Errata: Computing the solid angle subtended by&#8201;a&#8201;planar figure

John S. Asvestas

doi:10.1117/1.3582189

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

John S. Asvestas

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Optical Engineering, Vol. 50, Issue 5, 059801 (May 2011). https://doi.org/10.1117/1.3582189

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}\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}

\hat{ξ} = \hat{η} \times \hat{ζ} = - \frac{{\hat{R}}_{n} \times ({\hat{R}}_{n} \times {\hat{R}}_{n + 1})}{| {\hat{R}}_{n} \times {\hat{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{matrix} I_{n} & = & 2 \arctan [\frac{\frac{\hat{q} \cdot ({\hat{R}}_{n} \times {\hat{R}}_{n + 1})}{| {\hat{R}}_{n} \times {\hat{R}}_{n + 1} |} {(\frac{1 - {\hat{R}}_{n} \cdot {\hat{R}}_{n + 1}}{1 + {\hat{R}}_{n} \cdot {\hat{R}}_{n + 1}})}^{1 / 2}}{1 - \hat{q} \cdot {\hat{R}}_{n} + \frac{\hat{q} \cdot [{\hat{R}}_{n} \times ({\hat{R}}_{n} \times {\hat{R}}_{n + 1})]}{| {\hat{R}}_{n} \times {\hat{R}}_{n + 1} |} {(\frac{1 - {\hat{R}}_{n} \cdot {\hat{R}}_{n + 1}}{1 + {\hat{R}}_{n} \cdot {\hat{R}}_{n + 1}})}^{1 / 2}}] . \end{matrix}

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)

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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

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