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> The only way to create a proper circle is to have a raster image like the one above. facepalm. A raster image can't express a circle either - it's also just

> The only way to create a proper circle is to have a raster image like the one above. facepalm. A raster image can't express a circle either - it's also just an approximation. Same thing with a printout, for that matter. Maybe you could claim some kind of radial plotter is a true circle, but even there we're talking about pigment being absorbed along individual paper fibers. In reality, all the mathematical shapes are just abstractions to be approximated.

ars 3y ago
You completely missed his point. Pick a DPI, make a raster circle - it's accurate to the DPI you pick. Now use a Bezier to make that circle (and rasterize it to that DPI). It's not accurate to that DPI.
- mindslight 3y ago
  I don't see the conceptual difference between approximating a circle as a set of raster points, and approximating a circle as a Bezier curve. Also, it was the author's choice to frame the article in edgy universal terms "Circles do not exist" rather than something like a more pedestrian "Circles don't exist in PDF".
  - ars 3y ago
    > I don't see the conceptual difference between approximating a circle as a set of raster points, and approximating a circle as a Bezier curve. One approximates the circle to the limit of the chosen resultion, the other simply has an error. A small error, but an error nontheless.
    - mindslight 3y ago
      Neither rectangular pixels nor Bezier curves are bases capable of perfectly/natively representing a circle. Either one can approximate a circle to within some arbitrarily chosen amount of error, based on increasing the number of pixels or the number of Bezier curve segments. For a given circle and DPI, one can figure out the minimum number of Bezier curves necessary to represent the circle such that rendering those Bezier curves out to pixels will get you the exact same result as having rendered the circle in pixels directly.
deaddodo 3y ago
The article is arguing mathematical inaccuracy, you're arguing the limitations of compressing an infinite abstract concept on a finite field. You're not wrong, but it's like someone showing proof that a person can't fly unassisted and then responding with "well dogs can't meow".
- mindslight 3y ago
  But the mathematical inaccuracy is caused by the limitations of compressing an abstract concept on a finite field (of a different base abstraction). So no, it doesn't seem like your wacky analogy at all.
  - deaddodo 3y ago
    No, the Bézier inaccuracy has nothing to do with rasterization. Did you even read the article? They're saying that your arc for your circle will be incorrect before it's ever put into a concrete implementation. In other words, there's a base level inaccuracy and then your rasterization brings a whole new, different inaccuracy into play. So yes, my "wacky analogy" is exactly apropos here.
    - mindslight 3y ago
      You seem to be asserting some categorical difference between Bezier curves and pixels, likely because most output devices work in terms of pixels. But they're actually both just drawing primitives, and neither one is capable of representing perfect circles. > the Bézier inaccuracy has nothing to do with rasterization Obviously, because rasterization refers to pixels and has nothing to do with Bezier curves. On the other hand, that inaccuracy has everything to do with Bezierization. > there's a base level inaccuracy and then your rasterization brings a whole new, different inaccuracy into play. Yes, stacking two inexact transformations will result in more inaccuracy than a single one. Since you seem to be attached to your analogy - it's more like a person making the argument that humans can't fly unassisted, while showing a picture of a pig as an example of flying, and I'm pointing out that pigs can't fly either.

ars 3y ago

You completely missed his point. Pick a DPI, make a raster circle - it's accurate to the DPI you pick. Now use a Bezier to make that circle (and rasterize it to that DPI). It's not accurate to that DPI.

mindslight 3y ago
I don't see the conceptual difference between approximating a circle as a set of raster points, and approximating a circle as a Bezier curve. Also, it was the author's choice to frame the article in edgy universal terms "Circles do not exist" rather than something like a more pedestrian "Circles don't exist in PDF".
- ars 3y ago
  > I don't see the conceptual difference between approximating a circle as a set of raster points, and approximating a circle as a Bezier curve. One approximates the circle to the limit of the chosen resultion, the other simply has an error. A small error, but an error nontheless.
  - mindslight 3y ago
    Neither rectangular pixels nor Bezier curves are bases capable of perfectly/natively representing a circle. Either one can approximate a circle to within some arbitrarily chosen amount of error, based on increasing the number of pixels or the number of Bezier curve segments. For a given circle and DPI, one can figure out the minimum number of Bezier curves necessary to represent the circle such that rendering those Bezier curves out to pixels will get you the exact same result as having rendered the circle in pixels directly.

deaddodo 3y ago

The article is arguing mathematical inaccuracy, you're arguing the limitations of compressing an infinite abstract concept on a finite field. You're not wrong, but it's like someone showing proof that a person can't fly unassisted and then responding with "well dogs can't meow".

mindslight 3y ago
But the mathematical inaccuracy is caused by the limitations of compressing an abstract concept on a finite field (of a different base abstraction). So no, it doesn't seem like your wacky analogy at all.
- deaddodo 3y ago
  No, the Bézier inaccuracy has nothing to do with rasterization. Did you even read the article? They're saying that your arc for your circle will be incorrect before it's ever put into a concrete implementation. In other words, there's a base level inaccuracy and then your rasterization brings a whole new, different inaccuracy into play. So yes, my "wacky analogy" is exactly apropos here.
  - mindslight 3y ago
    You seem to be asserting some categorical difference between Bezier curves and pixels, likely because most output devices work in terms of pixels. But they're actually both just drawing primitives, and neither one is capable of representing perfect circles. > the Bézier inaccuracy has nothing to do with rasterization Obviously, because rasterization refers to pixels and has nothing to do with Bezier curves. On the other hand, that inaccuracy has everything to do with Bezierization. > there's a base level inaccuracy and then your rasterization brings a whole new, different inaccuracy into play. Yes, stacking two inexact transformations will result in more inaccuracy than a single one. Since you seem to be attached to your analogy - it's more like a person making the argument that humans can't fly unassisted, while showing a picture of a pig as an example of flying, and I'm pointing out that pigs can't fly either.