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1 May 2011 Errata: Computing the solid angle subtended by a planar figure

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

## Eq. 1

[TeX:] \documentclass[12pt]{minimal}\begin{document}$$\hat \xi = \hat \eta \times \hat \zeta = - \frac{{\hat R_n \times ( {\hat R_n \times \hat R_{n + 1} } )}}{{| {\hat R_n \times \hat R_{n + 1} } |}}.$$\end{document} $\stackrel{̂}{\xi }=\stackrel{̂}{\eta }×\stackrel{̂}{\zeta }=-\frac{{\stackrel{̂}{R}}_{n}×\left({\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}\right)}{|{\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}|}.$

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

## Eq. 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} $\begin{array}{ccc}\hfill {I}_{n}& =& 2\mathrm{arctan}\left[\frac{\frac{\stackrel{̂}{q}·\left({\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}\right)}{|{\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}|}{\left(\frac{1-{\stackrel{̂}{R}}_{n}·{\stackrel{̂}{R}}_{n+1}}{1+{\stackrel{̂}{R}}_{n}·{\stackrel{̂}{R}}_{n+1}}\right)}^{1/2}}{1-\stackrel{̂}{q}·{\stackrel{̂}{R}}_{n}+\frac{\stackrel{̂}{q}·\left[{\stackrel{̂}{R}}_{n}×\left({\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}\right)\right]}{|{\stackrel{̂}{R}}_{n}×{\stackrel{̂}{R}}_{n+1}|}{\left(\frac{1-{\stackrel{̂}{R}}_{n}·{\stackrel{̂}{R}}_{n+1}}{1+{\stackrel{̂}{R}}_{n}·{\stackrel{̂}{R}}_{n+1}}\right)}^{1/2}}\right].\hfill \end{array}$

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

## Notes

1 http://en.wikipedia.org/wiki/Atan2.

©(2011) Society of Photo-Optical Instrumentation Engineers (SPIE)
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
Published: 1 May 2011
JOURNAL ERRATUM
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