Reading ‘When Least is Best’ by Paul Nahin was quite a delight. This book has taken me on many tangents to explore the world of optimization. One path is the further exploration of the Regiomontanus angle maximization problem.

This problem was posed in the 15th century, where it was known that if you stand too close to a painting (that is above you), it will appear small (such as if one was against the wall looking vertically up, you wouldn’t actually see the painting). And, stand too far away, and the painting disappears from sight. Paul solves the problem using the AM-GM inequality, a commonly used math tool before the knowledge of calculus. The solution to maximize the viewing angle, θ, as the originally posed question is:

An issue with the above solution is it assumes an infinite picture width. In 1983, A. Tan and O. Castillo derived the solution to include a picture width as shown in the diagram above. This derivation showed that the original maximum θ and x-location were incorrect for a 2D painting. In their derivation, θ is the angle between the line-of-sight and the x-axis. Ω(x) is the angle that is swept across the image (i.e. the true viewing angle).

This is a complicated solution, no doubt. But this made me think. Most of the times when I view paintings on the wall or more commonly *presentations*, I rarely sit dead center. What would the viewing angle be if I was off-center?

Introducing a new variable, q, which is the distance away from the x-axis, the above limits of integration can be changed from ‘-b, b’ to ‘-b-q, b-q’, resulting in:

Contour plots of two examples are displayed at the end of this post where one can see that you just need to be in a small circle around the optimum point to see the largest image (the white area in the plot). The derivative of Ω with respect to x can also be made to obtain the maximum view for any given offset, q, which is presented as the purple line. This indicates that, at a minimum, you should stand or sit to the right of this zone. The angle of this region is about ±54° from the x-axis.

This is all fine and dandy, but digging deeper, when viewing a presentation, I find that I can move my body, neck, and head to various comfortable positions, and my eyes adjust to stare where needed. However, minimizing my eye movement seems to keep me awake during most presentations. Looking at anthropometric data posted on Learneasy.info, the optimum eye conical viewing angle is 30° and the approximate maximum eye conical viewing angle is 60°. Higher than 60°, and you will need to start rotating your head or body to view the screen. The limit of word recognition is 10-20°.

So, highlighting the 60°, 30°, and 10° viewing angles (the red line, yellow line, and green line respectively) can help understand where to sit during a presentation; which you are more likely to be viewing on a wall everyday. I personally like to sit in the 5° to 10° zone, I just never knew why until I started this derivation.

For the first example, a somewhat common projection screen size is used. The results show the screen will appear largest at 1.80 feet, which is awfully close. I would generally sit about 15 feet away, which corresponds approximately to a 10° viewing angle.

Looking at a common 60 inch television gives the below results.

Maybe the next step from here will be to include stadium seating. Then I could become a millionaire from creating an Optimal Conference Seating app.

References:

Nahin, Paul J. When Least is Best. Princeton University Press, 2011

Tan, A. and O. Castillo. Maximizing Paintings. The Mathematics Teacher, vol. 76, no. 7 (October 1983). Pg 472.