This article started as a curious question: how much taller a skyscraper needs to be in order to be perceived as twice its actual height? I assumed that the observer was at ground level at a horizontal distance d from the building, and that the real height is H. We all know that the building must be more than twice as tall due to perspective, in order to give the illusion that its height has doubled.
I used AI tools to find an answer but could not get anywhere. Then, looking at articles on the topic – some referenced in my paper – the focus was always on computing the exact height by correcting the perspective effect. All were rather trivial, involving simple trigonometry. Eventually I came up with a method to accurately quantify the induced illusion and correct it.
Highlights
I came up with an elegant mathematical solution, more advanced than what you will find in textbooks, including the hyperbolic sine function and its inverse, integration in polar coordinates, a fourth order linear recursion, and even a bit of non-trivial number theory.
Magically, each step and equation lead to beautiful closed-form mathematical formulas, without the need for approximations. This would make for an interesting math or physics problem for students familiar with advanced calculus. A problem that you cannot solve – at least for now – using OpenAI or similar tools. Or a job interview question for a scientist role, where the candidate is allowed to use AI to get an answer.
The 3-page paper, also featuring related computer vision problems such as assessing the speed and acceleration of a missile based on visual perception, is available as PDF #49, here.
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About the Author
Vincent Granville is a pioneering GenAI scientist and machine learning expert, co-founder of Data Science Central (acquired by a publicly traded company in 2020), Chief AI Scientist at MLTechniques.com and GenAItechLab.com, former VC-funded executive, author (Elsevier) and patent owner — one related to LLM. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Follow Vincent on LinkedIn.