If two single-valued continuous functions 
(curvature) and 
(torsion) are given for 
, then there exists exactly
one space curve, determined except for orientation
and position in space (i.e., up to a Euclidean motion),
where 
is the arc length, 
is the curvature, and 
is the torsion.
Fundamental Theorem of Space Curves
See also
Arc Length, Curvature, Euclidean Motion, Fundamental Theorem of Plane Curves, TorsionExplore with Wolfram|Alpha
References
Gray, A. "The Fundamental Theorem of Space Curves." §7.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 219-222, 1997.Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, p. 29, 1988.Referenced on Wolfram|Alpha
Fundamental Theorem of Space CurvesCite this as:
Weisstein, Eric W. "Fundamental Theorem of Space Curves." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofSpaceCurves.html