The importance of Regiomontanus' angle maximization problem in today's society is undeniable. Whether as a prominent figure in a specific field, as a topic of discussion in various contexts, or as a commemorative date, Regiomontanus' angle maximization problem plays a fundamental role in people's lives. Its influence ranges from politics to entertainment, and its relevance is reflected in the attention it receives from the media and society in general. In this article, we will explore the impact of Regiomontanus' angle maximization problem on different aspects of everyday life, and analyze its importance in the current context.
In 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:
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.
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.
Let
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, the distance from the wall to the point of tangency is therefore , i. e. the geometric mean of the heights of the top and bottom of the painting.
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 ).
Let
The angle we seek to maximize is β − α. The tangent of the angle increases as the angle increases; therefore it suffices to maximize
Since b − a is a positive constant, we only need to maximize the fraction that follows it. Differentiating, we get
Therefore the angle increases as x goes from 0 to √ab and decreases as x increases from √ab. The angle is therefore as large as possible precisely when x = √ab, the geometric mean of a and b.
We have seen that it suffices to maximize
This is equivalent to minimizing the reciprocal:
Observe that this last quantity is equal to
Recall that
Thus when we have u2 + v2, we can add the middle term −2uv to get a perfect square. We have
If we regard x as u2 and ab/x as v2, then u = √x and v = √ab/x, and so
Thus we have
This is as small as possible precisely when the square is 0, and that happens when x = √ab. Alternatively, we might cite this as an instance of the inequality between the arithmetic and geometric means.