Fundamental theorem of curves

In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size) completely determined by its curvature and torsion.^[1][2]

Use

A curve can be described, and thereby defined, by a pair of scalar fields: curvature $\kappa$ and torsion $\tau$, both of which depend on some parameter which parametrizes the curve but which can ideally be the arc length of the curve. From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.

Congruence

If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent to each other.

See also

• Gaussian curvature

References

1. Banchoff, Thomas F.; Lovett, Stephen T. (2010), Differential Geometry of Curves and Surfaces, CRC Press, p. 84, ISBN 9781568814568.
2. Agricola, Ilka; Friedrich, Thomas (2002), Global Analysis: Differential Forms in Analysis, Geometry, and Physics, Graduate Studies in Mathematics, 52, American Mathematical Society, p. 133, ISBN 9780821829516.

• do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7.