fundamental theorem of curves
{{short description|Regular 3-D curves are shape and size determined by their curvature and torsion}}
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 or scale) completely determined by its curvature and torsion.{{citation|title=Differential Geometry of Curves and Surfaces|first1=Thomas F.|last1=Banchoff|first2=Stephen T.|last2=Lovett|publisher=CRC Press|year=2010|isbn=9781568814568|page=84|url=https://books.google.com/books?id=EkfyHkB2ltMC&pg=PA84}}.{{citation|title=Global Analysis: Differential Forms in Analysis, Geometry, and Physics|volume=52|series=Graduate Studies in Mathematics|first1=Ilka|last1=Agricola|author1-link= Ilka Agricola |first2=Thomas|last2=Friedrich|publisher=American Mathematical Society|year=2002|isbn=9780821829516|page=133|url=https://books.google.com/books?id=4pA2P1HyTPoC&pg=PA133}}.
Use
A curve can be described, and thereby defined, by a pair of scalar fields: curvature and torsion , 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
References
{{Reflist}}
Further reading
- {{cite book |title = Differential Geometry of Curves and Surfaces|first = Manfredo|last = do Carmo|authorlink=Manfredo do Carmo | isbn = 0-13-212589-7 | year = 1976}}
{{DEFAULTSORT:Fundamental Theorem Of Curves}}