|
|
| Line 1: |
Line 1: |
| In [[Riemannian geometry]], the '''geodesic curvature''' <math>k_g</math> of a curve <math>\gamma</math> measures how far the curve is from being a [[geodesic]]. In a given manifold <math>\bar{M}</math>, the '''geodesic curvature''' is just the usual '''curvature''' of <math>\gamma</math> (see below), but when <math>\gamma</math> is restricted to lie on a submanifold <math>M</math> of <math>\bar{M}</math> (e.g. for [[Curvature#Curves on surfaces|curves on surfaces]]), geodesic curvature refers to the curvature of <math>\gamma</math> in <math>M</math> and it is different in general from the curvature of <math>\gamma</math> in the ambient manifold <math>\bar{M}</math>. The (ambient) curvature <math>k</math> of <math>\gamma</math> depends on two factors: the curvature of the submanifold <math>M</math> in the direction of <math>\gamma</math> (the [[normal curvature]] <math>k_n</math>), which depends only from the direction of the curve, and the curvature of <math>\gamma</math> seen in <math>M</math> (the geodesic curvature <math>k_g</math>), which is a second order quantity. The relation between these is <math>k = \sqrt{k_g^2+k_n^2}</math>. In particular geodesics on <math>M</math> have zero geodesic curvature (they are "straight"), so that <math>k=k_n</math>, which explains why they appear to be curved in ambient space whenever the submanifold is.
| | All person who wrote all of the article is called Leland but it's not those most masucline name out there. Managing people is undoubtedly where his primary wage comes from. His wife and him live back in Massachusetts and he has everything that he calls for there. [http://Dict.Leo.org/?search=Base+jumping Base jumping] is something that he or she is been doing for many. He could be running and maintaining a blog here: http://prometeu.net<br><br>my homepage: [http://prometeu.net clash of clans hack] |
| | |
| ==Definition==
| |
| Consider a curve <math>\gamma</math> in a manifold <math>\bar{M}</math>, parametrized by [[arclength]], with unit tangent vector <math>T=d\gamma/ds</math>. Its curvature is the norm of the [[Covariant derivative#Derivative along curve|covariant derivative]] of <math>T</math>: <math>k = \|DT/ds \|</math>. If <math>\gamma</math> lies on <math>M</math>, the '''geodesic curvature''' is the norm of the projection of the covariant derivative <math>DT/ds</math> on the tangent space to the submanifold. Conversely the '''normal curvature''' is the norm of the projection of <math>DT/ds</math> on the normal bundle to the submanifold at the point considered.
| |
| | |
| If the ambient manifold is the euclidean space <math>\mathbb{R}^n</math>, then the covariant derivative <math>DT/ds</math> is just the usual derivative <math>dT/ds</math>.
| |
| | |
| ==Example==
| |
| Let <math>M</math> be the unit sphere <math>S^2</math> in three dimensional Euclidean space. The normal curvature of <math>S^2</math> is identically 1, independently of the direction considered. Great circles have curvature <math>k=1</math>, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius <math>r</math> will have curvature <math>1/r</math> and geodesic curvature <math>k_g = \sqrt{1-r^2}/r</math>.
| |
| | |
| ==Some results involving geodesic curvature==
| |
| | |
| *The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold <math>M</math>. It does not depend on the way the submanifold <math>M</math> sits in <math>\bar{M}</math>.
| |
| | |
| * Geodesics of <math>M</math> have zero geodesic curvature, which is equivalent to saying that <math>DT/ds</math> is orthogonal to the tangent space to <math>M</math>.
| |
| | |
| *On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: <math>k_n</math> only depends on the point on the submanifold and the direction <math>T</math>, but not on <math>DT/ds</math>.
| |
| | |
| *In general Riemannian geometry, the derivative is computed using the [[Levi-Civita connection]] <math>\bar{\nabla}</math> of the ambient manifold: <math>DT/ds = \bar{\nabla}_T T</math>. It splits into a tangent part and a normal part to the submanifold: <math>\bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp</math>. The tangent part is the usual derivative <math>\nabla_T T</math> in <math>M</math> (it is a particular case of Gauss equation in the [[Gauss-Codazzi equations]]), while the normal part is <math>\mathrm{I\!I}(T,T)</math>, where <math>\mathrm{I\!I}</math> denotes the [[second fundamental form]].
| |
| | |
| *The [[Gauss–Bonnet theorem]].
| |
| | |
| ==See also==
| |
| * [[Curvature]]
| |
| * [[Darboux frame]]
| |
| * [[Gauss–Codazzi equations]]
| |
| | |
| == References ==
| |
| *{{citation | last = do Carmo|first =Manfredo P. | title=Differential Geometry of Curves and Surfaces | publisher=Prentice-Hall | year=1976 | isbn = 0-13-212589-7}}
| |
| * {{citation|first=Heinrich|last=Guggenheimer|author-link=Heinrich Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Surfaces|isbn=0-486-63433-7}}.
| |
| * {{springer|id=G/g044070|title=Geodesic curvature|first=Yu.S.|last=Slobodyan|year=2001}}.
| |
| | |
| ==External links==
| |
| * {{Mathworld|urlname=GeodesicCurvature|title=Geodesic curvature}}
| |
| | |
| | |
| | |
| [[Category:Geodesic (mathematics)]]
| |
| [[Category:Manifolds]]
| |
All person who wrote all of the article is called Leland but it's not those most masucline name out there. Managing people is undoubtedly where his primary wage comes from. His wife and him live back in Massachusetts and he has everything that he calls for there. Base jumping is something that he or she is been doing for many. He could be running and maintaining a blog here: http://prometeu.net
my homepage: clash of clans hack