Closed form arc length parametrization is impossible for quadratic Bézier curves

> It is well known in the computer graphics community that the arc length of cubic Bézier curves has no closed form and has to be computed numerically. Sadly, I’ve not yet seen a proof sketch for that, though.

On a somewhat related note: There are of course exceptions to this, such as Pythagorean-Hodograph curves, which do have closed form solutions and would be suitable for a huge number of use-cases. Sadly there's not too many mathematicians working in computer graphics so we just end up with numerical solutions to everything.

The linked article says:

> The arc length of quadratic Bézier curves actually can be computed with a closed form expression.

While indeed true, the article doesn't provide the closed form expression. The curious or unsatisfied reader can find the solution for the 2D case at the top of page 7 of this SIGGRAPH paper:

https://developer.download.nvidia.com/devzone/devcenter/game...

The quadratic function Q(t)=(x,y) is of the monomial form At^2 + Bt + C where A, B, and C are 2D coefficients (see page 5) where A is non-zero.

Simply convert your Bezier quadratic form to monomial form to apply this equation.

This equation still doesn't provide an arc length parameterization, the article's actual focus.

But if you did, say, want to move 26% (or N%, more generally) of the arc length along a quadratic Bezier segment, first compute the total (100%) arc length with the paper's formula (take care doing so as the paper suggests). Then split the Bezier at a halfway guess (try t=0.5). Again use the formula to evaluate the split quadratic. Repeating this in a divide and conquer fashion, you narrow in on the t value very close to 26% (or N%) of the arc length.

2D vector graphics standards expect to dash cubic & quadratic Bezier segments so some practical strategy to provide an arc length parameterization -- even if unavailable in closed form.

> It is well known in the computer graphics community that the arc length of cubic Bézier curves has no closed form and has to be computed numerically. Sadly, I’ve not yet seen a proof sketch for that, though.

A cubic Bezier curve B(t) is a cubic polynomial of t in [0, 1], parameterized by the four control points. Since it is continuously differentiable, its length is the integral from 0 to 1 of the square root of (1 + (B')^2), a quartic. Such an integral is well known to be reducible to the elliptic integrals, which have no closed form.

I love articles that walk a lay person through a mathematical discovery, piece by piece. An incredibly fun read.

Interesting topic!

However, I would like to point out that the dichotomy closed form <-> numerical methods is somewhat artificial. Even if one could express an arc length parametrization using exp and log, one would still need numerical methods to compute exp and log.

This somehow leads to the next question: What kind of functions are suitable to describe the arc length?

I'm not sure how much more "closed form" you need than elliptic integrals.

For another approach, expand sqrt(1 + B'(t)^2) in a Chebyshev series and you're off to the races.

Unless Schanuel’s conjecture is wrong, of course.

Yeah, interesting stuff. So, the whole debate between closed forms and numerical methods is kind of overplayed. Even if you had a closed form for arc length, you’d still need numerical methods to compute exp and log. What functions actually work best for arc length? That’s the real question.

Also, while some folks are hyped about Pythagorean-Hodograph curves, I think they’re kinda niche. Euler spirals seem more practical, even if you have to compute a special function for them. Numerical solutions tend to be more stable anyway, especially in cases where a closed form might break down, like near straight lines.