So I visualized the eight curve on Desmos and just tweaked it till I thought it was right. Turns out, I cut it and it feels great. I'll show you. I've been musing on luthiers who make big instruments and how it boils the whole art down to phallic magnification. I'm excited that this equation can represent the Persian tar as well as a comfortable laptop guitar shape in a diminutive form factor. Here's the desmos link: :it's the top equation, using the z slider to represent the different terraces. The essential LOG involves x squared and hypercubed as well as y in those exponents, but I also added x cubed to make the left side smaller than the right as in the tar.
I've always appreciated small instruments for the same reason I appreciate Miles Davis: you gotta be cool, and be boss. Bring a brass trumpet and summon your players to the jazz; bring a little synth and have a big sound. Although I applaud any friends who lift heavy weights for their art... A boxer does that too, but in the ring he's naked.
The third variable, Z, is the depth of the bit, so it traces a three dimensional, waisted and rounded shape. When I type it into equation solver (and I don't know how it does it) I can solve for y, in a very tight form: My first prototype cut great at the waist, in horizontal or diagonal waters. But the vertical segments should have precise and tight code because they are the end-grain and can hold fine woodwork. So I said, well then, instead of a left to right traversal, I want a spinner to cut the layers of this object. I attempted to substitute r*sin(t) for x and r*cos(t) for y and asked equation solver to solve for r, but the answer seems to go on forever: yield to stump.
One thing I thought of in the shower, was to do the waist in y's, and at the whim of an if..then (the hacking power of code doth insult the purity of math!) statement switch to the x solution and go up and down across the "butt" of the instrument. I conceive of the contour cuts in two separate parts, or regimes. First a planer rough cut, a terracing of the work piece. Then, a smoothing cut goes up and down in the z axis, and also should traverse the perimeter of the piece. It's a triangle wave that works so well from a woodworking standpoint. So I like it, but it is problematic without a polar representation.
I got stumped trying to convert the equation to x and also to polar in radius. Turns out that it was the x cubed term that was screwing up the polar solution, or somehow spinning it out into an infinite series. I poked at removing it, and the equation simplifies well into polar with only the even powers: x and y, squared and hypercubed. The desmos graph without x cubed.
You said an artist wants to control his materials. I said I'm not an artist but you said I am. I said that I just listen to the equations when they want to tell me something. This equation seems to be telling me: release asymmetry, since it solves for radius so well:
BTW all the equation solves are by NUMBER EMPIRE"S EQUATION SOLVER>