IMAGES IN CHRISTMAS BALLS: A.K.A. CRAZY OPTICS CALCULATIONS
By Eef van Beveren, Frieder Kleefeld, George Rupp | pdf
ABSTRACT: We describe light-reflection properties of spherically curved mirrors, like balls in the Christmas tree. In particular, we study the position of the image which is formed somewhere beyond the surface of a spherical mirror, when an eye observes the image of a pointlike light source. The considered problem, originally posed by Abu Ali Hasan Ibn al-Haitham — alias Alhazen — more than a millennium ago, turned out to have the now well known analytic solution of a biquadratic equation, being still of great relevance, e.g. for the aberration-free construction of telescopes. We do not attempt to perform an exhaustive survey of the rich historical and engineering literature on the subject, but develop a simple pedagogical approach to the issue, which we believe to be of continuing interest in view of its maltreating in many high-school textbooks.
Figure 6: The locations of the various images as seen by each of the ﬁve observers introduced in Fig. 1. We also indicate the angles of incidence and reﬂection, in order to make sure that they are equal.
Figure 7: Images of an extended object, as seen in a Christmas ball from diﬀerent angles.
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