### Parametric curves: Space curves

### Parametric curves in space

The lower-left spiral consists of all the points \(P(x,y,z)\) specified by the **coordinate functions** \[\left\{\;\begin{aligned} x &= \cos(8\pi t)\\ y &= \sin(8\pi t)\\ z &= t\end{aligned}\right.\] We speak of coordinate functions because the coordinates \(x\), \(y\) and \(z\) are functions of the variable \(t\). As \(t\) increases from \(-1\) to \(1\), the point \(P\) spirals counter clockwise in the upward direction, starting in \(P_{-1}=(1,0,-1)\) and terminating in \(P_1=(1,0,1)\).

The lower-right diagram is interactive; drag the right-mouse button to look at it from a different perspective or give it with the right-mouse button a push to animate the objects.

In general, you can describe a **space curve** mathematically by three functions \[\left\{\;\begin{aligned} x &= f(t)\\ y &= g(t)\\ z&= h(t)\end{aligned}\right.\] We speak of a **parameterisation** or **parameter equations** of the curve with **parameter** \(t\). Almost always one uses neat functions \(f, g\) and \(h\) as coordinate functions. The parameter \(t\) represents time in many applications.

The parameterisation of a curve in the space is not unique: thus the spiral in the above example, may also be parametrized by \[\left\{\;\begin{aligned}x &= \cos(2t^3)\\ y &= \sin(2t^3)\\ z &= t^3\end{aligned}\right.\] with the \(t\)-domain again equal to the interval \(-1\le t\le 1\). The only thing that has changed is the way the point \(P\) moves along the curve as \(t\) increases from \(-1\) to \(1\).