fundamental theorem of space curves
Informal summary.
The curvature and torsion
of a space
curve
are invariant with respect to Euclidean motions. Conversely, a
given space curve is determined up to a Euclidean motion, by its
curvature and torsion, expressed as functions of the arclength.
Theorem.
Let be a regular, parameterized space curve, without
points of inflection. Let be the
corresponding curvature and torsion functions. Let
be a Euclidean isometry. The curvature and
torsion of the transformed curve
are given by and , respectively.
Conversely, let be continuous functions,
defined on an interval , and suppose that
never vanishes. Then, there exists an arclength parameterization
of a regular, oriented space curve, without points of
inflection, such that and are the corresponding
curvature and torsion functions. If is another
such space curve, then there exists a Euclidean isometry
such that
Title | fundamental theorem of space curves![]() |
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Canonical name | FundamentalTheoremOfSpaceCurves |
Date of creation | 2013-03-22 13:23:28 |
Last modified on | 2013-03-22 13:23:28 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 5 |
Author | rmilson (146) |
Entry type | Theorem |
Classification | msc 53A04 |
Related topic | SpaceCurve |