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A coding error was found in calculating the optimal packing distribution of our geodesic array. The error was corrected and the new optimization results in slightly improved packing density. The overall approach and algorithm remain unchanged.
© 2013 Optical Society of America
In the article [1] we iteratively optimized the packing distribution of circles on a sphere using an icosahedral geodesic as the base and distorting the circle coordinates with a first order polynomial. The code used to project the vertices from the icosahedron onto the surface of a unit sphere was based off the method described by Kenner [2] on page 75, Eqs. (12).4-12.6. These Eqs. are listed below for reference.
Equations (1) through (3) are used to compute the Cartesian coordinates of the geodesic vertices (x 1, y 1, z 1) given the trilinear coordinates (x, y, z) used to describe the vertex locations on the triangular face of the icosahedron and the frequency (ν) of the geodesic [1].
These Eqs. were used in our Matlab code, but an error was found in the implementation of Eq. (2) where x 1 was used instead of x. Since the Eqs. were applied in the order shown above, the x value used in Eq. (2) was scaled by the sine term for non-zero values. This resulted in slightly decreased packing density and a violation of the constraint that edge vertices do not move normal to the edge [1]. We have corrected this error and rerun the optimization which produces an improvement in packing density and slightly different coefficients for the distortion polynomial from those shown in Fig. 3 and Table 1 . The revised comparison of packing density and chord ratio for baseline geodesic and distorted geodesic distributions are shown below in Fig. 3 and the new coefficients are listed in Table 1. Figure 3 shows that packing density and chord ratio have improved to 0.7666 and 0.1734, respectively, for a frequency 9 geodesic.
Table 1. First order distortion coefficient
Fig. 3 (a) Packing densities as a function of N. Blue line is baseline geodesic, red line is the distorted geodesic with 1st order correction, and black dashed line is the theoretical maximum. (b) Chord ratios as a function of N. Blue line is baseline geodesic and red line is the distorted geodesic with 1st order correction.
References and links
1. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef] [PubMed]
2. H. Kenner, Geodesic Math and How to Use It, 2nd ed. (University of California 2003).
