Describe 3D graph in plain English I draw the following graph with 3D plot, with the parametric function $(e^
{−t} \sin(t), e^{−t} \cos(t), e^{−t}).$ 
The problem is how do I describe this graph in plain English? It would be nice if someone could help me visualize it without having to plot it with computer (not just with this function, but for any function in general)

 A: So you have the following coordinates as a function of the parameter $t$:
$$
\begin{aligned}
x&=e^{-t} \sin t\\
y&=e^{-t} \cos t\\
z&=e^{-t} 
\end{aligned}
$$
The presence of the functions $\sin$, $\cos$ suggests that the variables $x$ and $y$ are turning with some sort of radius R, and indeed:
$$
R^2=x^2+y^2=e^{-2t}(\sin^2 t+\cos^2 t)= e^{-2t}= z^2
$$
But the function $e^{-t}$ is decreasing and tends to zero, so we have the composition of two movements:


*

*The variable $z$ moves towards the origin.

*At each instant $t$, there is a circular movement, with a radius that is also decreasing towards zero.


How would you describe that movement in plain English?
EDIT: The problem is that visually you cannot observe the trajectory because the convergence to zero is too fast. Try changing the exponentials to $e^{-\frac{t}{10}}$ and plot from $t=0$ to $t=6\pi$:

A: It's not clear how general your "in general" must be. Here's a start on how to think about the picture "in English".
The graph of
$$
(\sin(t),\cos(t),f(t))
$$
lives on the cylinder of radius $1$ above the unit circle in the $x$-$y$ plane. It winds around that cylinder every $2\pi$ time units. Its height at time $t$ is $f(t)$.
If $f(t) \to 0$ as $t \to \infty$ (as in your example) then the height is approaching $0$. The curve spirals around the cylinder, getting lower and lower on the cylinder. 
Now think about
$$
(f(t)\sin(t),f(t)\cos(t),f(t)) .
$$
Here the radius of what was a simple cylinder is shrinking just as the height is decreasing. That deforms the cylinder into a sort of cone: its radius at time $t$ is $f(t)$ which is shrinking to $0$. You can see that sort of cone by rotating the curve 
$$
(f(t),0,f(t))
$$
in the $x$-$z$ plane about the $z$ axis. Then imagine the spiral on that cone.
I think a picture would be worth the not quite 1000 words. @MiguelAtencia 's fine answer shows one.
A: Frankly, I'm not sure if this is what you have in mind, but here's a picture of a 3-D curve that I call Fermat's Beehive (for obvious reasons) for which the $z$-component increases and decreases smoothly (as opposed to a cone).

