The shape of a guitar pick

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I saw a post on X that plotted the function

(log x)² + (log y)² = 1.

Of course the plot of

x² + y² = 1

is a circle, but I never thought what taking logs would do to the shape.

Here’s what the contours look like setting the right hand side equal to 1, 2, 3, …, 10.

ContourPlot[Log[x]^2 + Log[y]^2, {x, 0, 10}, {y, 0, 10}, 
    Contours -> Range[10]]

The dark blue contour near the origin reminded me of a guitar pick, so I decided to take a stab at creating an equation for the shape of a guitar pick.

I wanted to rotate the image so the axis of symmetry for the pick is vertical, so I replaced x and y with x + y and x − y.

The aspect ratio was too wide, so I experimented with

log(y + kx)² + log(y − kx)² = r²

where increasing k increases the height-to-width ratio. After a little experimentation I settled on k = 1.5 and r = 1.

This has an aspect ratio of roughly 5:4, which is about what I measured from a photo of a guitar pick.

The post The shape of a guitar pick first appeared on John D. Cook.

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