Consider a curve $f$ that connects two arbitrary points on a torus. What are the equations that defines the curve $f_{min}$ whose such a distance is minimal?
2026-03-29 01:33:57.1774748037
Curve on a torus
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This depends strongly on the metric you choose in the torus $T$.
Since $T$ is uniformized by the plane $\Bbb R^2$ a canonical metric on $T$ is the flat metric with constant curvature 0. For this the geodesic arcs are image of line segments in the plane.