Namespaces
Variants
Actions

Difference between revisions of "Weingarten surface"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
A surface whose [[Mean curvature|mean curvature]] is in functional relationship with its [[Gaussian curvature|Gaussian curvature]]. A necessary and sufficient condition for a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097630/w0976301.png" /> to be a Weingarten surface is for both sheets of its focal set to be superimposed on a surface of revolution, and for the edges of regression (cuspidal edges) of the normals to the lines of curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097630/w0976302.png" /> to be superimposed on the meridians. Examples of Weingarten surfaces include surfaces of revolution and surfaces of constant mean or constant Gaussian curvature. Introduced by J. Weingarten [[#References|[1]]], [[#References|[2]]] in the context of the problem of finding all surfaces isometric to a given surface of revolution. The latter problem can be reduced to finding all Weingarten surfaces of this class.
+
{{TEX|done}}
 +
A surface whose [[Mean curvature|mean curvature]] is in functional relationship with its [[Gaussian curvature|Gaussian curvature]]. A necessary and sufficient condition for a surface $S$ to be a Weingarten surface is for both sheets of its focal set to be superimposed on a surface of revolution, and for the edges of regression (cuspidal edges) of the normals to the lines of curvature of $S$ to be superimposed on the meridians. Examples of Weingarten surfaces include surfaces of revolution and surfaces of constant mean or constant Gaussian curvature. Introduced by J. Weingarten [[#References|[1]]], [[#References|[2]]] in the context of the problem of finding all surfaces isometric to a given surface of revolution. The latter problem can be reduced to finding all Weingarten surfaces of this class.
  
 
====References====
 
====References====

Latest revision as of 16:58, 30 July 2014

A surface whose mean curvature is in functional relationship with its Gaussian curvature. A necessary and sufficient condition for a surface $S$ to be a Weingarten surface is for both sheets of its focal set to be superimposed on a surface of revolution, and for the edges of regression (cuspidal edges) of the normals to the lines of curvature of $S$ to be superimposed on the meridians. Examples of Weingarten surfaces include surfaces of revolution and surfaces of constant mean or constant Gaussian curvature. Introduced by J. Weingarten [1], [2] in the context of the problem of finding all surfaces isometric to a given surface of revolution. The latter problem can be reduced to finding all Weingarten surfaces of this class.

References

[1] J. Weingarten, "Ueber eine Klasse auf einander abwickelbarer Flächen" J. Reine Angew. Math. , 59 (1861) pp. 382–393
[2] J. Weingarten, "Ueber die Flächen, derer Normalen eine gegebene Fläche berühren" J. Reine Angew. Math. , 62 (1863) pp. 61–63
[3] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)


Comments

References

[a1] D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950)
How to Cite This Entry:
Weingarten surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weingarten_surface&oldid=12641
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article