View source for Normal curvature

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''of a regular surface''

A quantity that characterizes the deviation of the surface at a point  $  P $
in the direction  $  \mathbf l $
from its tangent plane and is the same in absolute value as the curvature of the corresponding [[Normal section|normal section]]. The normal curvature in the direction  $  \mathbf l $
is

$$ 
k _ {\mathbf l }   =  ( \mathbf n , \mathbf N ) k ,
$$

where  $  k $
is the curvature of the normal section in the direction  $  \mathbf l $, 
$  \mathbf n $
is the unit principal normal vector of the normal section and  $  \mathbf N $
is the unit normal vector to the surface. The normal curvature of a surface in a given direction is the same as that of the [[Osculating paraboloid|osculating paraboloid]] in this direction. The normal curvature of a surface parametrized by  $  u $
and  $  v $
can be expressed in terms of the values of the first and second fundamental forms of the surface (cf. [[Fundamental forms of a surface|Fundamental forms of a surface]]) computed for the values  $  ( d u , d v ) $
corresponding to the direction  $  \mathbf l $
by the formula

$$ 
k _ {\mathbf l }   = \ 

\frac{\textrm{ II } }{\textrm{ I } }
  = \ 

\frac{L  d u  ^ {2} + 2 M  d u  d v + N  d v  ^ {2} }{E  d u  ^ {2} + 2 F  d u  d v + G  d v  ^ {2} }
 .
$$

The curvature of a regular curve lying on a surface is connected with the normal curvature of the surface in the direction of the unit tangent  $  \mathbf l $
to the curve and with the geodesic curvature  $  k _ {g} $
of the curve by the relation

$$ 
k \mathbf n  =  k _ {g} \mathbf N \times \mathbf l + k _ {\mathbf l }  \mathbf N
$$

(see also [[Meusnier theorem|Meusnier theorem]]). By means of the normal curvature one can construct the [[Dupin indicatrix|Dupin indicatrix]], the [[Gaussian curvature|Gaussian curvature]] and the [[Mean curvature|mean curvature]] of the surface, as well as many other concepts of the local geometry of the surface.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>