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by delta_p_delta_x 1019 days ago
> Béziers can't represent circles. If you try to approximate one by hand, it'll look lopsided.

Many graphics algorithms (vector operations: scale, translate, rotate; raster operations like rasterisation and interpolation) specialise for straight lines (and hence also triangles), Bézier curves, circles, etc separately, rather than forcing everything to be a Bézier curve.
1 comments
slimsag 1019 days ago
"Béziers can't represent circles" is not exactly wrong, but is not really the full picture.

Often you don't need to _represent_ a circle. TrueType fonts (quadratics) and OpenType fonts (cubics) can't mathmatically _represent_ a circle, either, but does that mean no font has circles in it? Sometimes an approximation is pretty damn good.

Also conics (rational quadratic Beziers), which Microsoft has a surprisingly good article on[0], can represent circles.

[0] https://learn.microsoft.com/en-us/xamarin/xamarin-forms/user...

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cormullion 1019 days ago
Fortunately fonts usually don't contain perfect circles anyway, because the designer will optically correct any circles so that they don't appear to bulge. Kabel is the closest font I found with circular formas; LineTo's Circular is also quite close but slightly taller than wider.
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mkl 1018 days ago
Fonts are supposed to contain perfect circles; try searching for "circle" in a Unicode character map. They only don't in practice because quadratic and cubic Bezier splines can't do them.
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TheRealPomax 1018 days ago
And they don't have to. The approximation of a circle with a bunch of cubic curves is so accurate [1] that you'd need to print a huge circle and then sit there with a measuring tape to be able to even tell it's not a perfect circle. Which is why "you can't draw a circle with Bezier curves" is only a mathematical "problem", it's not an actual problem.

[1] https://pomax.github.io/bezierinfo/#circles_cubic

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ad8e 1018 days ago
It's a UI problem, not a mathematical problem. No set of curves can represent everything exactly, but they all can get close. The goal is just to make it easy to get close.

The computer can produce a near-perfect circle with Béziers for each 1/4 arc, and near-perfect circles are usually good enough. It's just inconvenient. It's also hard to manipulate the generated circle; the user might have a clear image of his head of the changed circle he wants, but pushing the control points to get there is not easy.

The mathematical "axioms" in the article are only attempting to provide a clean interface, so that the user can easily translate what is in his head to what the program creates. The circle axiom says, "The user often wants a circle, and also expects a circle, so let's produce the expected circle." There's no other mathematical purity involved.

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