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Circles do not exist

nibblestew.blogspot.com

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Re: Circles do not exist

#81

Post by raphlinus »

nine_k wrote: post #57
neilk wrote: post #20

If you use the standard way to approximate a circle with Bezier curves, it will differ from a true circle by about 0.03%. The human eye is not capable of seeing the difference, and, as the OP mentioned, most people can't accurately eyeball a perfect circle anyway. This post shows the math behind that, and then shows that it could be made even better with a few different choices. https://spencermortensen.com/articles/…

If a Bézier-based circles with four nodes are too rough, I suppose a circle with 16 or 24 or 128 nodes can be made smooth enough to exceed the laser cutter's margin of error. If you want to produce a highly regular circle in Inkscape, draw a polygon with many nodes (say, 24), convert it to path, then make all nodes smooth and symmetric.

The actual formula is O(n^6) scaling, and for circles it's very easy to achieve that. So a circle divided into 16ths has 4096 times less error than one in quarters and one made of 128 nodes is a (binary) billion times better.

This is true for curve fitting of arbitrary curves, not just circles, though arbitrary curve fitting is a bit tricky (it requires solving quartic equations). I personally think you can take this as evidence that cubic Béziers are "good enough" for all practical tasks requiring curve representation, though for specific applications there might be other representations that work better.

Re: Circles do not exist

#82

Post by susam »

The first programming language I learnt to program in was Logo, namely IBM/LCSI PC Logo. It too did not have any command to draw curves or circles. However, we were taught to draw a "circle" using code like this:

  REPEAT 360 [FD 1 RT 1]
But alas, circles did not exist! It is easy to realise that the above code does not attempt to draw a circle. These instructions specify a triacosiahexeacontagon instead.

Screenshot: https://susam.net/blog/fd-100.html#circles

Re: Circles do not exist

#83

Post by itishappy »

photonerd wrote: post #73
itishappy wrote: post #53

Earlier quoted context omitted.

Epilog claims (up to) 1200dpi resolution. Another comment mentioned the "standard bezier curve representation" is accurate to 0.03%. This means errors should start being observable with a circle as small as 3in. These are probably wildly optimistic assumptions, but it's still 10x smaller than the bed of their largest unit, and there's nothing to stop people from creating arcs of circles with radii even bigger than th…

Observable is pushing it. Technically observable, sure, but ~1dpi misplaced is… unlikely to be noticeable. Plus, if that level of accuracy is required simply adding more nodes improves accuracy exponentially. Not least, the “standard” approximation really should be updated in software, as modern 4-node approximations are down to 0.005%, almost an order of magnitude better.

Oh, I completely agree! "Observable" is pushing the limits of the word past absurdity. 1200dpi would put the resolution of the system at just twice the wavelength of their laser. I feel like that's about the limit of what's possible due to physics [citation needed]. I'm fairly confident their actual performance is less.

Re: Circles do not exist

#84

Post by namru1620 »

waffletower wrote: post #71
pkulak wrote: post #64

Earlier quoted context omitted.

That statement is just one of a million in the "calculus does not exist" realm, and is not nearly as interesting as this article was. EDIT: apologies for the tone; came off very snide on second read.

I don't think the statement attempted to invalidate the article, there are many gems inside, such as the idea that human perception elongates the vertical axis, etc.

You wouldn't happen to have a link handy about that vertical axis perception thing would you, I l'm super curious about it.

Re: Circles do not exist

#85

Post by Animats »

crote wrote: post #74
Animats wrote: post #12

PDF for CAD is silly. The problem is that laser cutters tend to take 2D vector image data as input, rather than G-Code like real CNC machines. You can definitely express a circular path in G-Code, and you expect a milling machine to cut one accurate to the precision of the mechanical parts of the machine. But laser cutter software is to some extent derived from printer technology. You can raster scan a picture onto a…

DXF seems to be reasonably common for laser cutters. It at least supports arcs! I don't think many laser cutters use raster scanning, though. That's going to leave some really nasty artifacts. You really want to cut continuous lines like, well, lines.

Laser engraving is usually raster scan, and most laser cutters can do both line cutting and raster scanning.

Re: Circles do not exist

#86

Post by A_D_E_P_T »

waffletower wrote: post #61

I thought the article would go further and posit that Circles do not exist in reality as they are approximations of a mathematical ideal. But the author does present the raster image as a bonafide circle, rather than the approximation that it truly is -- more resolved than others or not. A zoomed in excerpt of the author's rasterized circle approximation clearly shows that scale is key in perception of reified ideals…

That's true in a sense, but you've got to go way down. At a fine enough level --- say, at the level of atoms --- all material things are granular, and circular or spherical things are built stepwise from material building blocks, as though from pixels. Thus even very "smooth" things are rough and granular.

Further: There are no circular (2D) molecules; nature prefers hexagons and other polygons for that. Even the proton isn't perfectly spherical. Whether the electron and other subatomic particles are perfectly spherical cannot, as yet, be determined. (The electron seemingly has a diameter far below the Planck Length.)

There are no very large circles or spheres, either. All astronomical bodies are somewhat flattened.

So I dare say perfect circles and spheres do not really exist, save as idealized mathematical objects.

Re: Circles do not exist

#87

Post by Nevermark »

anamexis wrote: post #62
klysm wrote: post #51

Earlier quoted context omitted.

Using a PDF where 0.03% matters is insane no?

Yes, that's the point. 1. PDF is to become a standard CAD interchange format. 2. Using PDF where 0.03% matters is insane. 3. If CAD requires 0.03% tolerance, then using PDF as a standard CAD interchange format is insane. I would imagine you generally want arbitrary precision for CAD.

Absolutely!

I cannot imagine a CAD system that can't symbolically represent what is actually desired. The model and its spec should be one and the same.

Approximations also open the door to different precision versions. Not helpful when different parts from different sources need to be consistent.

Models that approximate/vary from the ideal spec, created intentionally to match manufacturing limitations, are ok.

Re: Circles do not exist

#88

Post by hwayne »

nine_k wrote: post #57
neilk wrote: post #20

If you use the standard way to approximate a circle with Bezier curves, it will differ from a true circle by about 0.03%. The human eye is not capable of seeing the difference, and, as the OP mentioned, most people can't accurately eyeball a perfect circle anyway. This post shows the math behind that, and then shows that it could be made even better with a few different choices. https://spencermortensen.com/articles/…

If a Bézier-based circles with four nodes are too rough, I suppose a circle with 16 or 24 or 128 nodes can be made smooth enough to exceed the laser cutter's margin of error. If you want to produce a highly regular circle in Inkscape, draw a polygon with many nodes (say, 24), convert it to path, then make all nodes smooth and symmetric.

Or just use the ellipse (E) in Inkscape, set the start/end to 0, and hold down ctrl. Then inkscape will generate a `` svg element.

Re: Circles do not exist

#89

Post by ks2048 »

Article says “Furthermore, the human vision system is a bit weird and perfect circles look vertically elongated. You have to make them non-circular for people to consider them properly circular.”

Anyone have a reference for this?

Re: Circles do not exist

#90

Post by karaterobot »

> The only way to create a proper circle is to have a raster image like the one above.

The raster image won't be a circle, it'll be a circle projected onto a square or rectangular grid. But, just like with the Bezier curves, nobody will notice. I remember the first week of Geometry, they told us that the drawings of squares and circles and whatnot printed in the textbook are not intended to be accurate, so don't bother measuring them to get the answer. The circle represents a circle, it is not a circle.

Wait til I tell you about this painting of a French pipe I saw one time.

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