When are cone geodesics planar I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section.
One asked whether if you 'unroll the cone' the conic section becomes a straight line on the resulting circular sector.
I can find examples that show this is false in general, but are there instances where it is true?
More specifically, given the cone defined by $\alpha\rho=z$ in cylindrical polars and the points $A=(\rho_0,0,\alpha\rho_0), B=(\rho_0,\phi_0,\alpha\rho_0)$, is the geodesic from $A$ to $B$ ever a planar curve?
 A: Let line $PH$ be a geodesic line on the lateral surface of a cone (I'll suppose $PH$ does not pass through the vertex $O$ of the cone), and $H$ its point nearest to $O$. Let $OH=d$ and $t=\angle POH$. Then: $OP=d/\cos t$.

If $a=OM=ON$ is the slant height of the cone, then (see figures):
$$
\text{arc}\ MN = ta=\theta a\sin\alpha, 
\quad\text{and:}\quad
\theta={t\over\sin\alpha}, 
$$
where $\alpha$ is the semi-aperture of the cone. It follows that we can write the coordinates of point $P$ as:
$$
P(t)=\left({d\over\cos t}\sin\alpha\cos\left({t\over\sin\alpha}\right), 
{d\over\cos t}\sin\alpha\sin\left({t\over\sin\alpha}\right),
{d\over\cos t}\cos\alpha\right).
$$
The above equation defines then the curve in space corresponding to geodesic $PH$; parameter $t$ can take any value in $(-\pi/2, \pi/2)$.
This is a plane curve only if its torsion $\tau(t)$ vanishes for all $t$. But a straightforward calculation gives:
$$
\tau(t)={1\over d}{\cot\alpha \sin t \cos^2 t},
$$
which vanishes only for $t=0$. Hence it cannot be a plane curve.

A: From differential geometry ( may not be suitable for high school students) consider the  definition of geodesic torsion ( $\psi $  is angle between conic arc and slant generator). The curve is planar when torsion vanishes.
$$ \tau_g = (\kappa_1-\kappa_2) \sin \psi \cos \psi $$
where $\kappa's$ are principal curvatures 
Since $$ \kappa_1=0 $$ and for straight generator minor curvature of Euler's relation 
$$\kappa_2 = \frac {\cos \alpha}{r} $$
and since by Clairaut's Law
$$ r_{min}= r\cdot \sin \psi$$
$$ \tau_g= \frac{\cos \alpha}{r_{min} }\, \sin^2\psi \cos \psi $$
can vanishes only for $\psi=0,\,$ i.e, when intersection plane contains the cone apex, all other terms are non-zero.
