Methods | Part IB, 2003

Consider the path between two arbitrary points on a cone of interior angle 2α2 \alpha. Show that the arc-length of the path r(θ)r(\theta) is given by

(r2+r2cosec2α)1/2dθ\int\left(r^{2}+r^{\prime 2} \operatorname{cosec}^{2} \alpha\right)^{1 / 2} d \theta

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

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