Next: Two-Dimensional Calculations: Circular
We can equally well use a right-circular-cylinder template of radius b and height h instead of the spherical template of radius b used in the preceding analysis. In this case, we position the center of one base of the cylinder on the point P so that the cylinder axis coincides with the surface normal vector n, and then require h to be sufficiently large that the other base lies entirely beneath the surface described by Eq. 5 (see Fig. 2). The portion of the scaled cylinder volume νc lying above the surface is then easily calculated, using Cartesian coordinates, by
where the origin of the Cartesian frame again coincides with the point P in question, G(X,Yi) is given by Eq. 5, and the z-axis coincides with the cylinder axis. Integrating Eq. 19 gives
Therefore, the only error incurred in computing the mean curvature by this method is due to any error in the 2nd-order approximation of the surface, O(ε), in Eq. 5. Implementation of this method all along an interface, however, is somewhat more complicated than that of the sphere method, since the cylinder method requires knowledge of the surface normal direction before the cylinder can be correctly placed. We therefore will restrict attention to the spherical template when discussing computer implementation in three dimensions.
Figure 2: Right circular cylinder template positioned such that one of its bases of radius b is tangent to the surface element S at P. The template height h is great enough to ensure that the other base of the cylinder does not intersect S. The Cartesian frame is oriented as in Fig. 1. Vc (see text) is the portion of the template volume that lies entirely to one side of S (in this figure, the upper side).