<story><title>Closed form arc length parametrization is impossible for quadratic Bézier curves</title><url>https://ninjakoa.la/curly_curves/posts/quadratic_bezier_arc_length_parametrization/</url></story><parent_chain></parent_chain><comment><author>LiamPowell</author><text>&amp;gt; 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.&lt;p&gt;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&amp;#x27;s not too many mathematicians working in computer graphics so we just end up with numerical solutions to everything.</text></comment>
<story><title>Closed form arc length parametrization is impossible for quadratic Bézier curves</title><url>https://ninjakoa.la/curly_curves/posts/quadratic_bezier_arc_length_parametrization/</url></story><parent_chain></parent_chain><comment><author>magnio</author><text>&amp;gt; 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.&lt;p&gt;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&amp;#x27;)^2), a quartic. Such an integral is well known to be reducible to the elliptic integrals, which have no closed form.</text></comment>