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 multili ...
, the two principal curvatures at a given point of a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
are the maximum and minimum values of the
curvature 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 denoted b ...
of the
shape operator
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 ...
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
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
one may choose a unit ''
normal vector''. 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
A normal plane is any plane containing the normal vector of a surface at a particular point.
The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see ...
. This curve will in general have different
curvatures 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 ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the
osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...
. 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 (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 ...
because these directions are as the
principal axes of a
symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
:T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_)
for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
—the
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
. A systematic analysis of the principal curvatures and principal directions was undertaken by
Gaston Darboux
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnigh ...
, using
Darboux frame In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...
s.
The product ''k''
1''k''
2 of the two principal curvatures is the
Gaussian curvature, ''K'', and the average (''k''
1 + ''k''
2)/2 is the
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
, ''H''.
If at least one of the principal curvatures is zero at every point, then the
Gaussian curvature will be 0 and the surface is a
developable surface
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
. For a
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
, the mean curvature is zero at every point.
Formal definition
Let ''M'' be a surface in Euclidean space with
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
. 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