In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
differential geometry, Euler's theorem is a result on the
curvature of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
who proved the theorem in .
More precisely, let ''M'' 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 Euclidean ...
, 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
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
)
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
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
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
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 () is sometimes called Euler's equation .
See also
*
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
*
Dupin indicatrix
In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. Th ...
References
Full 1909 text(now out of copyright)
*.
*
Differential geometry of surfaces
Theorems in differential geometry
Leonhard Euler
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