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# Fundamental Theorem of Space Curves

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.

Arc Length, Curvature, Euclidean Motion, Fundamental Theorem of Plane Curves, Torsion

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## 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.

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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