Interfaces, Capillarity, and Microdrops - Microfluidics for Biotechnology - page 82

Figure 3.10 Schematic view of the radius of curvature of a surface.

For a sphere of radius R , the two curvatures are equal to 1/ R and the mean
curvature is H = 1/ R . For a cylinder of base radius R , the maximum curvature is R
and the minimum curvature 0; hence, H = 1/2 R . For a plane, the two curvatures
are 0 and H = 0; a plane has no curvature. At a saddle point of a surface (Figure
3.11), one of the curvature radii is positive because it corresponds to a convex arc,
whereas the other one is negative, because it corresponds to a concave arc. If | R 1 | =
| R 2 |, the mean curvature H is zero, and we have what is called a minimal surface :

æ ö

1 æ 1 1 ö 1 1 1

H = + = - = 0

.

ç ÷

ç ÷

2 è R R ø 2 R R

è ø

1 2 1 2

Figure 3.11 Mean curvature at a saddle point is zero if | R 1 | = | R 2 |.