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Encyclopedia > 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...


The concept was introduced by Sophie Germain in her work on elasticity theory. [1][2] Marie-Sophie Germain (April 1, 1776 – June 27, 1831) was a French mathematician who made important contributions to the fields of differential geometry and number theory. ... Solid mechanics (also known as the theory of elasticity) is a branch of physics, which governs the response of solid material to applied stress (e. ...

Contents

Definition

Let p be a point on the surface S. Consider all curves Ci on S passing through the point p on the surface. Every such Ci has an associated curvature Ki given at p. Of those curvatures Ki, at least one is characterized as maximal κ1 and one as minimal κ2, and these two curvatures κ1,κ2 are known as the principal curvatures of S. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... Local and global maxima and minima for cos(3Ï€x)/x, 0. ... See: In mathematics, see Minimal element. ... Principal curvature is the inverse of the radius of the osculating circle. ...


The mean curvature at pin S is the average of curvatures (Spivak 1999, Volume 3, Chapter 2), hence the name:

H = {1 over 2} (kappa_1 + kappa_2).

More generally (Spivak 1999, Volume 4, Chapter 7), for a hypersurface T the mean curvature is given as In mathematics, a hypersurface is some kind of submanifold. ...

H=frac{1}{n}sum_{i=1}^{n} kappa_{i}.

More abstractly, the mean curvature is (frac{1}{n} times) the trace of the second fundamental form (or equivalently, the shape operator). In differential geometry, the second fundamental form is a quadratic form on the tangent space of a hypersurface, usually denoted by II. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface, where denoted covariant derivative and n a field of normal vectors on... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...


Additionally, the mean curvature H may be written in terms of the covariant derivative nabla as In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

Hvec{n} = g^{ij}nabla_inabla_j X,

using the Gauss-Weingarten relations, where X(x,t) is a family of smoothly embedded hypersurfaces, vec{n} a unit normal vector, and gij the metric tensor. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...


A surface is a minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface S, is said to obey a heat-type equation called the mean curvature flow equation. Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ... ↔ ⇔ ≡ logical symbols representing iff. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...


The sphere is the only surface of constant positive mean curvature without boundary or singularities. For other uses, see Sphere (disambiguation). ...


Surfaces in 3D space

For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface: A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

2 H = nabla cdot hat n

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 "away" from the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. For other uses, see Divergence (disambiguation). ...


For the special case of a surface defined as a function of two coordinates, eg z = S(x,y), and using downward pointing normal the (doubled) mean curvature expression is

begin{align}2 H & = nabla cdot left(frac{nabla(S - z)}{|nabla(S - z)|}right) & = nabla cdot left(frac{nabla S} {sqrt{1 + (nabla S)^2}}right) & = frac{ left(1 + left(frac{partial S}{partial x}right)^2right) frac{partial^2 S}{partial y^2} - 2 frac{partial S}{partial x} frac{partial S}{partial y} frac{partial^2 S}{partial x partial y} + left(1 + left(frac{partial S}{partial y}right)^2right) frac{partial^2 S}{partial x^2} }{left(1 + left(frac{partial S}{partial x}right)^2 + left(frac{partial S}{partial y}right)^2right)^{3/2}}. end{align}

If the surface is additionally known to be axisymmetric with z = S(r), Sphere symmetry group o. ...

2 H = frac{frac{partial^2 S}{partial r^2}}{left(1 + left(frac{partial S}{partial r}right)^2right)^{3/2}} + frac{frac{partial S}{partial r}}{r left(1 + left(frac{partial S}{partial r}right)^2right)^{1/2}}.

Mean curvature in fluid mechanics

An alternate definition is occasionally used in fluid mechanics to avoid factors of two: This box:      Fluid mechanics is the study of how fluids move and the forces on them. ...

Hf = (κ1 + κ2).

This results in the pressure according to the Young-Laplace equation inside an equilibrium spherical droplet being surface tension times Hf; the two curvatures are equal to the reciprocal of the droplet's radius: κ1 = κ2 = r − 1. In fluid dynamics, the Young–Laplace equation describes the pressure difference over a meniscus between two fluids, where is the pressure difference over the interface, the surface tension, and and are the principal radii of curvature at the interface. ... This box:      Surface tension is a property of the surface of a liquid that causes it to behave as an elastic sheet. ...


Minimal surfaces

A rendering of Costa's minimal surface.
A rendering of Costa's minimal surface.
Main article: Minimal surface

A minimal surface is a surface which has zero mean curvature at all points. Classic examples include the catenoid, helicoid and Enneper surface. Recent discoveries including Costa's minimal surface and the Gyroid. Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ... A catenoid A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. ... The helicoid is one of the first minimal surfaces discovered. ... In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is described parametrically by the following set of equations: It was introduced by Alfred Enneper in connection with minimal surface theory. ... A rendering of Costas minimal surface. ... Gyroid, a common card from the Cybernetic Revolution Booster Pack Gyroid Stats ATK: 1000 DEF: 1000 Attribute: Wind Type: Machine Effect: Booster Pack You get Gyroid in Cybernetic Revolution Booster Pack, card 7 Anime Appearance Gyroid is used and owned by Syrus Truesdale from Yu-Gi-Oh GX. Gyroid was...


An extension of the idea of a minimal surface are surfaces of constant mean curvature.


See also

  • Gaussian curvature
  • Mean curvature flow
  • Inverse mean curvature flow
  • First variation of area formula

Curvature is the amount by which a geometric object deviates from being flat. ...

Notes

  1. ^ Dubreil-Jacotin on Sophie Germain
  2. ^ http://links.jstor.org/sici?sici=0002-9890(200308%2F09)110%3A7%3C593%3ACITCC%3E2.0.CO%3B2-O

References

  • Spivak, Michael (1999), A comprehensive introduction to differential geometry (Volumes 3-4) (3rd ed.), Publish or Perish Press, ISBN 0-914098-72-1 (Volume 3), ISBN 0-914098-73-X (Volume 4) .
v • d • e
Different notions of curvature defined in differential geometry
Differential geometry of curves
Curvature · Torsion of curves · Frenet-Serret formulas · Radius of curvature (applications) · Affine curvature
Differential geometry of surfaces
Principal curvatures · Gaussian curvature  · Mean curvature  · Darboux frame  · Gauss-Codazzi equations  · First fundamental form  · Second fundamental form
Riemannian geometry
Curvature of connections
Curvature form  · Torsion tensor  · Cocurvature  · Holonomy
Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... Differential geometry of curves In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. ... In vector calculus, the Frenet-Serret formulas describe the dynamic properties of a particle which moves along a continuous, differentiable curve in three-dimensional space . ... Curvature is the amount by which a geometric object deviates from being flat. ... In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. ... In differential geometry, the second fundamental form is a quadratic form on the tangent space of a hypersurface, usually denoted by II. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface, where denoted covariant derivative and n a field of normal vectors on... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. ... In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. ... From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a...

  Results from FactBites:
 
Curvature - Wikipedia, the free encyclopedia (930 words)
For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
Mean curvature is closely related to the first variation of surface area, in particular a minimal surface like a soap film has mean curvature zero and soap bubble has constant mean curvature.
Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
  More results at FactBites »


 

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