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In the [[mathematics|mathematical]] field of [[differential geometry]], '''Euler's theorem''' is a result on the [[curvature]] of [[curve]]s on a surface.  The theorem establishes the existence of [[principal curvatures]] and associated ''principal directions'' which give the directions in which the surface curves the most and the least.  The theorem is named for [[Leonhard Euler]] who proved the theorem in {{harv|Euler|1760}}.


More precisely, let ''M'' be a surface in three-dimensional [[Euclidean space]], and ''p'' a point on ''M''.  A '''normal plane''' through ''p'' is a plane passing through the point ''p'' containing the [[normal vector]] to ''M''.  Through each ([[unit vector|unit]]) [[tangent vector]] to ''M'' at ''p'', there passes a normal plane ''P''<sub>''X''</sub> which cuts out a curve in ''M''.  That curve has a certain [[curvature]] &kappa;<sub>''X''</sub> when regarded as a curve inside ''P''<sub>''X''</sub>.  Provided not all &kappa;<sub>''X''</sub> are equal, there is some unit vector ''X''<sub>1</sub> for which ''k''<sub>1</sub>&nbsp;=&nbsp;&kappa;<sub>''X''<sub>1</sub></sub> is as large as possible, and another unit vector ''X''<sub>2</sub> for which ''k''<sub>2</sub>&nbsp;=&nbsp;&kappa;<sub>''X''<sub>2</sub></sub> is as small as possible.  Euler's theorem asserts that ''X''<sub>1</sub> and ''X''<sub>2</sub> are [[perpendicular]] and that, moreover, if ''X'' is any vector making an angle &theta; with ''X''<sub>1</sub>, then
{{NumBlk|:|<math>\kappa_X = k_1\cos^2\theta + k_2\sin^2\theta.\,</math>|{{EquationRef|1}}}}


The quantities ''k''<sub>1</sub> and ''k''<sub>2</sub> are called the '''principal curvatures''', and ''X''<sub>1</sub> and ''X''<sub>2</sub> are the corresponding '''principal directions'''. Equation ({{EquationNote|1}}) is sometimes called '''Euler's equation''' {{harv|Eisenhart|2004|p=124}}.
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==See also==
* [[Differential geometry of surfaces]]
* [[Dupin indicatrix]]
 
==References==
*{{citation | last = Eisenhart|first= Luther P.|authorlink=Luther Eisenhart | title=A Treatise on the Differential Geometry of Curves and Surfaces | publisher=Dover | year =2004 | isbn = 0-486-43820-1}} [http://www.archive.org/details/treatonthediffer00eiserich Full 1909 text] (now out of copyright)
*{{citation|title=Recherches sur la courbure des surfaces|first=Leonhard|last=Euler|authorlink=Leonhard Euler|year=1760|journal=Memoires de l'academie des sciences de Berlin|volume=16|pages=119&ndash;143|publication-date=1767|url=http://math.dartmouth.edu/~euler/pages/E333.html}}.
* {{citation|first=Michael|last=Spivak|authorlink=Michael Spivak|title=A comprehensive introduction to differential geometry, Volume II|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3}}
 
[[Category:Differential geometry of surfaces]]
[[Category:Theorems in differential geometry]]
 
{{differential-geometry-stub}}

Latest revision as of 13:45, 5 May 2014


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