Comment by mrb | Hacker News Reader

Comment by mrb | Hacker News Reader

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It doesn't matter. The error is 0.03% only with one of the most simplistic approximations of 1 Bézier curve per quarter circle. Bump this to 2 or 4 curves, and the error drops to 4.0e-6 or 6.2e-8 (https://pomax.github.io/bezierinfo/#circles_cubic)

You are going to have a really hard time finding an industrial application where a 6.2e-8 error matters.

In fact, check this out: the Epilog Fusion M2 40 laser machine that the author chooses to picture as the problem instance can engrave a circle of at most 28-inch diameter, and has a resolution of 1200 dpi. Therefore two (gasp!) Bézier curves per quarter circle offers far more precision (4.0e-6) than this machine's error (1/1200/28 = 3.0e-5).

Edit: or take the world's roundest object: a sphere made out of a single crystal of silicon-28 atoms, with a roundness delta of less than 50 nanometres over a 93.6 mm diameter (https://www.heason.com/news-media/technical-blog-archive/wor...) This is an error of 5.3e-7. Four puny Bézier curves per quarter circle (6.2e-8) are more perfect than this sphere.

That's the answer to a different question. I was just pointing out that, in most industrial applications, 0.03% tolerances are rarely seen but in the specific question of circular dimensions, it's a pretty common requirement.

And while it's possible to approximate with higher bezier curves, PDFs are auto generated with software libraries and it's going to be non-trivial to dive into the guts of a library of a library of a library and tweak the logic for generating circles for this one specific type of application.

When we're talking about a CO2 laser, the errors from taper, variations in laser power, rigidity, trueness of optical path, etc, etc, far exceed 0.03%.

If you want a circle that's better than a 0.03% tolerance, odds are you're not making it by tracing a circular path in g-code, but instead using something like a lathe.

> And while it's possible to approximate with higher bezier curves, PDFs are auto generated with software libraries and it's going to be non-trivial to dive into the guts of a library of a library of a library and tweak the logic for generating circles for this one specific type of application.

A lot of work on CO2 lasers you do with something like Coreldraw or Inkscape. You absolutely can pick the number of vertices with some slight work.

Lasers are a bit weird in that most of their work is for signmaking. Most end-user use cases are closer to a printer than that of a typical CNC machine. Indeed, it can be annoying to take something from CAD to lasers that want PDF, because you can end up with things like the resultant PDF not joining lines together properly into one shape and the laser "stopping" on each vertex. You end up pretty carefully picking tools and processes for this flow. Ideally you figure out how to give the laser a dxf...

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