error estimation

The method used to estimate the error of cubic Bézier is exactly the same as the one explained in section 3.3.2 for quadratic Bézier curves. The coefficients are given by tables 3 and 4.

As before, we introduce a safety rational fraction $s(b/a)$ to obtain an upper bound of the error: $\varepsilon_3\le s(b/a)\tilde{\varepsilon}_3$.

This bound correctly overestimates the error, but is not too conservative, the cumulative distribution function of $\varepsilon/(s(b/a)\tilde{\varepsilon})$ is quite smooth and the corresponding mean value is .

Table 3: error coefficients for cubic Bézier $(0 < b/a < 1/4)$

$i$	$j$	$\mu_{i,j,0}$	$\mu_{i,j,1}$	$\mu_{i,j,2}$	$\mu_{i,j,3}$
0	0	3.85268	-21.229	-0.330434	0.0127842
0	1	-1.61486	0.706564	0.225945	0.263682
0	2	-0.910164	0.388383	0.00551445	0.00671814
0	3	-0.630184	0.192402	0.0098871	0.0102527
1	0	-0.162211	9.94329	0.13723	0.0124084
1	1	-0.253135	0.00187735	0.0230286	0.01264
1	2	-0.0695069	-0.0437594	0.0120636	0.0163087
1	3	-0.0328856	-0.00926032	-0.00173573	0.00527385

Table 4: error coefficients for cubic Bézier $(1/4 \le b/a \le 1)$

$i$	$j$	$\mu_{i,j,0}$	$\mu_{i,j,1}$	$\mu_{i,j,2}$	$\mu_{i,j,3}$
0	0	0.0899116	-19.2349	-4.11711	0.183362
0	1	0.138148	-1.45804	1.32044	1.38474
0	2	0.230903	-0.450262	0.219963	0.414038
0	3	0.0590565	-0.101062	0.0430592	0.0204699
1	0	0.0164649	9.89394	0.0919496	0.00760802
1	1	0.0191603	-0.0322058	0.0134667	-0.0825018
1	2	0.0156192	-0.017535	0.00326508	-0.228157
1	3	-0.0236752	0.0405821	-0.0173086	0.176187