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.

Graph A	Graph B	Graph C

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 (not to be confused with the , which is something else).  It is a polar version of the , 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.