^n\rightarrow\mathbb{R}$$ and a $c\in\mathbb{R}$ we define the level curve as follows:

$$\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}$$

A more useful notation is $$\mathcal{C}_\alpha=\{x\in\mathbb{R}^n|f(x)=\alpha\}$$

Parameterisation

A parameterisation of a curve in $\mathbb{R}^n$ is a function:

$$\gamma:(a,b)\rightarrow\mathbb{R}^n$$ with $$-\infty\le a< b\le +\infty$$

Linking with Level curves

A parameterisation whos image is all (or a part of) a level curve is called a parameterisation (of part) of the level curve.