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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

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 In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...

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 sp ...

, and ''p'' a point on ''M''. A '' normal plane'' through ''p'' is a plane passing through the point ''p'' containing the ''

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 ...

'' 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 ele ...

to ''M'' at ''p'', there passes a normal plane ''P''_''X'' which cuts out a curve in ''M''. That curve has a certain

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 can ...

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 ''

'', and ''X''₁ and ''X''₂ are the corresponding '' principal directions''. Equation () is sometimes called Euler's equation .

References

Full 1909 text
(now out of copyright) *. * Differential geometry of surfaces Theorems in differential geometry Leonhard Euler {{differential-geometry-stub

See also

References