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Theory and Modern Applications

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

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

Constraint condition

Control points

Error

Under unrestricted condition

\(\mathrm{P}_{0} = ( - 4.9736,0.02432)\), \(\mathrm{P}_{1} = ( - 7.9384,4.08)\), \(\mathrm{P}_{2} = (2.3687,7.887)\), \(\mathrm{P}_{3} = (10.202,3.969)\), \(\mathrm{P}_{4} = (6.971, - 0.03129)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.13928 \times 10^{ - 3}\)

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

\(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 7.9148,4.1)\), \(\mathrm{P}_{2} = (2.3725,7.8947)\), \(\mathrm{P}_{3} = (10.16,3.9215)\), \(\mathrm{P}_{4} = (7,0)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.28678 \times 10^{ - 3}\)

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

\(\mathrm{P}_{0} = ( - 5,0)\), \(\mathrm{P}_{1} = ( - 7.8,4.2)\), \(\mathrm{P}_{2} = (2.3841,7.921)\), \(\mathrm{P}_{3} = (10.0,3.75)\), \(\mathrm{P}_{4} = (7,0)\)

\(d^{2}(\mathrm{r}_{6}^{*}(\theta ),\mathrm{r}_{4}(\theta )) = 0.31312 \times 10^{ - 2}\)