Parametric curve

A parametric curve &gamma;(t) in Rn is one which is defined by an n-dimensional function of one real parameter t.

Definition
Let n be a natural number, r a natural number or &infin;, 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 Cr (i.e. &gamma; is r times continuously differentiable is called a parametric curve of class Cr or a Cr parametrization of the curve &gamma;. t is called the parameter of the curve &gamma;. &gamma;(I) is called the image of the curve.

It is important to distinguish between a curve &gamma; and the image of a curve &gamma;(I) because a given image can be described by several different Cr curves.

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

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

If &gamma;(a) = &gamma;(b) we say &gamma; is closed or a loop. Furthermore we call &gamma; a closed Cr-curve if &gamma;(k)(a) = &gamma;(k)(b) for all k &le; r.

If &gamma;:(a,b) &rarr; Rn is injective, we call the curve simple.

If &gamma; 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 -&gamma; to say the curve is traversed in opposite direction.

Example
Some parametrizatios of a circle


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