Surfaces in 3D space

For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface:

${\displaystyle 2H=-\nabla \cdot {\hat {n}}}$

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 "towards" 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. Mean Curvature may also be calculated

${\displaystyle 2H={\text{Trace}}((\mathrm {II} )(\mathrm {I} ^{-1}))}$

where I and II denote first and second quadratic form matrices, respectively.

If ${\displaystyle S(x,y)}$ is a parametrization of the surface and ${\displaystyle u,v}$ are two linearly independent vectors in parameter space then the mean curvature can be written in terms of the first and second fundamental forms as

$${\displaystyle {\frac {lG-2mF+nE}{2(EG-F^{2})}}}$$

where ${\displaystyle E=\mathrm {I} (u,u)}$, ${\displaystyle F=\mathrm {I} (u,v)}$, ${\displaystyle G=\mathrm {I} (v,v)}$, ${\displaystyle l=\mathrm {II} (u,u)}$, ${\displaystyle m=\mathrm {II} (u,v)}$, ${\displaystyle n=\mathrm {II} (v,v)}$.^[4]

For the special case of a surface defined as a function of two coordinates, e.g. ${\displaystyle z=S(x,y)}$, and using the upward pointing normal the (doubled) mean curvature expression is

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

In particular at a point where ${\displaystyle \nabla S=0}$, the mean curvature is half the trace of the Hessian matrix of ${\displaystyle S}$.

If the surface is additionally known to be axisymmetric with ${\displaystyle z=S(r)}$,

${\displaystyle 2H={\frac {\frac {\partial ^{2}S}{\partial r^{2}}}{\left(1+\left({\frac {\partial S}{\partial r}}\right)^{2}\right)^{3/2}}}+{\frac {\partial S}{\partial r}}{\frac {1}{r\left(1+\left({\frac {\partial S}{\partial r}}\right)^{2}\right)^{1/2}}},}$

where ${\displaystyle {\frac {\partial S}{\partial r}}{\frac {1}{r}}}$ comes from the derivative of ${\textstyle z=S(r)=S\left({\sqrt {x^{2}+y^{2}}}\right)}$.

Implicit form of mean curvature

The mean curvature of a surface specified by an equation ${\displaystyle F(x,y,z)=0}$ can be calculated by using the gradient ${\displaystyle \nabla F=\left({\frac {\partial F}{\partial x}},{\frac {\partial F}{\partial y}},{\frac {\partial F}{\partial z}}\right)}$ and the Hessian matrix

${\displaystyle \textstyle {\mbox{Hess}}(F)={\begin{pmatrix}{\frac {\partial ^{2}F}{\partial x^{2}}}&{\frac {\partial ^{2}F}{\partial x\partial y}}&{\frac {\partial ^{2}F}{\partial x\partial z}}\\{\frac {\partial ^{2}F}{\partial y\partial x}}&{\frac {\partial ^{2}F}{\partial y^{2}}}&{\frac {\partial ^{2}F}{\partial y\partial z}}\\{\frac {\partial ^{2}F}{\partial z\partial x}}&{\frac {\partial ^{2}F}{\partial z\partial y}}&{\frac {\partial ^{2}F}{\partial z^{2}}}\end{pmatrix}}.}$

The mean curvature is given by:^[5][6]

${\displaystyle H={\frac {\nabla F\ {\mbox{Hess}}(F)\ \nabla F^{\mathsf {T}}-|\nabla F|^{2}\,{\text{Trace}}({\mbox{Hess}}(F))}{2|\nabla F|^{3}}}}$

Another form is as the divergence of the unit normal. A unit normal is given by ${\displaystyle {\frac {\nabla F}{|\nabla F|}}}$ and the mean curvature is

${\displaystyle H=-{\frac {1}{2}}\nabla \cdot \left({\frac {\nabla F}{|\nabla F|}}\right).}$

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 are called Bryant surfaces.^[7]