Dan Freve of DMC wins this month’s Geek Challenge by correctly selecting D as the correct combination!
Dan saw that at the maximum radius, 2, Graph A appeared to be zero while graphs B and C were clearly negative. Upon evaluating the formulas, only Formula 2 evaluates to zero at R=2, so 2→A was a necessary match. Of the multiple choice answers, only D had the 2→A paring, so this was the correct answer.
Dan also provided his own sombrero formula:
His Darth Vader-inspired sombrero has two formulas for the inner and outer sections:
Some more about the formulas used in the challenge:
Formula 2, or Graph A, is recognized in an obscure Wikipedia stub as the Sombrero Function (not to be confused with the Mexican Hat Wavelet, which is something else). It is a polar version of the sinc function, and apart from looking like a hat, it apparently has a use in vision processing. I found that compared to the Mexican sombrero I was trying to match it’s outside slope was too flat and the center cone was too wide.
Formula 3, or Graph C, is a simple cosine function, with less than one full period shown. It gave the steep curl at the edge I was looking for, but the center cone is too tall and wide.
Formula 1, or Graph B, is a formula I created to match my inspiration photo as closely as possible. I built it using two variations of a function that looks like a smoothed step function. I used the function 1/(x^n +1), scaled it, and widened it as necessary to get my desired hat shape.