A parametric curve γ(*t*) in **R**^{n} is one which is defined by an *n*-dimensional function of one real parameter *t*.

## Definition Edit

Let *n* be a natural number, *r* a natural number or ∞, *I* be a non-empty interval of real numbers and *t* in *I*. A vector valued function

- $ \mathbf{\gamma}:I \to {\mathbb R}^n $

of class *C*^{r} (i.e. γ is *r* times continuously differentiable is called a **parametric curve of class C ^{r}** or a

*C*

^{r}parametrization of the curve γ.

*t*is called the parameter of the curve γ. γ(

*I*) is called the

**image**of the curve.

It is important to distinguish between a curve γ and the image of a curve γ(*I*) because a given image can be described by several different *C*^{r} curves.

One may think of the parameter *t* as representing time and the curve γ(*t*) as the trajectory of a moving particle in space.

If *I* is a closed interval [*a*, *b*] we call γ(*a*) the **starting point** and γ(*b*) the **endpoint** of the curve γ.

If γ(*a*) = γ(*b*) we say γ is **closed** or a **loop**. Furthermore we call γ a **closed C ^{r}-curve** if γ

^{(k)}(a) = γ

^{(k)}(

*b*) for all

*k*≤

*r*.

If γ:(*a*,*b*) → **R**^{n} is injective, we call the curve **simple**.

If γ is a parametric curve which can be locally described as a power series, we call the curve **analytic** or of class $ C^\omega $.

We write -γ to say the curve is traversed in opposite direction.

## Example Edit

Some parametrizatios of a circle

- $ \gamma_1(t) = \langle r \cos(t), r \sin(t)\rangle $
- $ \gamma_2(t) = \langle r \frac{1-t^2}{1+t^2}, r \frac{2 t}{1+t^2}\rangle $