In
mathematics
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 ...
, the mean curvature
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 ...
is an ''extrinsic'' measure of
curvature that comes from
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 ...
and that locally describes the curvature of an
embedded surface in some ambient space such as
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 ...
.
The concept was used by
Sophie Germain
Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's lib ...
in her work on
elasticity theory
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ...
.
Jean Baptiste Marie Meusnier
Jean Baptiste Marie Charles Meusnier de la Place (Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature o ...
used it in 1776, in his studies of
minimal surfaces
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 ...
. It is important in the analysis of
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 that ...
s, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as
soap film
Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s) which, for example, have constant mean curvature in static flows, by the
Young-Laplace equation.
Definition
Let
be a point on the surface
inside the three dimensional Euclidean space . Each plane through
containing the normal line to
cuts
in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle
(always containing the normal line) that curvature can vary. The
maximal curvature
and
minimal curvature
are known as the ''
principal curvature
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 ...
s'' of
.
The mean curvature at
is then the average of the signed curvature over all angles
:
:
.
By applying
Euler's theorem, this is equal to the average of the principal curvatures :
:
More generally , for a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
the mean curvature is given as
:
More abstractly, the mean curvature is the trace of 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 ...
divided by ''n'' (or equivalently, 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 ...
).
Additionally, the mean curvature
may be written in terms of the
covariant derivative as
:
using the ''Gauss-Weingarten relations,'' where
is a smoothly embedded hypersurface,
a unit normal vector, and
the
metric tensor.
A surface is 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 that ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface
, is said to obey a
heat-type equation called the
mean curvature flow equation.
The
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is weakened to "immersed surface".
Surfaces in 3D space
For a surface defined in 3D space, the mean curvature is related to a unit
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
of the surface:
:
where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of the unit normal may be calculated. Mean Curvature may also be calculated
:
where I and II denote first and second quadratic form matrices, respectively.
If
is a parametrization of the surface and
are two linearly independent vectors in parameter space then the mean curvature can be written in terms of the
first
First or 1st is the ordinal form of the number one (#1).
First or 1st may also refer to:
*World record, specifically the first instance of a particular achievement
Arts and media Music
* 1$T, American rapper, singer-songwriter, DJ, and rec ...
and
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 ...
s as
where
,
,
,
,
,
.
For the special case of a surface defined as a function of two coordinates, e.g.
, and using the upward pointing normal the (doubled) mean curvature expression is
:
In particular at a point where
, the mean curvature is half the trace of the Hessian matrix of
.
If the surface is additionally known to be
axisymmetric with
,
:
where
comes from the derivative of
.
Implicit form of mean curvature
The mean curvature of a surface specified by an equation
can be calculated by using the gradient
and the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
:
The mean curvature is given by:
:
Another form is as the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of the unit normal. A unit normal is given by
and the mean curvature is
:
Mean curvature in fluid mechanics
An alternate definition is occasionally used in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
to avoid factors of two:
:
.
This results in the pressure according to the
Young-Laplace equation inside an equilibrium spherical droplet being
surface tension times
; the two curvatures are equal to the reciprocal of the droplet's radius
:
.
Minimal surfaces
A minimal surface is a surface which has zero mean curvature at all points. Classic examples include the
catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
,
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...
and
Enneper surface
In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by:
\begin
x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\
y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...
. Recent discoveries include
Costa's minimal surface
In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact s ...
and the
Gyroid
A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.
History and properties
The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces. I ...
.
CMC surfaces
An extension of the idea of a minimal surface are surfaces of constant mean curvature. The surfaces of unit constant mean curvature in
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
are called
Bryant surfaces.
[.]
See also
*
Gaussian curvature
*
Mean curvature flow
*
Inverse mean curvature flow
In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the R ...
*
First variation of area formula
*
Stretched grid method
Notes
References
*.
*
{{curvature
Differential geometry
Differential geometry of surfaces
Surfaces
Curvature (mathematics)