Regiomontanus' angle maximization problem

[3] 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.

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 [6]).

We have seen that it suffices to maximize This is equivalent to minimizing the reciprocal: Observe that this last quantity is equal to Thus when we have u2 + v2, we can add the middle term −2uv to get a perfect square.