## Abstract

A closed-form solution for the shape of an aspheric mirror with a constant angular magnification is presented. The focal surface of such a mirror is derived. The advantages of the proposed design over the spherical mirror for application in the all-sky camera are discussed.

© 1992 Optical Society of America

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### Equations (8)

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(1)
$$\text{d}y/\text{d}x=-\text{tan}(N\hspace{0.17em}\text{arctan}[x/(d-y)],$$
(2)
$$\begin{array}{l}X=x/d,\\ Y=y/d,\\ Z=1-Y,\\ U=X/Z.\end{array}$$
(3)
$$\frac{dZ}{Z}=\frac{\text{tan}[N\hspace{0.17em}\text{arctan}(U)]}{1-U\hspace{0.17em}\text{tan}[N\hspace{0.17em}\text{arctan}(U)]}\text{d}U,$$
(4)
$$\begin{array}{l}x(\mathrm{\alpha})=\frac{d\hspace{0.17em}\text{sin}(\mathrm{\alpha})}{{\{\text{cos}[(N+1)\mathrm{\alpha}]\}}^{1/(N+1)}},\\ y(\mathrm{\alpha})=d-\frac{d\hspace{0.17em}\text{cos}(\mathrm{\alpha})}{{\{\text{cos}[(N+1)\mathrm{\alpha}]\}}^{1/(N+1)}},\end{array}$$
(5)
$$\begin{array}{l}{x}_{f}(\mathrm{\alpha})=\frac{M}{M-1}x(\mathrm{\alpha}),\\ {y}_{f}(\mathrm{\alpha})=\frac{M}{M-1}y(\mathrm{\alpha})-\frac{d}{M-1}.\end{array}$$
(7)
$${\mathrm{\beta}}^{*}=2\hspace{0.17em}\text{arcsin}[(M+1)\text{sin}(\mathrm{\alpha})/2]-\mathrm{\alpha}.$$
(8)
$$\text{RAD}(\mathrm{\alpha})=({\mathrm{\beta}}^{*}-\mathrm{\beta})/\mathrm{\beta}.$$