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Comment by undershirt | Hacker News Reader

Is this an accurate summary?

Spiro curves are necessarily smooth, but beziers aren't. To show what makes beziers smooth, you created a viz that maps spiro curves to their bezier equivalents:

  *   x,y = spiro curve angles
  * color = bezier handle lengths (to best approximate spiro)

The abrupt changes in color in the image reveal the difficulty in creating smooth bezier curves—because it forces abrupt changes in handle lengths for small angle changes.

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Yes, that's about right. The only thing I'd qualify is that Spiro contains Euler spiral segments but also other things, especially smooth straight-to-curved transitions. This is only about the first, but the others are also important to font design. I wanted to keep it simple though.

I don’t think that’s the right assessment about the abrupt changes in color.

What I think those instead reveal is areas of the parameter space where there is a collapse of a degree of freedom: if your curve is close to a parabola, then you can make many different cubic curves which are close approximations, and it doesn’t much matter which among them you choose.

Raph’s optimizer picks opposite choices for which arm should be the short one when you go across that boundary, but if you smoothed the map out so there were no creases I don’t think you’d get significantly worse results. (I’d be interested to see him try this ;)

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