Consider the path between two arbitrary points on a cone of interior angle $2 \alpha$. Show that the arc-length of the path $r(\theta)$ is given by

$$\int\left(r^{2}+r^{\prime 2} \operatorname{cosec}^{2} \alpha\right)^{1 / 2} d \theta$$

where $r^{\prime}=\frac{d r}{d \theta}$. By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.