How to intuitivly think of graphing a function in $\Bbb{R}^3$

Question

How to intuitivly think of graphing a function in $\mathbb R^3$?

Let there be $f: \mathbb R \rightarrow \mathbb R^3$ $$f(t)=\begin{bmatrix} \cos(t) \\ \sin(t) \\ t \\ \end{bmatrix}$$

Graph $f$. How do I even think about it? I thought that it's a cylinder with made of unit circles at different levels of $t$, but that doesn't seem right anymore...

Also, just a question about a term: What's a parameterized curve and what is it different from other curves?

Thanks!

Answer 1

The pair $\begin{bmatrix} \cos t \\ \sin t\end{bmatrix}$ just goes around in a circle. If you think of that circle as being in a horizontal plane, then $t$ is the height above that plane. So the height steadily increases as the point goes around a circle. You've seen that as an architectural element.

A parametrized curve is a curve with each point of which a number, called the parameter, is associated. So as that number changes, the point on the curve moves. In your example, $t$ is the parameter.

Answer 2

It would be a cylinder if we had two variables, so $f:\mathbb{R^2}\rightarrow\mathbb{R^3}$, since then we can let:

$f(t,\theta) = (\cos(\theta),\sin(\theta),t)$ for say $0\leq\theta\leq2\pi$, $0\leq t \leq 1$

But since we only have one variable to work with, as $t$ varies from $0$ to $2\pi$, the $\cos(t)$ and $\sin(t)$ perform a full rotation, making our graph a spiral.