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I need to place points on a quadratic Bézier curve at length intervals l.

Found a pretty good resource at quadratic Bézier curve length and used it to calculate the length of the entire curve. It covers the theory and the resulting formula is perfect for adoption into a piece of software.

However, doing the reverse - finding the coordinates or t from the path length is not a trivial task.

There is a great Bézier curve primer covering, among several subjects, tracing of curves at equal distances, much like my case. I understood that there is no generic solution with radicals for Bézier curves. However, quadratic ones are just a subset...

Could there be a solution just for the quadratic case, or perhaps a good approximation? For instance, length of the curve, above, the formula was adopted to the point that there were no more integrals. Coders are left to write a function that has only a logarithm and a power of operations. No need for numeric methods.

Is it possible to come up with a formula/function, where instead of t, we pass l and the curve points, and get $$C(l) \quad l\in[0,L]$$ or a good approximation of it? I have only seen Newton solutions for this.

Jean-Claude Arbaut
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Igor Shmukler
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    There are techniques and code in the answers here. They deal with cubic Bézier curves. Quadratics are simpler, of course, but I don’t see any way to take advantage of the simplicity. https://math.stackexchange.com/questions/321293/find-coordinates-of-equidistant-points-in-bezier-curve?noredirect=1&lq=1 – bubba Nov 30 '22 at 22:41
  • Yes, I have seen this. Both lookup tables and Runge-Kutta and covered extensively in a few places. I am looking for an approximation to solve the problem "with radicals." – Igor Shmukler Nov 30 '22 at 23:37
  • Highly unlikely that any simple closed-form formula exists. And formulas are over-rated, anyway. The polyline solution is fast is accurate enough for every application that I’m aware of. – bubba Dec 01 '22 at 02:13
  • If you want arclength formulae, use bi-arc curves or Pythagorean hodograph curves instead of general Bézier curves. – bubba Dec 01 '22 at 02:15

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