Euler's theorem (differential geometry)
Encyclopedia
In the 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 Leonard Euler who proved the theorem in .
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 PX which cuts out a curve in M. That curve has a certain curvature
κX when regarded as a curve inside PX. Provided not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small as possible. Euler's theorem asserts that X1 and X2 are perpendicular
and that, moreover, if X is any vector making an angle θ with X1, then
The quantities k1 and k2 are called the principal curvatures, and X1 and X2 are the corresponding principal directions. Equation is sometimes called Euler's equation .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of differential geometry, Euler's theorem is a result on the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
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 Leonard Euler who proved the theorem in .
More precisely, let M be a surface in three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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
Tangent vector
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 Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
to M at p, there passes a normal plane PX which cuts out a curve in M. That curve has a certain curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
κX when regarded as a curve inside PX. Provided not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small as possible. Euler's theorem asserts that X1 and X2 are perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
and that, moreover, if X is any vector making an angle θ with X1, then
The quantities k1 and k2 are called the principal curvatures, and X1 and X2 are the corresponding principal directions. Equation is sometimes called Euler's equation .