In a certain exercise I have been asked to find the area of the triangle of vertices $ABC$, given the metric $ds^2 = \frac{du^2+dv^2}{v^2}$. The coordinates of $A$ are $u=0,v=1$, $B$ are $u=10,v=1$ and $C$ are $u=0,v=1000$. I have also been asked to find the length of the sides of the triangle.
The thing is, however, that I don't have any idea of how to find these values outside planar or spherical geometry.
I first thought about finding the length of the segments that go from $(0,1)$ to $(10,1)$ and so on with the first fundamental form, which enables me to find these lengths and even the area of the region. However, I'm not sure about the success of these approach.
Besides, I have also been asked to say which is the shortest curve from $A$ to $B$ (the geodesic). This seems complicated to me because I thought the sides of a triangle in any geometry were the shortest paths (for example, they are in planar and in spherical geometry), but it seems that I was wrong. I would really appreciate any help or hint.
2026-05-16 19:09:45.1778958585