Varying "speed" in parametric equation

Question

Is it possible to vary the "speed" at which a function is drawn in a parametric equation? It is to my understanding that a parametric function produces the same amount of points per time. So, is it possible to accelerate/decelerate the rate at which the functions is "drawn"?

Answer 1

Yes, functions may be parameterized many different ways, for example: let$ \lambda \in \mathbb{R}, \overline{f}(\lambda) = \lambda\begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} + \begin{pmatrix} 3 \\ 4 \end{pmatrix} $ and $\overline{g}(\lambda) = \lambda \begin{pmatrix}2 \\ 4 \end{pmatrix} + \begin{pmatrix} 3 \\ 4\end{pmatrix}$, where $-\infty < \lambda < \infty$ are both paramatarizations of the line $h(x) = 2x - 2$. Notice, of course that the point $(4,6)$ is of coursed reached by both $\overline{f}$ and $\overline{g}$, yet with $\overline{f}$ it is reached when $\lambda = 1$, and with $\overline{g}$ it is reached when $\lambda = \frac{1}{2}$

Answer 2

Your parametric functions will be of the form x(t)=... y(t)=... z(t)=...

Let's say you want the function (which represents a particle in this example) to move twice as fast.

You would then use {x(2t),y(2t),z(2t)}.

Let's say you wanted to move half as fast. {x(t/3),y(t/3),z(t/3)}

Just substitute a multiple of t, ct, where c is the time scalar (how many times as fast you want the function to move) for t.