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In mathematics, the Regiomontanus' angle maximization problem, is a famous optimization problem〔Heinrich Dörrie,''100 Great Problems of Elementary Mathematics: Their History And Solution'', Dover, 1965, pp. 369–370〕 posed by the 15th-century German mathematician Johannes Müller〔Eli Maor, ''Trigonometric Delights'', Princeton University Press, 2002, pages 46–48〕 (also known as Regiomontanus). The problem is as follows: : A painting hangs from a wall. Given the heights of the top and bottom of the painting above the viewer's eye level, how far from the wall should the viewer stand in order to maximize the angle subtended by the painting and whose vertex is at the viewer's eye? If the viewer stands too close to the wall or too far from the wall, the angle is small; somewhere in between it is as large as possible. The same approach applies to finding the optimal place from which to kick a ball in rugby.〔.〕 For that matter, it is not necessary that the alignment of the picture be at right angles: we might be looking at a window of the Leaning Tower of Pisa or a realtor showing off the advantages of a sky-light in a sloping attic roof. == Solution by elementary geometry == There is a unique circle passing through the top and bottom of the painting and tangent to the eye-level line. By elementary geometry, if the viewer's position were to move along the circle, the angle subtended by the painting would remain constant. All positions on the eye-level line except the point of tangency are outside of the circle, and therefore the angle subtended by the painting from those points is smaller. By ''Elements'' III.36 (alternatively the power-of-a-point theorem), the distance from the wall to the point of tangency is the geometric mean of the heights of the top and bottom of the painting. This means, in turn, that if we reflect the bottom of the picture in the line at eye-level and draw the circle with the segment between the top of the picture and this reflected point as diameter, the circle intersects the line at eye-level in the required position (by Elements II.14). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Regiomontanus' angle maximization problem」の詳細全文を読む スポンサード リンク
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