Re: [Inkscape-devel] Recent change to ellipse to path conversion method?

The behaviour of the tail_error(B, n) function in build_from_sbasis() in the file sbasis-to-bezier.cpp has been reevaluated. The term tail_error(B, 2) is too large since it is of order 4 in theta, for an arc, while the error currently produced is of order 6 in theta. For this reason the term tail_error(B, 3) was investigated numerically to see how it behaves. By numerical printouts at different theta, it was found that for a unit circle the tail_error(B, 3) term is roughly given by : 0.000022*(theta)^6. This agrees surprisingly well with the formula given above for the case where the sbasis is fit at t = 0.5. The formula given above was: error = (2/27)*(theta/4)^6. The difference between these results is only 20%, which I find surprising given the very different histories of the two calculations. In any event, I would like to propose that the term tail_error(B, 3) be used to determine the error estimate when bisecting the sbasis curves to meet the required tolerance. If this is done, numerical tests indicate that the dividing point in switching from 4 Beziers to 8 Beziers for a full circle will occur at r = 407 instead of the current r = 8.18.