I noticed a submit on X that plotted the perform
(log x)² + (log y)² = 1.
In fact the plot of
x² + y² = 1
is a circle, however I by no means thought what taking logs would do to the form.
Right here’s what the contours appear to be setting the suitable hand facet equal to 1, 2, 3, …, 10.
ContourPlot[Log[x]^2 + Log[y]^2, {x, 0, 10}, {y, 0, 10},
Contours -> Vary[10]]
The darkish blue contour close to the origin jogged my memory of a guitar decide, so I made a decision to take a stab at creating an equation for the form of a guitar decide.
I needed to rotate the picture so the axis of symmetry for the decide is vertical, so I changed x and y with x + y and x − y.
The side ratio was too broad, so I experimented with
log(y + kx)² + log(y − kx)² = r²
the place rising ok will increase the height-to-width ratio. After slightly experimentation I settled on ok = 1.5 and r = 1.
This has a facet ratio of roughly 5:4, which is about what I measured from a photograph of a guitar decide.
Updating: refining the match
After posting this text on X, Paul Graham replied with a photograph of a Fender guitar decide with the picture above overlaid. The match was pretty good, however the side ratio wasn’t fairly proper.
So then I did some research. The form referred to on this submit is called the “351,” however even for the 351 form the side ratio varies barely between picks.
Setting ok = 1.6 provides a greater match to Paul Graham’s decide.
The blue line represents my match utilizing ok = 1.5 and the pink line represents my match utilizing ok = 1.6.
