Sam Derbyshire

@samderbyshire.bsky.social

38 Followers 54 Following

Cohomology enthusiast. I've seen rings you people wouldn't believe.

A colourful density plot, with a yellow-red-purple-black gradient for high error --> low error. The plot is reddish in colour (error 10^-10), with wispy purple curves where the integrator does better (for no good reason), and two curves whose vicinity is yellow (corresponding to families of cubic Béziers with a cusp). There are also some half-disk-like yellow areas where the integrator performs worse, due to the derivative of the speed function having non-real roots that are close to the real axis for such Béziers.
A collection of plots. Each plot is for a specific cubic Bézier curve (straight line, simple Bézier, Bézier with a cusp, Bézier with a near-cusp, degenerate Bézier). Each plot contains coloured histograms, in which different methods for estimating the arc length are displayed with different colours. Above the x-axis, the histograms show computation time, while below the x-axis they show accuracy of the method, on a log scale. Methods compared: poly-line, Romberg's method, Clenshaw-Curtis, tanh-sinh, Gauss-Legendre and Gravesen. The naive chord-length methods (poly-line and Romberg's method) are predictable but start taking a long time to converge for higher accuracy. Gravesen (which does adaptive refinment) is similarly underwhelming. Clenshaw-Curtis, tanh-sinh, Gauss-Legendre all converge with very high accuracy on simple examples, but struggle in the presence of cusps or near cusps, unless one subdivides at cusps first. With subdivision, tanh-sinh seems to do the best at dealing with difficult Béziers.