I calculated that the geodesics in a cone satisfy the following formula:
$$r=\frac{1}{Acos(\omega \phi + \alpha)}$$
The cone is parametrised by taking spherical coordinates and fixing $\theta=\theta_0$ such that $\omega=sin(\theta_0)$, and both $A$ and $\alpha$ are parameters of the geodesics equation that will have to be determined by boundary conditions.
Now, I've seen on WolframAlpha that the geodesics in the cone have the following shape:
I see how $r$ goes to infinity when the cosine is 0, but I can't see why does it has this sort of "8" shape, and how is it that we can get a negative $r$ value from the formula?
It also can be seen in the picture that, for a certain value of $phi$, two $r$ values can be given, and I don't understand that either.
I've also seen that these geodesics are equivalent to a straight line when we slice the cone and "unroll" it into a plane, but I don't see how can I prove that from my geodesics formula.