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A theoretical practice question for you: are Bezier curves really important? I'm not at all a "graphic-one", that's for sure, but sometimes used Inkscape (or other software that had Bezier curves among its tools) for simple tasks and I never felt the need to use them... I'm wondering what am I missing? I mean, what's most common use for them? How would your life be worse without them? And what couldn't be done at all without them? I think you got the point about what I'm asking about, thank you in advance!

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Not really. We could use any number of other parametric curves. However when Adobe defined their specification for printers they chose to use Bézier curves for the job. Since Adobe decided to use Béziers everybody used those for printer output. Since then we have just inherited the choice of Béziers, in PostScript in PDF in SVG etc.

Like I said the choice is pretty arbitrary. Engineers made a different choice and chose rational B-splines instead. But again we could have chosen other basis curves as well but at that time Béziers and B-splines were the top contenders. It has since been shown that they in fact both are reducible to rational B-splines (but not vice versa since the B-spline space is a super set of Béziers). Today we would probably choose some other parametrisation, that would preferably have closed form solutions for length.

Basically we use Béziers to be able to use standard formats.

Do we need parametric curves?

Yes, without parametric curves it would be pretty hard to evolve the surfaces and curves we use in day to day design (be it 2D or 3D). If we simply used a bunch of lines we would have all sorts of problems with booleans and manipulation of curves.

Splines make forming of curves easy even to people who aren't so mathematically oriented.

When do you need one?

Whenever you need something to actually curve. Fonts would be much much larger files and such things as modern cars couldn't be designed digitally.

But like i said that curve does not have to be a Bézier. Its just that if you wish to share the data across applications or to your printer then you don't have much choice its either Bézier or rational B-splines. That or then flatting the curve so its useless for re-editing which is fine for output but not archival.

The nice thing of this approach is that even the things you do not think of as Béziers probably are internally Béziers anyway. So you only really have to implement one shape primitive, lines are most likely Beziers. Circles are in many apps, although you can not make exact circles with Béziers, but no one would notice (and this is why engineers use rational splines instead). So, in fact, you're probably using Bézier curves all the time.

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