1 May 2011 Errata: Computing the solid angle subtended by a planar figure
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Abstract
This PDF file contains the errata for “OE Vol. 50 Issue 05 Paper OE ERR-MAY2011-1” for OE Vol. 50 Issue 05

This article [ Opt. Eng. 33(12), 4055–4059 (1994)  10.1117/12.183402 ] was published in December 1994. We correct two mathematical expressions.

In the original paper, a negative sign is missing in Eq. (6). The correct expression is

1

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \hat \xi = \hat \eta \times \hat \zeta = - \frac{{\hat R_n \times ( {\hat R_n \times \hat R_{n + 1} } )}}{{| {\hat R_n \times \hat R_{n + 1} } |}}. \end{equation}\end{document} ξ̂=η̂×ζ̂=R̂n×(R̂n×R̂n+1)|R̂n×R̂n+1|.

The same mistake appears in Eq. (16). The correct expression is

2

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} I_n &=& 2\arctan \left[ {\frac{{\displaystyle\frac{{\hat q \cdot ( {\hat R_n \times \hat R_{n + 1} } )}}{{| {\hat R_n \times \hat R_{n + 1} } |}}\left( {\frac{{1 - \hat R_n \cdot \hat R_{n + 1} }}{{1 + \hat R_n \cdot \hat R_{n + 1} }}} \right)^{1/2} }}{{1 - \hat q \cdot \hat R_n + \displaystyle\frac{{\hat q \cdot [ {\hat R_n \times ( {\hat R_n \times \hat R_{n + 1} } )} ]}}{{| {\hat R_n \times \hat R_{n + 1} } |}}\left( {\displaystyle\frac{{1 - \hat R_n \cdot \hat R_{n {+} 1} }}{{1 {+} \hat R_n \cdot \hat R_{n + 1} }}} \right)^{1/2} }}} \right]. \end{eqnarray}\end{document} In=2arctanq̂·(R̂n×R̂n+1)|R̂n×R̂n+1|1R̂n·R̂n+11+R̂n·R̂n+11/21q̂·R̂n+q̂·[R̂n×(R̂n×R̂n+1)]|R̂n×R̂n+1|1R̂n·R̂n+11+R̂n·R̂n+11/2.

All the results presented in the original paper are correct. In numerical implementations1 1 the atan2 function should be used for the arctangent.

© (2011) Society of Photo-Optical Instrumentation Engineers (SPIE)
John S. Asvestas, John S. Asvestas, } "Errata: Computing the solid angle subtended by a planar figure," Optical Engineering 50(5), 059801 (1 May 2011). https://doi.org/10.1117/1.3582189 . Submission:
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