The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
of a point on the surface. These formulas were established in 1861 by the German mathematician
Julius Weingarten
Julius Weingarten (2 March 1836 – 16 June 1910) was a German mathematician. He received his doctorate in 1864 from Martin-Luther-Universität Halle-Wittenberg.
He made some important contributions to the differential geometry of surfaces, s ...
.
Statement in classical differential geometry
Let ''S'' be a surface in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
that is parametrized by the position vector r(''u'', ''v''). Let ''P'' = ''P''(''u'', ''v'') be a point on the surface. Then
:
are two tangent vectors at point ''P''.
Let n(''u'', ''v'') be the unit
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
and let (''E'', ''F'', ''G'') and (''L'', ''M'', ''N'') be the coefficients of the
first and
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
s of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector n at point ''P'' in terms of the tangent vectors r
''u'' and r
''v'':
:
:
This can be expressed compactly in index notation as
:
,
where ''K
ab'' are the components of the surface's second fundamental form (shape tensor).
Notes
References
*
* Springer ''Encyclopedia of Mathematics''
Weingarten derivational formulas*
*
Erwin Kreyszig
Erwin Otto Kreyszig (January 6, 1922 in Pirna, Germany – December 12, 2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of appli ...
, ''Differential Geometry'', Dover Publications, 1991, {{isbn, 0-486-66721-9, section 45.
Differential geometry of surfaces