On sunny days, we may notice bright elliptical patches on the ground while walking under a tree [(Fig. 1(a)]. They are caused by small gaps in the leaf canopy, which act as “pinhole cameras.”

Suppose a typical elliptical patch has major and minor axes a and b, respectively (Fig. 2). The particular pinhole (approximately circular) is at height H in the canopy, and the distance from there to the center of the patch is L. The angular diameter of the Sun at the pinhole is α radians; the diameter of the Sun is D; its distance is R, and the solar altitude is β. Note that αβ.

Pick several different isolated elliptical patches in Fig. 1(a), and find a typical aspect ratio for b/a.

The solar crescents in Fig. 1(b) were photographed on August 21, 2017,...

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