Fundamental Theorem of Space Curves

If two single-valued continuous functions $\kappa(s)$ (Curvature) and $\tau(s)$ (Torsion) are given for $s>0$, then there exists Exactly One Space Curve, determined except for orientation and position in space (i.e., up to a Euclidean Motion), where $s$ is the Arc Length, $\kappa$ is the Curvature, and $\tau$ is the Torsion.

See also Arc Length, Curvature, Euclidean Motion, Fundamental Theorem of Plane Curves, Torsion (Differential Geometry)

References

Gray, A. ``The Fundamental Theorem of Space Curves.'' §7.7 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 123 and 142-145, 1993.

Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, p. 29, 1988.