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Theory and Modern Applications

Table 1 Control points and errors for Q-Bézier curve of degree six to degree quartic (\(\lambda _{1}=1\), \(\lambda _{2}=1\), \(\lambda _{3}=2\), \(\lambda _{4}=0\))

From: Approximate multi-degree reduction of Q-Bézier curves via generalized Bernstein polynomial functions

Constraint condition

Control points

Error

Under unrestricted condition

\(\mathrm{P}_{0} = ( - 5.0207,0.007382)\), \(\mathrm{P}_{1} = ( - 7.6988,4.167)\), \(\mathrm{P}_{2} = (3.0487,8.14)\), \(\mathrm{P}_{3} = (9.8918,3.853)\), \(\mathrm{P}_{4} = (7.0179, - 0.01358)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.50153 \times 10^{ - 4}\)

Under \(C^{0}\) constraint condition

\(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 7.7184,4.1712)\), \(\mathrm{P}_{2} = (3.0507,8.1466)\), \(\mathrm{P}_{3} = (9.9157,3.8302)\), \(\mathrm{P}_{4} = (7,0)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.97175 \times 10^{ - 4}\)

Under \(C^{1}\) constraint condition

\(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 7.8,4.2)\), \(\mathrm{P}_{2} = (3.0613,8.1701)\), \(\mathrm{P}_{3} = (10.0,3.75)\), \(\mathrm{P}_{4} = (7,0)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.8934 \times 10^{ - 3}\)