Consider a right semicircular cone with height $h$ and radius $r$ given by $\mathcal{C}=\left\{\left(\frac{h-u}{h}r\cos(\theta),\frac{h-u}{h}r\sin(\theta),u\right)\,:\,u\in[0,h],\,\theta\in[0,\pi]\right\}$.


Letting $A=(r,0,0)$ and $B=(-r,0,0)$, find the path $\gamma$ whose initial and final points are $A$ and $B$ and whose trace is over $\mathcal{C}$ and such that its arclength is minimum.


Note that $\gamma$ is not necesarily differentiable (only rectificable). Also, it is almost clear that $\gamma$ is simple.


How do you would solve this problem? Since $\gamma$ is not necesarily differentiable, i think that Calculus of variations is not enought.



Geodesic in half cone
To elaborate on the comment: Let $A$ be the point at $(+r,0,0)$, $B$ be the point at $(-r,0,0)$, $O$ be the vertex of the cone; and $P$ be the point at $(0,r,0)$. For a right circular cone, $OA = OP = OB = r$ (and we know $AB = 2r$).


In a plane, draw the circular wedge diagram $OAPB$, with angle $AOB =\frac\pi 2$, whihc it must be since the cone is right circular).


Curves on this wedge diagram map to curves on the cone, preserving length, because the surface of the cone has zero intrinsic curvature (Reimann tensor would be zero). So for example the circular arc $APB$ corresponds to travelling along the base of the cone, and that would be a distance of $\frac\pi 2 r$.


Of course, you can do better; in particular, a straight line from $A$ to $B$ has length $\sqrt{2} r$ and this is the minimum possible curve length on the cone from $A$ to $B$.


Parameterize the line by $\theta$ of the original code; then at any given $\theta$ we have a triangle with two sides $r$ and $\rho$ ($\rho$ is the distance from the centerline of the cone to the point on the curve) and their corresponding opposite angles $\frac\pi 4$ and $\frac{3\pi}{4}-\frac{\theta}{2}$. By the law of sines, with only a little trig simplification,


$$
\rho = r \frac{\sin (\pi/4)}{\sin(\pi/4 + \theta/2)}
$$
and we know that $z = h \frac{r-\rho}r$. So we have the equation of the line, in parametric form.


