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Interesting contrast to this is that most modern vector 2D graphics API use PostScript/PDF drawing model that cannot represent exact circles and elipses and the

by dfox 2y ago

Interesting contrast to this is that most modern vector 2D graphics API use PostScript/PDF drawing model that cannot represent exact circles and elipses and these are instead interpolated by a bunch of bezier curve segments.

fluoridation 2y ago
I don't think that's true. Drawing Bezier curves is not any easier than drawing ellipses, and in fact I'm pretty sure it's even more difficult. If someone wants to approximate an ellipse the easiest solution is to just draw a bunch of short segments.
brulard 2y ago
Can you provide examples? I have not seen an API not to support circles and arcs. SVG is AFAIK the most common API, you have there circles, arcs, ellipsis, quadratic and cubic beziers etc.
- LegionMammal978 2y ago
  At least the PostScript variant used in PDF 1.7 only supports straight lines and cubic Bézier curves as path segments [0]. You may be able to hack together true circular disks using round line caps and joins. [0] https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandards/PDF32000_2008.pdf#page=140 https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandard..., sec. 8.5.2, "Path Construction Operators"
- shaftway 2y ago
  This is how SVG and most other drawing programs are implemented under the hood. Down at the lowest level the engine can only really do straight lines. You may ask for a circle or a bezier, but the engine only approximates this with enough fidelity that you wouldn't notice the flaws. Beziers end up being approximated as a bunch of tiny line segments. Circles and ellipses can be approximated pretty easily as a handful of beziers. Beziers are particularly useful because they have some handy properties. You don't need to do any trig to calculate them (like you do for arcs). They're extremely simple to compute, and any bezier curve can be split into two smaller bezier curves. That last fact makes it easier to have varying fidelity along the curve. Source: was on the original SVG Working Group team @ Adobe.
  - shaftway 2y ago
    FWIW, here's the bezier algorithm in Python: def bezier(t, *handles): """Computes a single point of a bezier of any order where 0 <= t <= 1.""" if len(handles) == 1: return handles[0] else: pairs = zip(handles[:-1], handles[1:]) return bezier(t, *[_bezier_interpolate(t, a, b) for a, b in pairs]) def _bezier_interpolate(t, a, b): return [a * (1-t) + b * t for a, b in zip(a, b)] assert bezier(0.25, [1, 1, 1], [2, 2, 2], [3, 4, 5]) == [1.5, 1.5625, 1.625] You can pass it any order of bezier (cubic, quadratic, etc.) for any number of dimensions, and how for along the interpolation you are. It will give you a single point. Connect a bunch together to get your curve. You can actually stop when `len(handles) == 2` and you get line segments instead of points, but it's slightly less accurate. You can see the math for computing a point (or segment) is trivial (addition and multiplication) and it can be easily parallelized. It can even be a one-liner, but that isn't very readable. Here's a good website for visualizing the math: https://www.summbit.com/blog/bezier-curve-guide/ https://www.summbit.com/blog/bezier-curve-guide/
ttd 2y ago
Bezier curves (the regular kind) cannot represent circles and ellipses, this is true. However, rational Bezier curves can [1]. I don't know if PostScript has a rational Bezier primitive, but some drawing libraries do. A prominent example of a library that does is Skia, the rendering engine behind Chrome and Android. [1] - e.g. https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B%C3%A9zier_curves https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B%C...

by dfox 2y ago

fluoridation 2y ago

I don't think that's true. Drawing Bezier curves is not any easier than drawing ellipses, and in fact I'm pretty sure it's even more difficult. If someone wants to approximate an ellipse the easiest solution is to just draw a bunch of short segments.

brulard 2y ago

Can you provide examples? I have not seen an API not to support circles and arcs. SVG is AFAIK the most common API, you have there circles, arcs, ellipsis, quadratic and cubic beziers etc.

LegionMammal978 2y ago
At least the PostScript variant used in PDF 1.7 only supports straight lines and cubic Bézier curves as path segments [0]. You may be able to hack together true circular disks using round line caps and joins. [0] https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandards/PDF32000_2008.pdf#page=140 https://opensource.adobe.com/dc-acrobat-sdk-docs/pdfstandard..., sec. 8.5.2, "Path Construction Operators"
shaftway 2y ago
This is how SVG and most other drawing programs are implemented under the hood. Down at the lowest level the engine can only really do straight lines. You may ask for a circle or a bezier, but the engine only approximates this with enough fidelity that you wouldn't notice the flaws. Beziers end up being approximated as a bunch of tiny line segments. Circles and ellipses can be approximated pretty easily as a handful of beziers. Beziers are particularly useful because they have some handy properties. You don't need to do any trig to calculate them (like you do for arcs). They're extremely simple to compute, and any bezier curve can be split into two smaller bezier curves. That last fact makes it easier to have varying fidelity along the curve. Source: was on the original SVG Working Group team @ Adobe.
- shaftway 2y ago
  FWIW, here's the bezier algorithm in Python: def bezier(t, *handles): """Computes a single point of a bezier of any order where 0 <= t <= 1.""" if len(handles) == 1: return handles[0] else: pairs = zip(handles[:-1], handles[1:]) return bezier(t, *[_bezier_interpolate(t, a, b) for a, b in pairs]) def _bezier_interpolate(t, a, b): return [a * (1-t) + b * t for a, b in zip(a, b)] assert bezier(0.25, [1, 1, 1], [2, 2, 2], [3, 4, 5]) == [1.5, 1.5625, 1.625] You can pass it any order of bezier (cubic, quadratic, etc.) for any number of dimensions, and how for along the interpolation you are. It will give you a single point. Connect a bunch together to get your curve. You can actually stop when `len(handles) == 2` and you get line segments instead of points, but it's slightly less accurate. You can see the math for computing a point (or segment) is trivial (addition and multiplication) and it can be easily parallelized. It can even be a one-liner, but that isn't very readable. Here's a good website for visualizing the math: https://www.summbit.com/blog/bezier-curve-guide/ https://www.summbit.com/blog/bezier-curve-guide/

ttd 2y ago

Bezier curves (the regular kind) cannot represent circles and ellipses, this is true. However, rational Bezier curves can [1]. I don't know if PostScript has a rational Bezier primitive, but some drawing libraries do. A prominent example of a library that does is Skia, the rendering engine behind Chrome and Android. [1] - e.g. https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B%C3%A9zier_curves https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Rational_B%C...