Theory and Modern Applications
Table 5 Control points and errors for Q-Bézier curve of degree eight to degree quartic (\(\lambda _{1}=1\), \(\lambda _{2}=-1\), \(\lambda _{3}=0\), \(\lambda _{4}=1\))
From: Approximate multi-degree reduction of Q-Bézier curves via generalized Bernstein polynomial functions
Constraint condition | Control points | Error |
|---|---|---|
Under unrestricted condition | \(\mathrm{P}_{0} = ( - 4.9809,0.02524)\), \(\mathrm{P}_{1} = ( - 10.503,5.207)\), \(\mathrm{P}_{2} = (6.0274,16.48)\), \(\mathrm{P}_{3} = (22.471,5.225)\), \(\mathrm{P}_{4} = (16.987,0.02064)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.13315 \times 10^{ - 2}\) |
Under \(C^{0}\) constraint condition | \(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 10.484,5.2502)\), \(\mathrm{P}_{2} = (6.0191,16.414)\), \(\mathrm{P}_{3} = (22.463,5.2641)\), \(\mathrm{P}_{4} = (17,0)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.11689 \times 10^{ - 3}\) |
Under \(C^{1}\) constraint condition | \(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 10.4,5.4)\), \(\mathrm{P}_{2} = (5.9985,16.14)\), \(\mathrm{P}_{3} = (22.4,5.4)\), \(\mathrm{P}_{4} = (17,0)\) | \(d^{2}(\mathrm{r}_{8}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.13315 \times 10^{ - 2}\) |