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Comment by quickthrower2 | Hacker News Reader

He didn’t answer the question the first come to my mind: Is the curve from the physical spline a bezier?

No - it's an elastica, which cannot be represented exactly as a bezier. Raph Levien's work is a great read on this https://www.levien.com/phd/thesis.pdf

and

https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-...

IIRC, generally speaking a beam with a transverse point load (think like a diver on the tip of a diving board) deforms into a parabolic curve, and beams with more point loads end up taking higher order polynomial curves. Engineering beam math usually much assumes that the beam is acting under small deflections though.

I don't remember as much about beziers but I think they are also often essentially a series of cubic curves. There are a couple of different types of splines- one of them is controlled locally only by it's local control points (I think that one's beziers) and one requires all control points to control the whole curve.

That's incorrect. That curve is Bernoulli's elastica, which is different from a parabola (though for small deformations they can be a reasonably close approximation).

There is a neat connection between a rolling hyperbola and an elastica though: https://twitter.com/KangarooPhysics/status/14337538508177858...

You should be able to describe (within any reasonable tolerance) the same curve with a bezier, however the control points will obviously be different.

Any imperfections in the physical spline, say a knot if it's wooden, would translate to a slight curvature difference.

Sounds like the physical curves are a subset of bezier? So there might be some bezier curves that cannot be replicated physically

Not really, you can just add more and more control points to a Bézier and it will be possible to describe a curve to a greater accuracy.

However, in reality when using Bézier curves you want to really want to use the minimum number of control point neccesery to ensure a nice smooth curve.

I mean vice versa. Can the physical thing do any CAD Bezier?

A Bézier curve can have a cusp, which you can’t do with a physical spine.

> Not really, you can just add more and more control points to a Bézier and it will be possible to describe a curve to a greater accuracy.

The more control points you add to your curve the more computationally expensive it becomes, though.

This is just the law of spline demand.

Excellently integrated pun!

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