= Mean curvature of a sphere and cylinder
{title2=$H_{\mathrm{sphere}}=-1/R,\quad H_{\mathrm{cylinder}}=-1/(2R)$}
For a <sphere> of <radius> $R>0$, its outward <unit normal> is $N(p)=p/R$. Applying the <shape operator> gives $S(v)=-v/R$, so the two <principal curvatures> coincide:
$$
k_1=k_2=-\frac1R,\qquad H=-\frac1R,\qquad K=\frac1{R^2}.
$$
For the lateral surface of a <circular cylinder> with outward <unit normal>, moving around a circle turns $N$ at rate $1/R$, while moving along the axis leaves $N$ constant. Thus the circumferential and axial <principal curvatures> give
$$
k_{\mathrm{circ}}=-\frac1R,\qquad k_{\mathrm{axial}}=0,\qquad H=-\frac1{2R},\qquad K=0.
$$
Here $K$ is <Gaussian curvature>. The <circular cylinder> bends in only one <principal direction>, so its <mean curvature> is half that of the equally sized <sphere>. These formulas concern the smooth lateral surface, excluding any edges where end caps meet it.
\Image[/principal-and-mean-curvature-comparison.png]
{title=Signed principal curvatures and their mean on an outward sphere, outward cylinder, and catenoid waist}
The common <length> scale is $\ell$, equal to the <radius> $R$ for the <sphere> and <circular cylinder>, and to the waist <radius> $a$ for the <catenoid>. The <mean curvature> marker is halfway between the two <principal curvature> markers. <Gaussian curvature> has a different unit and is listed separately beneath each panel.
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