Theory and Modern Applications
Table 6 Control points and errors for Q-Bézier curve of degree eight to degree quartic (\(\lambda _{1}=1\), \(\lambda _{2}=0\), \(\lambda _{3}=0\), \(\lambda _{4}=1\))
From: Approximate multi-degree reduction of Q-Bézier curves via generalized Bernstein polynomial functions
Constraint condition | Control points | Error |
|---|---|---|
Under unrestricted condition | \(\mathrm{P}_{0} = ( - 4.9836,0.03464)\), \(\mathrm{P}_{1} = ( - 10.505,5.127)\), \(\mathrm{P}_{2} = (5.9596,16.49)\), \(\mathrm{P}_{3} = (22.447,5.298)\), \(\mathrm{P}_{4} = (16.991,0.01223)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.5008 \times 10^{ - 4}\) |
Under \(C^{0}\) constraint condition | \(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 10.487,5.1792)\), \(\mathrm{P}_{2} = (5.9488,16.427)\), \(\mathrm{P}_{3} = (22.443,5.3304)\), \(\mathrm{P}_{4} = (17,0)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.1153 \times 10^{ - 3}\) |
Under \(C^{1}\) constraint condition | \(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 10.4,5.4)\), \(\mathrm{P}_{2} = (5.9059,16.149)\), \(\mathrm{P}_{3} = (22.4,5.4)\), \(\mathrm{P}_{4} = (17,0)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.1863 \times 10^{ - 2}\) |