In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, the two principal curvatures at a given point of a
surface are the maximum and minimum values of 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 ...
as expressed by the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of the
shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.
Discussion
At each point ''p'' of a
differentiable surface in 3-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 Euclidea ...
one may choose a unit ''
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 ...
''. A ''
normal plane'' at ''p'' is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called
normal section. This curve will in general have different
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 ...
s for different normal planes at ''p''. The principal curvatures at ''p'', denoted ''k''
1 and ''k''
2, are the maximum and minimum values of this curvature.
Here the curvature of a curve is by definition the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of the
osculating circle. The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if ''k''
1 does not equal ''k''
2, a result of
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 ...
(1760), and are called principal directions. From a modern perspective, this theorem follows from the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
because these directions are as the
principal axes of a
symmetric tensor—the
second fundamental form. A systematic analysis of the principal curvatures and principal directions was undertaken by
Gaston Darboux, using
Darboux frames.
The product ''k''
1''k''
2 of the two principal curvatures is the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
, ''K'', and the average (''k''
1 + ''k''
2)/2 is the
mean curvature, ''H''.
If at least one of the principal curvatures is zero at every point, then the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
will be 0 and the surface is a
developable surface. For a
minimal surface, the mean curvature is zero at every point.
Formal definition
Let ''M'' be a surface in Euclidean space with
second fundamental form . Fix a point ''p''∈''M'', and an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
''X''
1, ''X''
2 of tangent vectors at ''p''. Then the principal curvatures are the eigenvalues of the symmetric matrix
:
If ''X''
1 and ''X''
2 are selected so that the matrix