*Euler's formula* relates the complex exponential function
to the circular trig functions:
\[
\exp(ix) = \cos x + i\sin x.
\]
A (*period-$1$*) *Fourier polynomial* is a function
of the form
\begin{align*}
f(x) &= \sum_{k=1}^{N} a_{k}\exp(2\pi kix) \\
&= a_{1}\exp(2\pi ix) + a_{2}\exp(4\pi ix) + \dots + a_{N}\exp(2N\pi ix).
\end{align*}
In the diagram above, $N = 18$, the coefficients $a_{k}$ are
real numbers between $-0.5$ and $0.5$, in steps of
size $0.005$, and the graph $\bigl(x, f(x)\bigr)$ is plotted (as a
space curve) over $-1 \le x \le 1$.

The sliders set the coefficients. The left and right arrow keys “pan” the view point, the up and down arrow keys “roll” the view point, and the buttons reset the view point to lie on one of the coordinate axes.