Convexity-Preserving Scattered Data Interpolation Scheme Using Side-Vertex Method

Abstract

A convexity-preserving interpolation scheme is developed for convex scattered data. The interpolation is carried out by rational side-vertex interpolant. Twenty-four parameters arise in each triangular patch. Constraints are derived on half of the parameters to preserve the convexity of the scattered data. Remaining unconstrained parameters are the free parameters for shape refinement. The proposed scheme is verified graphically with some numerical convex scattered data sets and found ideally suitable for data as well as data with derivatives.

Appendix

$$ \begin{aligned} \tilde{Q}_{4} & = \frac{{P_{2} }}{{\left( {Q_{2d} } \right)^{2} }},\quad P_{2} = \mathop \sum \limits_{k = 0}^{7} \left( {1 - b} \right)^{7 - k} b^{k} P_{2,k} , \\ P_{2,0} & = 2\omega_{5}^{2} \mu_{5} \left( {f_{2} - F\left( {S_{2} } \right)} \right) + 2\omega_{5}^{2} \eta_{5} R_{3} + \left( {2\omega_{5}^{2} v_{5} - 2\omega_{5}^{3} } \right)R_{4} , \\ P_{2,1} & = \left( {6\omega_{5}^{2} \eta_{5} + 8\omega_{5}^{2} \mu_{5} } \right)\left( {f_{2} - F\left( {S_{2} } \right)} \right) + 8\omega_{5}^{2} \eta_{5} R_{3} + \left( {6\omega_{5}^{2} \mu_{5} - 2\omega_{5}^{3} + 2\omega_{5}^{2} v_{5} } \right)R_{4} , \\ P_{2,2} & = 6\omega_{5}^{2} \left\{ {\left( {3\eta_{5} + \mu_{5} } \right)\left( {f_{2} - F\left( {S_{2} } \right)} \right) + 2\left( {\mu_{5} + \eta_{5} } \right)} \right.R_{4} + \left. {\eta_{5} R_{3} } \right\} + 6\omega_{5} \left\{ {\left( {\eta_{5} v_{5} + \mu_{5} v_{5} - \mu_{5}^{2} } \right)} \right. \\ & \quad \times \,\left( {f_{2} - F\left( {S_{2} } \right)} \right) + \eta_{5} v_{5} R_{3} - \left. {\omega_{5} \eta_{5} R_{4} } \right\}, \\ P_{2,3} & = \left( {12\omega_{5}^{2} \eta_{5} } \right. + 2v_{5}^{2} \mu_{5} + 2v_{5}^{2} \eta_{5} + 6\omega_{5} v_{5} \mu_{5} + 22\omega_{5} v_{5} \eta_{5} - 8\omega_{5} \mu_{5}^{2} - 16\mu_{5} \omega_{5} \eta_{5} \\ & \quad \left. { - \,2v_{5} \mu_{5}^{2} } \right)\left( {f_{2} - F\left( {S_{2} } \right)} \right) + \left( {6\mu_{5} \omega_{5}^{2} + 26\eta_{5} \omega_{5}^{2} + 2} \right.v_{5} \omega_{5} \mu_{5} + 2\omega_{5} v_{5} \eta_{5} \left. { - 2\omega_{5} \mu_{5}^{2} } \right)R_{4} \\ & \quad + \,\left( { - 14\omega_{5} \eta_{5}^{2} + 6\omega_{5} v_{5} \eta_{5} - 8\mu_{5} \omega_{5} \eta_{5} - 2\mu_{5} v_{5} \eta_{5} + 2v_{5}^{2} \eta_{5} } \right)R_{3} , \\ P_{2,4} & = \left( {12\eta_{5}^{2} \omega_{5} } \right. + 2\mu_{5}^{2} \omega_{5} + 2\mu_{5}^{2} v_{5} + 6\eta_{5} v_{5} \mu_{5} + 22\mu_{5} \omega_{5} \eta_{5} - 8\eta_{5} v_{5}^{2} - 16v_{5} \omega_{5} \eta_{5} \\ & \quad \left. { - \,2\mu_{5} v_{5}^{2} } \right)\left( {F\left( {S_{2} } \right) - f_{2} } \right) + \left( { - 6v_{5} \eta_{5}^{2} - 26\omega_{5} \eta_{5}^{2} - 2} \right.\eta_{5} \omega_{5} \mu_{5} - 2\omega_{5} \mu_{5} \eta_{5} \left. { + 2\eta_{5} v_{5}^{2} } \right)R_{3} \\ & \quad + \,\left( {14\eta_{5} \omega_{5}^{2} + 2\omega_{5} v_{5} \mu_{5} + 8v_{5} \omega_{5} \eta_{5} - 6\mu_{5} \omega_{5} \eta_{5} - 2\mu_{5}^{2} \omega_{5} } \right)R_{4} , \\ P_{2,5} & = 6\eta_{5}^{2} \left\{ {\left( {3\omega_{5} + v_{5} } \right)\left( {F\left( {S_{2} } \right) - f_{2} } \right) - \left( {\omega_{5} + 2v_{5} } \right)} \right.R_{3} - 2\left. {\omega_{5} R_{4} } \right\} + 6\eta_{5} \left\{ {\omega_{5} v_{5} R_{4} - \omega_{5} \mu_{5} R_{3} } \right. \\ & \quad \left. { + \,\left( {\omega_{5} \mu_{5} + \mu_{5} v_{5} - v_{5}^{2} } \right)\left( {F\left( {S_{2} } \right) - f_{2} } \right)} \right\}, \\ P_{2,6} & = \left( {6\eta_{5}^{2} \omega_{5} + 8\eta_{5}^{2} v_{5} } \right)\left( {F\left( {S_{2} } \right) - f_{2} } \right) - 8\eta_{5}^{2} \omega_{5} R_{4} + \left( { - 2\eta_{5}^{2} \mu_{5} + 2\eta_{5}^{3} - 6\eta_{5}^{2} v_{5} } \right)R_{3} , \\ P_{2,7} & = 2\eta_{5}^{2} v_{5} \left( {F\left( {S_{2} } \right) - f_{2} } \right) - 2\eta_{5}^{2} \omega_{5} R_{4} + \left( {2\eta_{5}^{3} - 2\eta_{5}^{2} \mu_{5} } \right)R_{3} ,\, \\ \end{aligned} $$

$$ \tilde{Q}_{5} = \frac{{\mathop \sum \nolimits_{k = 0}^{7} c^{7 - k} a^{k} \tilde{\beta }_{k} }}{{\left( {\eta_{2} c^{3} + \mu_{2} c^{2} a + v_{2} ca^{2} + \omega_{2} a^{3} } \right)^{3} }}, $$

$$ \begin{aligned} \tilde{\beta }_{0} & = \varLambda_{1}^{2} \left\{ {2\eta_{2}^{2} v_{2} \left( {f_{1} - f_{3} } \right) - 2\eta_{2}^{2} \omega_{2} d_{6} - \left. {2\eta_{2}^{2} \mu_{2} d_{5} } \right\} - 2\varLambda_{3} \varLambda_{1} } \right.\eta_{2}^{3} d_{5} , \\ \tilde{\beta }_{1} & = 2\eta_{2}^{3} \varLambda_{3}^{2} d_{5} + \varLambda_{1}^{2} \left\{ {6\eta_{2}^{2} \omega_{2} \left( {f_{1} - f_{3} } \right) - } \right.\left. {6\eta_{2}^{2} v_{2} d_{5} } \right\} + 2\varLambda_{3} \varLambda_{1} \left\{ {4\eta_{2}^{2} v_{2} \left( {f_{3} - f_{1} } \right) + 4\eta_{2}^{2} \omega_{2} d_{6} + \eta_{2}^{2} \mu_{2} d_{5} } \right\}, \\ \tilde{\beta }_{2} & = 6\eta_{2}^{2} \varLambda_{1}^{2} \left\{ {\eta_{2} \left( {f_{1} - f_{3} } \right) - \omega_{2} d_{6} } \right\} + 6\varLambda_{1}^{2} \left\{ {\eta_{2} \left( {\omega_{2} \mu_{2} - v_{2}^{2} } \right)\left( {f_{1} - f_{1} } \right) - 2\eta_{2}^{2} \omega_{2} d_{5} + \eta_{2} v_{2} \omega_{2} d_{6} } \right\} \\ & \quad + \,2\varLambda_{3} \varLambda_{1} \left\{ {\left( {3\eta_{2} \mu_{2} v_{2} - 9\eta_{2}^{2} \omega_{2} } \right)\left( {f_{3} - f_{1} } \right)\left. { + 3\eta_{2} \mu_{2} \omega_{2} d_{6} + 9\eta_{2}^{2} v_{2} d_{5} } \right\},} \right. \\ \tilde{\beta }_{3} & = 6\varLambda_{3}^{2} \left\{ {\left( {\eta_{2} \mu_{2} v_{2} + 2\eta_{2}^{2} \omega_{2} } \right)\left( {f_{1} - f_{3} } \right)\left. { - \eta_{2} \mu_{2} \omega_{2} d_{6} - \eta_{2}^{2} v_{2} d_{5} } \right\}} \right. + \varLambda_{1}^{2} \left\{ {\left( {2\omega_{2} \mu_{2}^{2} } \right.} \right. - 26\eta_{2} v_{2} \omega_{2} \\ & \quad \left. { - \,2\mu_{2} v_{2}^{2} } \right)\left( {f_{1} - f_{3} } \right) + 2\left( {\eta_{2} v_{2}^{2} - \eta_{2} \mu_{2} \omega_{2} } \right)d_{5} + \left( {2\mu_{2} v_{2} \omega_{2} + 24\eta_{2} \omega_{2}^{2} } \right)\left. {d_{6} } \right\} + 2\varLambda_{3} \varLambda_{1} \\ & \quad \times \,\left\{ {\left( {11\eta_{2} \mu_{2} \omega_{2} + v_{2} \mu_{2}^{2} - 4\eta_{2} v_{2}^{2} } \right)\left( {f_{1} - f_{3} } \right) + \left( {13\omega_{2} \eta_{2}^{2} + \eta_{2} \mu_{2} v_{2} } \right)} \right.d_{5} \left. { - 4\eta_{2} v_{2} \omega_{2} d_{6} } \right\}, \\ \tilde{\beta }_{4} & = \varLambda_{3}^{2} \left\{ {\left( {16\eta_{2} \mu_{2} \omega_{2} + 2\mu_{2}^{2} v_{2} } \right)\left( {f_{1} - f_{3} } \right) - \left( {14\omega_{2} \eta_{2}^{2} + 2\eta_{2} \mu_{2} v_{2} } \right)d_{5} + 2\left. {\omega_{2} \left( {v_{2} \eta_{2} - \mu_{2}^{2} } \right)d_{6} } \right\}} \right. \\ & \quad + \,\varLambda_{1}^{2} \left\{ {\left( {6v_{2} \mu_{2} \omega_{2} - 12\omega_{2}^{2} \eta_{2} } \right)\left( {f_{3} - f_{1} } \right) - 14\omega_{2}^{2} \mu_{2} d_{6} + 6\left. {\omega_{2} v_{2} \eta_{2} d_{5} } \right\}} \right. + 2\varLambda_{3} \varLambda_{1} \left\{ { - \left( {13\eta_{2} } \right.} \right.\omega_{2}^{2} \\ & \quad \left. { + \,v_{2} \mu_{2} \omega_{2} } \right)d_{6} + \left( {4\eta_{2} \mu_{2} \omega_{2} - v_{2}^{2} \eta_{2} } \right)d_{5} \left. { + \left( {4\mu_{2}^{2} \omega_{2} - \mu_{2} v_{2}^{2} - 11\omega_{2} v_{2} \eta_{2} } \right)\left( {f_{3} - f_{1} } \right)} \right\}, \\ \tilde{\beta }_{5} & = 6\varLambda_{3}^{2} \left\{ {\left( {\eta_{2} v_{2} \omega_{2} - \mu_{2}^{2} \omega_{2} } \right)\left( {f_{3} - f_{1} } \right) + 2\eta_{2} \omega_{2}^{2} d_{6} \left. { - \eta_{2} \mu_{2} \omega_{2} d_{5} } \right\} + 6\varLambda_{1}^{2} \omega_{2}^{2} \left\{ {\mu_{2} \left( {f_{3} - f_{1} } \right)} \right.} \right. \\ & \quad \left. { + \,\eta_{2} d_{5} } \right\} + 2\varLambda_{3} \varLambda_{1} \left\{ {\left( {9\eta_{2} \omega_{2}^{2} + 3v_{2} \mu_{2} \omega_{2} } \right)\left( {f_{1} - f_{3} } \right) - 6\omega_{2}^{2} \mu_{2} d_{6} - 3\eta_{2} v_{2} \omega_{2} d_{5} } \right\}, \\ \tilde{\beta }_{6} & = 6\varLambda_{3}^{2} \omega_{2}^{2} \left\{ {\eta_{2} \left( {f_{3} - f_{1} } \right)\left. { - \mu_{2} d_{6} } \right\} - 2\varLambda_{1}^{2} \omega_{2}^{3} d_{6} + 2\varLambda_{3} \varLambda_{1} \left\{ {4\mu_{2} \omega_{2}^{2} \left( {f_{1} - f_{3} } \right) - v_{2} \omega_{2}^{2} d_{6} - 4\eta_{2} \omega_{2}^{2} d_{5} } \right\},} \right. \\ \tilde{\beta }_{7} & = 2\varLambda_{3}^{2} \omega_{2}^{2} \left\{ {\mu_{2} \left( {f_{3} - f_{1} } \right)\left. { + \eta_{2} d_{5} + v_{2} d_{6} } \right\} + 2\varLambda_{3} \varLambda_{1} \omega_{2}^{3} d_{6} } \right., \\ \end{aligned} $$

$$ \tilde{Q}_{6} = \frac{{\mathop \sum \nolimits_{k = 0}^{5} c^{5 - k} a^{k} \beta_{k} }}{{\left( {\eta_{2} c^{3} + \mu_{2} c^{2} a + v_{2} ca^{2} + \omega_{2} a^{3} } \right)^{2} }}, $$

$$ \begin{aligned} \beta_{0} & = - \varLambda_{3} \varLambda_{2} \eta_{2}^{2} d_{5} , \\ \beta_{1} & = - 2\varLambda_{3} \varLambda_{2} \eta_{2} v_{2} \left( {f_{1} - f_{3} } \right) + 2\varLambda_{3} \varLambda_{2} \eta_{2} \omega_{2} d_{6} + 2\varLambda_{3} \varLambda_{1} \eta_{2}^{2} d_{5} , \\ \beta_{2} & = \varLambda_{3} \varLambda_{1} \left\{ {2\eta_{2} v_{2} \left( {f_{1} - f_{3} } \right) - 2\eta_{2} \omega_{2} d_{6} } \right\} + \varLambda_{1} \varLambda_{2} \left\{ {\left( {\mu_{2} v_{2} + 3\eta_{2} \omega_{2} } \right)\left( {f_{1} - f_{3} } \right) + \mu_{2} \omega_{2} d_{6} + v_{2} \eta_{2} d_{5} } \right\}, \\ \beta_{3} & = \varLambda_{3} \varLambda_{2} \left\{ { - 2\mu_{2} \omega_{2} \left( {f_{1} - f_{3} } \right) + 2\eta_{2} \omega_{2} d_{5} } \right\} + \varLambda_{3} \varLambda_{1} \left\{ { - \left( {\mu_{2} v_{2} + 2\eta_{2} \omega_{2} } \right)\left( {f_{1} - f_{3} } \right) - \mu_{2} \omega_{2} d_{6} - v_{2} \eta_{2} d_{5} } \right\}, \\ \beta_{4} & = \varLambda_{3} \varLambda_{1} \left\{ {2\mu_{2} \omega_{2} \left( {f_{1} - f_{3} } \right) - 2\eta_{2} \omega_{2} d_{5} } \right\} - \varLambda_{3} \varLambda_{2} \omega_{2}^{2} d_{6} , \\ \beta_{5} & = \varLambda_{3} \varLambda_{1} \omega_{2}^{2} d_{6} . \\ D_{\varvec{u}}^{2} Q_{3} \left( {a,b,c} \right) & = \varLambda_{1}^{2} \frac{{\partial^{2} Q_{3} }}{{\partial a^{2} }} + \varLambda_{2}^{2} \frac{{\partial^{2} Q_{3} }}{{\partial b^{2} }} + \varLambda_{3}^{2} \frac{{\partial^{2} Q_{3} }}{{\partial c^{2} }} + 2\varLambda_{1} \varLambda_{2} \frac{{\partial^{2} Q_{3} }}{\partial a\partial b} + 2\varLambda_{1} \varLambda_{3} \frac{{\partial^{2} Q_{3} }}{\partial a\partial c} + 2\varLambda_{2} \varLambda_{3} \frac{{\partial^{2} Q_{3} }}{\partial b\partial c}, \\ D_{\varvec{u}}^{2} Q_{3} \left( {a,b,c} \right) & = \varLambda_{3}^{2} \tilde{Q}_{7} + \left\{ {v_{6} c\left( {1 - c} \right)^{2} + \omega_{6} \left( {1 - c} \right)^{3} } \right\}\tilde{Q}_{8} + 2\{ 2c^{4} \left( {1 - c} \right)v_{6} \eta_{6} + c^{3} \left( {1 - c} \right)^{2} \\ & \quad \times \,\left( {2v_{6} \eta_{6} + v_{6} \mu_{6} + 3\omega_{6} \eta_{6} } \right) + \left. {c^{2} \left( {1 - c} \right)^{3} \left( {v_{6} \mu_{6} + 2\mu_{6} \omega_{6} + 3\omega_{6} \eta_{6} } \right) + 2c\left( {1 - c} \right)^{4} \mu_{6} \omega_{6} } \right\}\tilde{Q}_{9} , \\ \end{aligned} $$

$$ \begin{aligned} \tilde{Q}_{7} & = \frac{{P_{3} }}{{\left( {Q_{3d} } \right)^{2} }},\quad P_{3} = \mathop \sum \limits_{k = 0}^{7} \left( {1 - c} \right)^{7 - k} c^{k} P_{3,k} , \\ P_{3,0} & = 2\omega_{5}^{2} \mu_{5} \left( {f_{3} - F\left( {S_{3} } \right)} \right) + 2\omega_{6}^{2} \eta_{6} R_{5} + \left( {2\omega_{6}^{2} v_{6} - 2\omega_{6}^{3} } \right)R_{6} , \\ P_{3,1} & = \left( {6\omega_{6}^{2} \eta_{6} + 8\omega_{6}^{2} \mu_{6} } \right)\left( {f_{3} - F\left( {S_{3} } \right)} \right) + 8\omega_{6}^{2} \eta_{6} R_{5} + \left( {6\omega_{6}^{2} \mu_{6} - 2\omega_{6}^{3} + 2\omega_{6}^{2} v_{6} } \right)R_{6} , \\ P_{3,2} & = 6\omega_{6}^{2} \left\{ {\left( {3\eta_{6} + \mu_{6} } \right)\left( {f_{3} - F\left( {S_{3} } \right)} \right) + 2\left( {\mu_{6} + \eta_{6} } \right)} \right.R_{6} + \left. {\eta_{6} R_{5} } \right\} + 6\omega_{6} \left\{ {\left( {\eta_{6} v_{6} + \mu_{6} v_{6} - \mu_{6}^{2} } \right)} \right. \\ & \quad \times \,\left( {f_{3} - F\left( {S_{3} } \right)} \right) + \eta_{6} v_{6} R_{5} - \left. {\omega_{6} \eta_{6} R_{6} } \right\}, \\ P_{3,3} & = \left( {12\omega_{6}^{2} \eta_{6} } \right. + 2v_{6}^{2} \mu_{6} + 2v_{6}^{2} \eta_{6} + 6\omega_{6} v_{6} \mu_{6} + 22\omega_{6} v_{6} \eta_{6} - 8\omega_{6} \mu_{6}^{2} - 16\mu_{6} \omega_{6} \eta_{6} \\ & \quad \left. { - 2v_{6} \mu_{6}^{2} } \right)\left( {f_{3} - F\left( {S_{3} } \right)} \right) + \left( {6\mu_{6} \omega_{6}^{2} + 26\eta_{6} \omega_{6}^{2} + 2} \right.v_{6} \omega_{6} \mu_{6} + 2\omega_{6} v_{6} \eta_{6} \left. { - 2\omega_{6} \mu_{6}^{2} } \right)R_{6} \\ & \quad + \,\left( { - 14\omega_{6} \eta_{6}^{2} + 6\omega_{6} v_{6} \eta_{6} - 8\mu_{6} \omega_{6} \eta_{6} - 2\mu_{6} v_{6} \eta_{6} + 2v_{6}^{2} \eta_{6} } \right)R_{5} , \\ P_{3,4} & = \left( {12\eta_{6}^{2} \omega_{6} } \right. + 2\mu_{6}^{2} \omega_{6} + 2\mu_{6}^{2} v_{6} + 6\eta_{6} v_{6} \mu_{6} + 22\mu_{6} \omega_{6} \eta_{6} - 8\eta_{6} v_{6}^{2} - 16v_{6} \omega_{6} \eta_{6} \\ & \quad \left. { - \,2\mu_{6} v_{6}^{2} } \right)\left( {F\left( {S_{3} } \right) - f_{3} } \right) + \left( { - 6v_{6} \eta_{6}^{2} - 26\omega_{6} \eta_{6}^{2} - 2} \right.\eta_{6} \omega_{6} \mu_{6} - 2\omega_{6} \mu_{6} \eta_{6} \left. { + 2\eta_{6} v_{6}^{2} } \right)R_{5} \\ & \quad + \,\left( {14\eta_{6} \omega_{6}^{2} + 2\omega_{6} v_{6} \mu_{6} + 8v_{6} \omega_{6} \eta_{6} - 6\mu_{6} \omega_{6} \eta_{6} - 2\mu_{6}^{2} \omega_{6} } \right)R_{6} , \\ P_{3,5} & \quad = 6\eta_{6}^{2} \left\{ {\left( {3\omega_{6} + v_{6} } \right)\left( {F\left( {S_{3} } \right) - f_{3} } \right) - \left( {\omega_{6} + 2v_{6} } \right)} \right.R_{5} - 2\left. {\omega_{6} R_{6} } \right\} + 6\eta_{6} \left\{ {\omega_{6} v_{6} R_{6} - \omega_{6} \mu_{6} R_{5} } \right. \\ & \quad \left. { + \,\left( {\omega_{6} \mu_{6} + \mu_{6} v_{6} - v_{6}^{2} } \right)\left( {F\left( {S_{3} } \right) - f_{3} } \right)} \right\}, \\ P_{3,6} & = \left( {6\eta_{6}^{2} \omega_{6} + 8\eta_{6}^{2} v_{6} } \right)\left( {F\left( {S_{3} } \right) - f_{3} } \right) - 8\eta_{6}^{2} \omega_{6} R_{6} + \left( { - 2\eta_{6}^{2} \mu_{6} + 2\eta_{6}^{3} - 6\eta_{6}^{2} v_{6} } \right)R_{5} , \\ P_{3,7} & = 2\eta_{6}^{2} v_{6} \left( {F\left( {S_{3} } \right) - f_{3} } \right) - 2\eta_{6}^{2} \omega_{6} R_{6} + \left( {2\eta_{6}^{3} - 2\eta_{6}^{2} \mu_{6} } \right)R_{5} , \\ \end{aligned} $$

$$ \tilde{Q}_{8} = \frac{{\mathop \sum \nolimits_{k = 0}^{7} a^{7 - k} b^{k} \tilde{\gamma }_{k} }}{{\left( {\eta_{3} a^{3} + \mu_{3} a^{2} b + v_{3} ab^{2} + \omega_{3} b^{3} } \right)^{3} }}, $$

$$ \begin{aligned} \tilde{\gamma }_{0} & = \varLambda_{2}^{2} \left\{ {2\eta_{3}^{2} v_{3} \left( {f_{2} - f_{1} } \right) - 2\eta_{3}^{2} \omega_{3} d_{2} - \left. {2\eta_{3}^{2} \mu_{3} d_{1} } \right\} - 2\varLambda_{1} \varLambda_{2} } \right.\eta_{3}^{3} d_{1} , \\ \tilde{\gamma }_{1} & = 2\eta_{3}^{3} \varLambda_{1}^{2} d_{1} + \varLambda_{2}^{2} \left\{ {6\eta_{3}^{2} \omega_{3} \left( {f_{2} - f_{1} } \right) - } \right.\left. {6\eta_{3}^{2} v_{3} d_{1} } \right\} + 2\varLambda_{1} \varLambda_{2} \left\{ {4\eta_{3}^{2} v_{3} \left( {f_{1} - f_{2} } \right) + 4\eta_{3}^{2} \omega_{3} d_{2} + \eta_{3}^{2} \mu_{3} d_{1} } \right\}, \\ \tilde{\gamma }_{2} & = 6\eta_{3}^{2} \varLambda_{2}^{2} \left\{ {\eta_{3} \left( {f_{2} - f_{1} } \right) - \omega_{3} d_{2} } \right\} + 6\varLambda_{2}^{2} \left\{ {\eta_{3} \left( {\omega_{3} \mu_{3} - v_{3}^{2} } \right)\left( {f_{2} - f_{1} } \right) - 2\eta_{3}^{2} \omega_{3} d_{1} + \eta_{3} v_{3} \omega_{3} d_{2} } \right\} \\ & \quad + \,2\varLambda_{1} \varLambda_{2} \left\{ {\left( {3\eta_{3} \mu_{3} v_{3} - 9\eta_{3}^{2} \omega_{3} } \right)\left( {f_{1} - f_{2} } \right)\left. { + 3\eta_{3} \mu_{3} \omega_{3} d_{2} + 9\eta_{3}^{2} v_{3} d_{1} } \right\},} \right. \\ \tilde{\gamma }_{3} & = 6\varLambda_{1}^{2} \left\{ {\left( {\eta_{3} \mu_{3} v_{3} + 2\eta_{3}^{2} \omega_{3} } \right)\left( {f_{2} - f_{1} } \right)\left. { - \eta_{3} \mu_{3} \omega_{3} d_{2} - \eta_{3}^{2} v_{3} d_{1} } \right\}} \right. + \varLambda_{2}^{2} \left\{ {\left( {2\omega_{3} \mu_{3}^{2} } \right.} \right. - 26\eta_{3} v_{3} \omega_{3} \\ & \quad \left. { - \,2\mu_{3} v_{3}^{2} } \right)\left( {f_{2} - f_{1} } \right) + 2\left( {\eta_{3} v_{3}^{2} - \eta_{3} \mu_{3} \omega_{3} } \right)d_{1} + \left( {2\mu_{3} v_{3} \omega_{3} + 24\eta_{3} \omega_{3}^{2} } \right)\left. {d_{2} } \right\} + 2\varLambda_{1} \varLambda_{2} \\ & \quad \times \,\left\{ {\left( {11\eta_{3} \mu_{3} \omega_{3} + v_{3} \mu_{3}^{2} - 4\eta_{3} v_{3}^{2} } \right)\left( {f_{2} - f_{1} } \right) + \left( {13\omega_{3} \eta_{3}^{2} + \eta_{3} \mu_{3} v_{3} } \right)} \right.d_{1} \left. { - 4\eta_{3} v_{3} \omega_{3} d_{2} } \right\}, \\ \tilde{\gamma }_{4} & = \varLambda_{1}^{2} \left\{ {\left( {16\eta_{3} \mu_{3} \omega_{3} + 2\mu_{3}^{2} v_{3} } \right)\left( {f_{2} - f_{1} } \right) - \left( {14\omega_{3} \eta_{3}^{2} + 2\eta_{3} \mu_{3} v_{3} } \right)d_{1} + 2\left. {\omega_{3} \left( {v_{3} \eta_{3} - \mu_{3}^{2} } \right)d_{2} } \right\}} \right. \\ & \quad + \,\varLambda_{2}^{2} \left\{ {\left( {6v_{3} \mu_{3} \omega_{3} - 12\omega_{3}^{2} \eta_{3} } \right)\left( {f_{1} - f_{2} } \right) - 14\omega_{3}^{2} \mu_{3} d_{2} + 6\left. {\omega_{3} v_{3} \eta_{3} d_{1} } \right\}} \right. + 2\varLambda_{1} \varLambda_{2} \left\{ { - \left( {13\eta_{3} } \right.} \right.\omega_{3}^{2} \\ & \quad \left. { + \,v_{3} \mu_{3} \omega_{3} } \right)d_{2} + \left( {4\eta_{3} \mu_{3} \omega_{3} - v_{3}^{2} \eta_{3} } \right)d_{1} \left. { + \left( {4\mu_{3}^{2} \omega_{3} - \mu_{3} v_{3}^{2} - 11\omega_{3} v_{3} \eta_{3} } \right)\left( {f_{1} - f_{2} } \right)} \right\}, \\ \tilde{\gamma }_{5} & = 6\varLambda_{1}^{2} \left\{ {\left( {\eta_{3} v_{3} \omega_{3} - \mu_{3}^{2} \omega_{3} } \right)\left( {f_{1} - f_{2} } \right) + 2\eta_{3} \omega_{3}^{2} d_{2} \left. { - \eta_{3} \mu_{3} \omega_{3} d_{1} } \right\} + 6\varLambda_{2}^{2} \omega_{3}^{2} \left\{ {\mu_{3} \left( {f_{1} - f_{2} } \right)} \right.} \right. \\ & \quad \left. { + \,\eta_{3} d_{1} } \right\} + 2\varLambda_{1} \varLambda_{2} \left\{ {\left( {9\eta_{3} \omega_{3}^{2} + 3v_{3} \mu_{3} \omega_{3} } \right)\left( {f_{2} - f_{1} } \right) - 6\omega_{3}^{2} \mu_{3} d_{2} - 3\eta_{3} v_{3} \omega_{3} d_{1} } \right\}, \\ \tilde{\gamma }_{6} & = 6\varLambda_{1}^{2} \omega_{3}^{2} \left\{ {\eta_{3} \left( {f_{1} - f_{2} } \right)\left. { - \mu_{3} d_{2} } \right\} - 2\varLambda_{2}^{2} \omega_{3}^{3} d_{2} + 2\varLambda_{1} \varLambda_{2} \left\{ {4\mu_{3} \omega_{3}^{2} \left( {f_{2} - f_{1} } \right) - v_{3} \omega_{3}^{2} d_{2} - 4\eta_{3} \omega_{3}^{2} d_{1} } \right\}} \right., \\ \tilde{\gamma }_{7} & = 2\varLambda_{1}^{2} \omega_{3}^{2} \left\{ {\mu_{3} \left( {f_{1} - f_{2} } \right)\left. { + \eta_{3} d_{1} + v_{3} d_{2} } \right\} + 2\varLambda_{1} \varLambda_{2} \omega_{3}^{3} d_{2} } \right., \\ \end{aligned} $$

$$ \begin{aligned} \tilde{Q}_{9} & = \frac{{\mathop \sum \nolimits_{k = 0}^{5} a^{5 - k} b^{k} \gamma_{k} }}{{\left( {\eta_{3} a^{3} + \mu_{3} a^{2} b + v_{3} ab^{2} + \omega_{3} b^{3} } \right)^{2} }}, \\ \gamma_{0} & = - \varLambda_{3} \varLambda_{2} \eta_{3}^{2} d_{1} , \\ \gamma_{1} & = - 2\varLambda_{3} \varLambda_{2} \eta_{3} v_{3} \left( {f_{2} - f_{1} } \right) + 2\varLambda_{3} \varLambda_{2} \eta_{3} \omega_{3} d_{2} + 2\varLambda_{3} \varLambda_{1} \eta_{3}^{2} d_{1} , \\ \gamma_{2} & = \varLambda_{3} \varLambda_{1} \left\{ {2\eta_{3} v_{3} \left( {f_{2} - f_{1} } \right) - 2\eta_{3} \omega_{3} d_{2} } \right\} + \varLambda_{1} \varLambda_{2} \left\{ {\left( {\mu_{3} v_{3} + 3\eta_{3} \omega_{3} } \right)\left( {f_{1} - f_{2} } \right) + \mu_{3} \omega_{3} d_{2} + v_{3} \eta_{3} d_{1} } \right\}, \\ \gamma_{3} & = \varLambda_{3} \varLambda_{2} \left\{ { - 2\mu_{3} \omega_{3} \left( {f_{2} - f_{1} } \right) + 2\eta_{3} \omega_{3} d_{1} } \right\} + \varLambda_{3} \varLambda_{1} \left\{ { - \left( {\mu_{3} v_{3} + 3\eta_{3} \omega_{3} } \right)\left( {f_{1} - f_{2} } \right) - \mu_{3} \omega_{3} d_{2} - v_{3} \eta_{3} d_{1} } \right\}, \\ \gamma_{4} & = \varLambda_{3} \varLambda_{1} \left\{ {2\mu_{3} \omega_{3} \left( {f_{2} - f_{1} } \right) - 2\eta_{3} \omega_{3} d_{1} } \right\} - \varLambda_{3} \varLambda_{2} \omega_{3}^{2} d_{2} , \\ \gamma_{5} & = \varLambda_{3} \varLambda_{1} \omega_{3}^{2} d_{2} . \\ \end{aligned} $$