Euler's theorem (differential geometry)
- Euler's theorem (differential geometry)
In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves 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 Leonard 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) tangent vector to "M" at "p", there passes a normal plane "P""X" which cuts out a curve in "M". That curve has a certain curvature κ"X" when regarded as a curve inside "P""X". Provided not all κ"X" are equal, there is some unit vector "X"1 for which "k"1 = κ"X"1 is as large as possible, and another unit vector "X"2 for which "k"2 = κ"X"2 is as small as possible. Euler's theorem asserts that "X"1 and "X"2 are perpendicular and that, moreover, if "X" is any vector making an angle θ with "X"1, then
The quantities "k"1 and "k"2 are called the principal curvatures, and "X"1 and "X"2 are the corresponding principal directions. Equation (EquationNote|1) is sometimes called Euler's equation harv|Eisenhart|2004|p=124.
ee 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 | id=ISBN 0486438201 [http://www.archive.org/details/treatonthediffer00eiserich Full 1909 text] (now out of copyright)
*citation|title=Recherches sur la courbure des surfaces|first=Leonard|last=Euler|authorlink=Leonard Euler|year=1760|journal=Memoires de l'academie des sciences de Berlin|volume=16|year=1760|pages=119–143|publication-date=1767|url=http://math.dartmouth.edu/~euler/pages/E333.html.
*
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