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Two-Dimensional Calculations: Circular Template

The curvature of an arc bounding a 2D "membrane" of phase α can be computed by using a circular template in the same manner as the spherical template is used in three dimensions. Any infinitesimal arc element possessing curvature κ at the origin of a polar coordinate system can be represented as a 2nd-order Taylor expansion about the origin. Proceeding as we did to arrive at Eq. 7 in 3D,

where θ is the angle from the tangent line passing through P, K = κb, and ε is the 2D analogue to ε appearing in Eq. 7. The portion of scaled area, a = Ab⁻², of the template circle that excludes phase α is then given by

which integrates to give, after expanding θ(R) to 3rd order in K,

or, equivalently

Again, the error in the linear approximation, the term enclosed in square brackets, is usually small compared to the true curvature. For example, taking the surface as a circle for which K = 0.5, the error is 1.2%.

As a further generalization of the method in 2D (omitted from the 3D description due to mathematical complexity), one can bias the area elements over which integration is performed by including a dimensionless weighting function of the form

where w is a positive real constant and n an integer. A weighting factor like Eq. 26 can potentially be useful because it reduces the dependence of the area integral, Eq. 23, on portions of the surface that are further removed from P. By biasing the computation to that part of the surface very near P, weighting the integral can increase accuracy. Using Eq. 26, Eq. 23 then becomes

Examination of Eq. 27 reveals that n is less than or equal to 0 for the integral to remain bounded. Of course, for n = −1 and w = 1, Eq. 27 reduces to Eq. 23. Increasingly negative values of n will bias away from P. Since curvature depends on highly localized variations in the surface at P, we will perform the integration for n = 0. In that case,

Comparison of this result to Eq. 25 shows that the error in the linear approximation due to higher-order curvature terms can be reduced by almost 20% if 1/R weighting is incorporated, but 1/R weighting is ineffective for reducing the error caused by approximating the surface to second order. In the remainder of this paper, we will focus on the unweighted result, Eq. 25, since it is easier to implement computationally, and the error term in Eq. 25 is usually small enough already.