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 [math]f:\mathbb{R}^n\rightarrow\mathbb{R}[/math] and a [ilmath]c\in\mathbb{R} [/ilmath] we define the level curve as follows:

[math]\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}[/math]

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

Parameterisation

A parameterisation of a curve in [ilmath]\mathbb{R}^n[/ilmath] is a function:

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

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.