The Regiomontanus Problem

摘要

Do you know the name "Regiomontanus"? You undoubtedly know his problem. What is the optimal place to stand to view a painting? The usual assumptions are in effect, namely, the painting hangs on a wall above eye level, and optimal refers to maximizing the angle from the observer's eye. The problem is ubiquitous in modern calculus books, but Regiomontanus, born Johann Müller (1436-76) in Konigsberg, formulated the problem in 1471, some 200 years before the discovery of calculus. Not surprisingly then, the first known solution is geometric and is shown in Figure 1. A circle is drawn passing through the top and bottom of the painting, tangent to the horizontal line at eye level. The point of tangency is the optimal point. That the so-constructed angle is maximal follows from the intersecting secants theorem in geometry. It isn't known if Regiomontanus found the solution. A careful construction is a nontrivial Appolonian problem. See [3], [5], and [6] for additional background. Modern calculus students (and mathematicians) are routinely asked to solve the Regiomontanus problem. Whatever their solution, if it involves a derivative, critical points, etc., it reveals nothing of the original geometry. What follows is an approach that uses ideas from differential geometry to connect the calculus-based solution to the geometric.