Suitable curvature functions determine a regular Frenet curve up to a Euclidean rigid motion. In the plane one signed curvature suffices; in three dimensions positive curvature and torsion determine a unit-speed curve, as in the fundamental theorem of regular space curves. The result is local on the parameter interval and assumes the regularity needed by the frame equations.
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The Fundamental Theorem of Curves is a concept in differential geometry that establishes a relationship between curves in Euclidean space and the properties of their curvature and torsion. While the specific formulation can vary depending on the context, generally, the theorem addresses the representation of a curve based on its intrinsic geometric properties.