Blending using ODE swept surfaces with shape control and $C^1$ continuity

Abstract

Surface blending with tangential continuity is most widely applied in computer-aided design, manufacturing systems, and geometric modeling. In this paper, we propose a new blending method to effectively control the shape of blending surfaces, which can also satisfy the blending constraints of tangent continuity exactly. This new blending method is based on the concept of swept surfaces controlled by a vector-valued fourth order ordinary differential equation (ODE). It creates blending surfaces by sweeping a generator along two trimlines and making the generator exactly satisfy the tangential constraints at the trimlines. The shape of blending surfaces is controlled by manipulating the generator with the solution to a vector-valued fourth order ODE. This new blending methods have the following advantages: (1) exact satisfaction of $C^1$ continuous blending boundary constraints, (2) effective shape control of blending surfaces, (3) high computing efficiency due to explicit mathematical representation of blending surfaces, and (4) ability to blend multiple (more than two) primary surfaces.

Appendix A: Other analytical solutions of Eq. (2)

For $\beta ^2=4\alpha \gamma$ and $\alpha /\beta >0$, solving the nonlinear algebra equation (4), the following roots are found,

$$\begin{aligned} r_{1,2,3,4} =\pm iq_2 \end{aligned}$$

(24)

where $i$ is an imaginary unit and,

$$\begin{aligned} q_2 =\sqrt{\beta /(2\alpha )} \end{aligned}$$

(25)

With the roots given in Eq. (24), the analytical solution to Eq. (2) becomes,

$$\begin{aligned} {\mathbf G}(u)={\mathbf d}_1 \cos q_2 u+{\mathbf d}_2 \sin q_2 u+{\mathbf d}_3 u\cos q_2 u+{\mathbf d}_4 u\sin q_2 u \end{aligned}$$

(26)

where ${\mathbf d}_1$, ${\mathbf d}_2$, ${\mathbf d}_3$ and ${\mathbf d}_4$ are vector-valued unknown constants.

To determine the unknown constants in Eq. (26), we perform the same sweeping operation by substituting it into Eq. (1), and solving for the four unknown constants ${\mathbf d}_1$, ${\mathbf d}_2$, ${\mathbf d}_3$ and ${\mathbf d}_4$. Then, we substitute the unknown constants back into Eq. (26), and obtain,

$$\begin{aligned} \begin{array}{l} {\mathbf S}(u,v)=( \cos q_2 u+A_2 \sin q_2 u-q_2 A_2 u\cos q_2 u\\ \qquad \qquad \quad -\,\, uctgq_2 \sin q_2 u \\ \qquad \qquad \quad +\,\, {A_2 A_5 u\sin q_2 u}){\mathbf C}_0 (v)\\ \qquad \qquad \quad +\,\, ( {A_3 \sin q_2 u+u\cos q_2 u-q_2 A_3 u\cos q_2 u} \\ \qquad \qquad \quad -\,\, uctgq_2 \sin q_2 u+ {A_3 A_5 u\sin q_2 u})\bar{{\mathbf {C}}}_0 (v)\\ \qquad \qquad \quad +\,\, ( {-A_4 \sin q_2 u+q_2 A_4 }u\cos q_2 u\\ \qquad \qquad \quad +\,\, {u\sin q_2 u} / {\sin q_2 }- {A_4 A_5 u\sin q_2 u}){\mathbf C}_1 (v)\\ \qquad \qquad \quad +\, \,( {\sin q_2 }u-q_2 u\cos q_2 u\!+\!A_5 u\sin q_2 u){\bar{{\mathbf {C}}}_1 (v)} / {A_1 } \\ \end{array} \end{aligned}$$

(27)

where,

$$\begin{aligned} \begin{array}{l} A_1 ={q_2 } / {\sin q_2 }-\sin q_2 \\ A_2 ={\left[ {q_2 \sin q_2 +ctgq_2 ( {\sin q_2 +q_2 \cos q_2 })} \right] } / {A_1 } \\ A_3 =\big [ -( {\cos q_2 -q_2 \sin q_2 })\\ \qquad \qquad +\, ctgq_2 ( {\sin q_2 +q_2 \cos q_2 }) \big ] / {A_1 } \\ A_4 ={( {\sin q_2 +q_2 \cos q_2 })} / {( {A_1 \sin q_2 })} \\ A_5 ={( {q_2 \cos q_2 -\sin q_2 })} / {\sin q_2 } \\ \end{array} \end{aligned}$$

(28)

For $\beta ^2>4\alpha \gamma$, solving the nonlinear algebra equation (4) gives the following roots,

$$\begin{aligned} \begin{array}{l} r_{1,2} =\pm iq_3 \\ r_{3,4} =\pm iq_4 \\ \end{array} \end{aligned}$$

(29)

where$i$ is an imaginary unit and,

$$\begin{aligned} \begin{array}{l} q_3 =\sqrt{\beta (1-\sqrt{1-4\alpha \gamma /\beta ^2} )/(2\alpha )} \\ q_4 =\sqrt{\beta (1+\sqrt{1-4\alpha \gamma /\beta ^2} )/(2\alpha )} \\ \end{array} \end{aligned}$$

(30)

With the roots given in Eq. (30), the analytical solution to Eq. (2) takes the form of,

$$\begin{aligned} {\mathbf G}(u)={\mathbf d}_1 \cos q_3 u+{\mathbf d}_2 \sin q_3 u+{\mathbf d}_3 \cos q_4 u+{\mathbf d}_4 \sin q_4 u\nonumber \\ \end{aligned}$$

(31)

where ${\mathbf d}_1$, ${\mathbf d}_2$, ${\mathbf d}_3$ and ${\mathbf d}_4$ are vector-valued unknown constants.

Same as above, we substitute Eq. (31) into Eq. (1), carry out the sweeping operation, and determine the four unknown constants ${\mathbf d}_1$, ${\mathbf d}_2$, ${\mathbf d}_3$ and ${\mathbf d}_4$. After substituting these unknown constants back to Eq. (31), the mathematical equation of blending surfaces becomes,

$$\begin{aligned} \begin{array}{l} {\mathbf S}(u,v)=\big [ -g_3 (u){\cos q_4 } / A_6 -g_4 (u){q_4 \sin q_4 } / A_6 \\ \quad \ \qquad \qquad +\cos q_4 u \big ]\\ \qquad \quad \qquad \ \times \, {\mathbf C}_0 (v)\big [ -g_3 (u){\sin q_4 } / {q_4 } / {A_6 }\\ \qquad \qquad \quad \ +\, g_4 (u){\cos q_4 } /{A_6 +{\sin q_4 u} / {q_4 }} \big ] \bar{{\mathbf {C}}}_0 (v)\\ \qquad \qquad \quad \ +\, {g_3 (u){\mathbf C}_1 (v)} / {A_6 -{g_4 (u)\bar{{\mathbf {C}}}_1 (v)} / {A_6 }} \\ \end{array}\nonumber \\ \end{aligned}$$

(32)

where,

$$\begin{aligned} \begin{array}{l} A_6 =q_3 ( {\cos q_3 -\cos q_4 })^2-( {\sin q_3 -{q_3 \sin q_4 }/ {q_4 }}) \\ ( {q_4 \sin q_4 -q_3 \sin q_3 }) \\ \end{array} \end{aligned}$$

(33)

and,

$$\begin{aligned} \begin{array}{l} g_3 (u)=q_3 ( {\cos q_3 -\cos q_4 })( {\cos q_3 u-\cos q_4 u}) \\ \qquad \qquad \quad +\, ( {q_4 \sin q_4 }- {q_3 \sin q_3 })( {q_3 {\sin q_4 u} / {q_4 }-\sin q_3 u}) \\ g_4 (u)=( {\cos q_3 -\cos q_4 })( {q_3 {\sin q_4 u} / {q_4 }-\sin q_3 u}) \\ \qquad \qquad \quad +\, ( {\sin q_3 }- q_3 {{\sin q_4 } / {q_4 }})( {\cos q_3 u-\cos q_4 u}) \\ \end{array}\nonumber \\ \end{aligned}$$

(34)

For $\beta ^2<4\alpha \gamma$, solving the nonlinear algebra equation (4) generates the four roots below,

$$\begin{aligned} r_{1,2,3,4} =\pm q_5 \pm iq_6 \end{aligned}$$

(35)

where$i$ is an imaginary unit and,

$$\begin{aligned} \begin{array}{l} q_5 =\root 4 \of {\gamma / \alpha }\sqrt{0.5-\beta / {( {4\sqrt{\alpha \gamma } })}} \\ q_6 =\root 4 \of {\gamma / \alpha }\sqrt{0.5+\beta / {( {4\sqrt{\alpha \gamma } })}} \\ \end{array} \end{aligned}$$

(36)

With the roots given in Eq. (36), the solution to Eq. (2) is found to be,

$$\begin{aligned} {\mathbf G}(u)&= {\mathbf d}_1 e^{q_5 u}\cos q_6 u+{\mathbf d}_2 e^{q_5 u}\sin q_6 u+{\mathbf d}_3 e^{-q_5 u}\cos q_6 u \nonumber \\&+\, {\mathbf d}_4 e^{-q_5 u}\sin q_6 u \end{aligned}$$

(37)

where ${\mathbf d}_1$ , ${\mathbf d}_2$, ${\mathbf d}_3$and ${\mathbf d}_4$ are vector-valued unknown constants.

Substituting Eq. (37) into Eq. (1) and doing the sweeping operation, the 4 unknown constants are determined and blending surfaces satisfying Eqs. (1) and (2) are found to be,

$$\begin{aligned} {\mathbf S}(u,v)&= \big [ {-A_9 g_5 (u)}/ q_6 -( {{A_7 q_5 } /{q_6 -A_8 }})g_6 (u)\nonumber \\&\quad +\, e^{-q_5 u}\cos q_6 u {+q_5 e^{-q_5 u}{\sin q_6 u} / {q_6 }}\big ]{\mathbf C}_0 (v)\nonumber \\&\quad +\, \left[ {{-A_7 g_6 (u)} / {q_6 -e^{-q_5 }}} {\sin q_6 g_5 (u)} / {q_6 }\right. \nonumber \\&\quad \left. {+e^{-q_5 u}{\sin q_6 u} / {q_6 }} \right] \bar{\mathbf {C}}_0 (v)+\, g_5 (u){\mathbf C}_1 (v) \nonumber \\&\quad +g_6 (u)\bar{{\mathbf {C}}}_1 (v) \end{aligned}$$

(38)

where,

$$\begin{aligned} \begin{array}{l} A_7 =q_6 e^{-q_5 }\cos q_6 -q_5 e^{-q_5 }\sin q_6 \\ A_8 =q_5 e^{-q_5 }\cos q_6 +q_6 e^{-q_5 }\sin q_6 \\ A_9 =q_5 e^{-q_5 }\sin q_6 +q_6 e^{-q_5 }\cos q_6 \\ g_5 (u)=\left\{ {A_{12} \left[ {( {e^{-q_5 u}-e^{q_5 u}})\sin q_6 u} \right] } \right. \\ \qquad \qquad +A_{13} \left[ {( {e^{q_5 u}-e^{-q_5 u}})\cos q_6 u} \right. \\ \qquad \qquad {\left. {\left. {-2q_5 e^{-q_5 u}{\sin q_6 u} /{q_6 }} \right] } \right\} } / {A_{14} } \\ g_6 (u)=\left\{ {A_{10} \left[ {( {e^{q_5 u}-e^{-q_5 u}})\sin q_6 u} \right] } \right. \\ \qquad \qquad +A_{11} \left[ {( {e^{-q_5 u}-e^{q_5 u}})\cos q_6 u} \right. \\ \qquad \qquad {\left. {\left. {+2q_5 e^{-q_5 u}{\sin q_6 u} /{q_6 }} \right] } \right\} } / {A_{14} } \\ \end{array} \end{aligned}$$

(39)

and,

$$\begin{aligned} \begin{array}{l} A_{10} =( {e^{q_5 }-e^{-q_5 }})\cos q_6 -2q_5 e^{-q_5 }{\sin q_6 }/ {q_6 } \\ A_{11} =( {e^{q_5 }-e^{-q_5 }})\sin q_6 \\ A_{12} =( {q_5 \cos q_6 -q_6 \sin q_6 })e^{q_5 }+A_8 -2q_5 {A_7 }/{q_6 } \\ A_{13} =( {q_5 \sin q_6 +q_6 \cos q_6 })e^{q_5 }-A_9 \\ A_{14} =A_{10} A_{13} -A_{11} A_{12} \\ \end{array} \end{aligned}$$

(40)