Regiomontanus' angle maximization problem
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Regiomontanus's angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller (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 In geometry, an angle is subtended by an arc, line segment or any other section of a curve when its two rays pass through the endpoints of that arc, line segment or curve section. Conversely, the arc, line segment or curve section confined wi ...
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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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. Let : ''a'' = the height of the painting´s bottom above eye level; : ''b'' = the height of the painting´s top above eye level; A right triangle is formed from the centre of the circle, the centre of the picture and the bottom of the picture. The hypotenuse has the length of the circle´s radius a+(b-a)/2, the length of the two legs are the distance from the wall to the point of tangency and (b-a)/2 respectively. According to the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, the distance from the wall to the point of tangency is therefore \sqrt , i. e. the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of the heights of the top and bottom of the painting.


Solution by calculus

In the present day, this problem is widely known because it appears as an exercise in many first-year calculus textbooks (for example that of Stewart James Stewart, ''Calculus: Early Transcendentals'', Fifth Edition, Brooks/Cole, 2003, page 340, exercise 58). Let : ''a'' = the height of the bottom of the painting above eye level; : ''b'' = the height of the top of the painting above eye level; : ''x'' = the viewer's distance from the wall; : ''α'' = the angle of elevation of the bottom of the painting, seen from the viewer's position; : ''β'' = the angle of elevation of the top of the painting, seen from the viewer's position. The angle we seek to maximize is ''β − α''. The
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of the angle increases as the angle increases; therefore it suffices to maximize : \tan(\beta - \alpha) = \frac = \frac = (b-a)\frac. Since ''b'' − ''a'' is a positive constant, we only need to maximize the fraction that follows it. Differentiating, we get : \left(\frac\right) = \frac \qquad \begin > 0 & \text 0 \le x < \sqrt, \\ = 0 & \text x = \sqrt, \\ < 0 & \text x > \sqrt. \end Therefore the angle increases as ''x'' goes from 0 to and decreases as ''x'' increases from . The angle is therefore as large as possible precisely when ''x'' = , the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of ''a'' and ''b''.


Solution by algebra

We have seen that it suffices to maximize : \frac. This is equivalent to ''minimizing'' the reciprocal: : \frac = x + \frac. Observe that this last quantity is equal to : \left( \sqrt - \sqrt\frac\, \right)^2 + 2\sqrt. This is as small as possible precisely when the square is 0, and that happens when ''x'' = . Alternatively, we might cite this as an instance of the inequality between the arithmetic and geometric means.


References

{{Calculus topics Trigonometry Circles Calculus History of mathematics Mathematical problems