Weingarten derivational formulas
Formulas yielding the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. Let $ \mathbf r = {\mathbf r } ( u, v) $
be the position vector of the surface, let $ \mathbf n $
be the unit normal vector and let $ E $,
$ F $,
$ G $,
$ L $,
$ M $,
$ N $
be the coefficients of the first and second fundamental forms of the surface, respectively; the Weingarten derivational formulas will then take the form
$$ \mathbf n _ {u} = \frac{FM- GL }{EG - F ^ { 2 } } \mathbf r _ {u} + FL- \frac{EM}{EG- F ^ { 2 } } \mathbf r _ {v} , $$
$$ \mathbf n _ {v} = FN- \frac{GM}{EG- F ^ { 2 } } \mathbf r _ {u} + FM- \frac{EN}{EG- F ^ { 2 } } \mathbf r _ {v} . $$
The formulas were established in 1861 by J. Weingarten.
References
[1] | P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian) |
Comments
References
[a1] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |
[a2] | N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) |
Weingarten derivational formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weingarten_derivational_formulas&oldid=49201