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Suppose cone $C=\{(x,y,z)| x^2+y^2=z^2, z\geq 0\}$ is given and I want to compute the distance of two points $p,q \in C$. Let $d_C(p,q) := \operatorname{inf}_{\gamma} l(\gamma)$ where $l(\gamma)$ is the length of the curve $\gamma$ in $C$ from $p$ to $q$.

How does one compute this length, explicitly?

Followings are my trial;

Let $p = (p_1, p_2, p_3)$ with $p_1^2+p_2^2=p_3^2$ and $q=(q_1, q_2, q_3)$ with $q_1^2+q_2^2=q_3^2$.

Let $\gamma$ be the curve $\gamma(0)=p, \gamma(1)=q$, Then the length $l(\gamma) = \int_0^1 ||\gamma'(t)|| dt$. Assume $\gamma$ is unit speed curve, then I can parametrize $\gamma(t) = X(u,v) = (u,v,\sqrt{u^2+v^2})$, then $l=\int\sqrt{E (u')^2 + 2 Fu'v'+ G(v')^2} dt $, then....

I think this approach is not good for computing $d_C(p,q)$. I wonder there is a better way to computing this distance.

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1 Answers1

1

Hint

The cone is a developable surface, whose flattening is a sector of a disk. The length of a geodesic on a developable surface is the length of the line segment in its flattening.

Hence just compute the image of the two points $p,q$ in the flattened sector of disk and the distance between those in the plane. This distance is equal to $d_C(p,q)$.

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