**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Mean curvature

Summary

In mathematics, the mean curvature of a surface 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 concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation.
Let be a point on the surface inside the three dimensional Euclidean space R3. 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 curvatures 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 the mean curvature is given as
More abstractly, the mean curvature is the trace of the second fundamental form divided by n (or equivalently, the shape operator).
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 if and only if 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 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".

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (34)

Related people (5)

Related concepts (14)

Related courses (7)

Related lectures (35)

Differential geometry of surfaces

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 perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

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 minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame.

Soap bubble

A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also used in artistic performances. Assembling many bubbles results in foam. When light shines onto a bubble it appears to change colour.

MATH-213: Differential geometry

Ce cours est une introduction à la géométrie différentielle classique des courbes et des surfaces, principalement dans le plan et l'espace euclidien.

ME-411: Mechanics of slender structures

Analysis of the mechanical response and deformation of slender structural elements.

CS-457: Geometric computing

This course will cover mathematical concepts and efficient numerical methods for geometric computing. We will explore the beauty of geometry and develop algorithms to simulate and optimize 2D and 3D g

Explores minimal surfaces and asymptotic lines on curved surfaces.

Covers course recap, shape optimization, asymptotic grid shells, and differential geometry concepts.

Covers the Brown-York stress tensor and its relation to AdS/CFT correspondence.

In this paper we study Weingarten surfaces and explore their potential for fabrication-aware design in freeform architecture. Weingarten surfaces are characterized by a functional relation between their principal curvatures that implicitly defines approxim ...

Pascal Fua, Mathieu Salzmann, Victor Constantin, Shaifali Parashar, Erhan Gündogdu

In this paper, we tackle the problem of static 3D cloth draping on virtual human bodies. We introduce a two-stream deep network model that produces a visually plausible draping of a template cloth on virtual 3D bodies by extracting features from both the b ...

2020Mark Pauly, Florin Isvoranu, Uday Kusupati, Tian Chen, Yingying Ren, Davide Pellis

We present a computational inverse design framework for a new class of volumetric deployable structures that have compact rest states and deploy into bending-active 3D target surfaces. Umbrella meshes consist of elastic beams, rigid plates, and hinge joint ...

2022