Editing 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 (mathematics)|scale]]) completely determined by its curvature and [[Torsion of curves|torsion]].<ref>{{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}}.</ref><ref>{{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}}.</ref>

==Use==
A curve can be described, and thereby defined, by a pair of [[scalar field]]s: curvature <math>\kappa</math> and torsion <math>\tau</math>, both of which depend on some parameter which [[parametric equation|parametrizes]] the curve but which can ideally be the [[arc length]] of the curve.  From just the curvature and torsion, the [[vector field]]s for the tangent, normal, and binormal vectors can be derived using the [[Frenet–Serret formulas]].  Then, [[Integral|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 [[congruence (geometry)|congruent]] to each other.

==See also==
*[[Differential geometry of curves]]
*[[Gaussian curvature]]

==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}}
[[Category:Theorems about curves]]
[[Category:Theorems in differential geometry]]