This question's not getting anywhere on math.stackexchange.com, so let's see if someone here can say something:
In the $xyz$-space imagine a circle of radius $r>0$ in the $xz$-plane, whose center is on the $x$-axis at a distance $R>r$ from the $z$-axis, and revolve it about the $z$-axis, getting a torus embedded in $\mathbb R^3$.
Its intersection with planes parallel to the $xy$-planes let us call "parallel circles".
A curve winds $m$ times around the long way and $n$ times around the short way (and $m$ and $n$ are not both $0$), and returns to its starting point. It is situated so that it meets all parallel circles at the same angle.
As a function of $m$ and $n$ and $r$ and $R$ and the position on the curve, what are
- the length of the curve;
- the angle at which it meets the parallel circles;
- the curvature;
- the torsion?
When $m=0$ or $n=0$ the answers are obvious.
When $m=n=1$ the answer is surprising: the torsion is everywhere $0$ and the curvature is constant (and equal to $1/R$, so the arc length is $2\pi R$).
Are there other cases where the answer is surprising or elegant or of interest for other reasons?