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, Torsion## Explore 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 Curves## Cite this as:

Weisstein, Eric W. "Fundamental Theorem of Space Curves." From *MathWorld*--A Wolfram Web Resource.
https://mathworld.wolfram.com/FundamentalTheoremofSpaceCurves.html