Interactive PDE patch-based surface modeling from vertex-frames

Abstract

Polygon, subdivision, and NURBS are three mainstream modeling techniques widely applied in commercial software packages. They require heavy manual operations, and involve a lot of design variables leading to big data, high storage costs and slow network transmissions. In this paper, we integrate the strengths of boundary-based surface creation and partial differential equation (PDE)-based geometric modeling to obtain the first analytical $C^0$ continuous 4-sided PDE patches involving sculpting force-based shape creation and manipulation and use them to develop an interactive modeling technique for easy and quick creation of 3D models with small data from vertex-frames. With this modeling technique, a vertex frame is defined by eight vertices, and a $C^0$ continuous 4-sided PDE patch is created from the vertex-frame through an analytical solution to a vector-valued second-order PDE subjected to the boundary conditions determined by the eight vertices of a vertex-frame. A user-friendly interface is developed from the obtained analytical solution, which enables users to interactively input and modify vertex-frame models easily and create 3D models in real time. Different surface modeling tasks are carried out to test the developed interactive tool and compare our proposed method with polygon and NURBS modeling and Coons surfaces. The results demonstrate the effectiveness of our proposed method and its advantages in reducing design variables, saving storage costs, and effective shape creation and manipulation.

Appendix A: Analytical formulae of \({\bar{P}}_{{\bar{m}},{\bar{n}}}^{(k)}\) (\({\bar{m}}=i-1\), i, \(i+1\); \({\bar{n}}=j-1\), j, \(j+1\))

For $S_{i,j}$ patch, ${\bar{P}}_{i,j}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i,j}^{(0)}=&{}P_{i,j}^{(1)},{\bar{P}}_{i,j}^{(1)}=-3P_{i,j}^{(1)} +4P_{i,j}^{(2)}-P_{i,j}^{(3)}+\frac{1}{2}{\bar{P}}_{i,j}^{(3)}\\ {\bar{P}}_{i,j}^{(2)}=&{}2P_{i,j}^{(1)}-4P_{i,j}^{(2)}+2P_{i,j}^{(3)}-\frac{3}{2}{\bar{P}}_{i,j}^{(3)}, {\bar{P}}_{i,j}^{(3)}=\frac{{\bar{A}}_{i,j}D_{i,j}-A_{i,j}{\bar{B}}_{i,j}}{{\bar{B}}_{i,j}B_{i,j}-D_{i,j}D_{i,j}},\\ {\bar{P}}_{i,j}^{(4)}=&{}4P_{i,j}^{(8)}-P_{i,j}^{(7)}-3P_{i,j}^{(1)}+\frac{1}{2}{\bar{P}}_{i,j}^{(6)}\\ {\bar{P}}_{i,j}^{(5)}=&{}-4P_{i,j}^{(8)}+2P_{i,j}^{(7)}+2P_{i,j}^{(1)}-\frac{3}{2}{\bar{P}}_{i,j}^{(6)}, {\bar{P}}_{i,j}^{(6)}=\frac{A_{i,j}D_{i,j}-{\bar{A}}_{i,j}B_{i,j}}{{\bar{B}}_{i,j}B_{i,j}-D_{i,j}D_{i,j}}\\ {\bar{P}}_{i,j}^{(7)}=&{}5P_{i,j}^{(1)}-4P_{i,j}^{(2)}-P_{i,j}^{(3)}+4P_{i,j}^{(4)}-3P_{i,j}^{(5)}+4P_{i,j}^{(6)}-P_{i,j}^{(7)}-4P_{i,j}^{(8)}\\ {\bar{P}}_{i,j}^{(8)}=&{}-2P_{i,j}^{(1)}+4P_{i,j}^{(2)}-2P_{i,j}^{(3)}+2P_{i,j}^{(5)}-4P_{i,j}^{(6)}+2P_{i,j}^{(7)}\\ {\bar{P}}_{i,j}^{(9)}=&{}-2P_{i,j}^{(1)}+2P_{i,j}^{(3)}-4P_{i,j}^{(4)}+2P_{i,j}^{(5)}-2P_{i,j}^{(7)}+4P_{i,j}^{(8)}\\ \end{array}\right. } \end{aligned}$$

((A.1))

where

$$\begin{aligned} \begin{aligned}&B_{i,j}\!\!=\!\!\sum _{m=1}^M\!\sum _{n=1}^N\!a_1^2(6u_m\!-\!3)^2, {\bar{B}}_{i,j}\!\!=\!\!\sum _{m=1}^M\!\sum _{n=1}^N\!a_2^2(6v_n\!-\!3)^2, {\bar{A}}_{i,j}\!\!=\!\!\sum _{m=1}^M\!\sum _{n=1}^N\!a_2(6v_n\!-\!3)F_{i,j}^{m,n}\\&D_{i,j}=\sum _{m=1}^M\sum _{n=1}^N a_1a_2(6u_m-3)(6v_n-3), A_{i,j}=\sum _{m=1}^M\sum _{n=1}^N a_1(6u_m-3)F_{i,j}^{m,n}\\ \end{aligned} \end{aligned}$$

((A.2))

and $F_{i,j}^{m,n}$ is determined by

$$\begin{aligned} \begin{aligned} F_{i,j}^{m,n}=&4(a_1-a_1v_n+a_2-a_2u_m)P_{i,j}^{(1)}+8a_1(v_n-1)P_{i,j}^{(2)}+4(a_1-a_1v_n+a_2u_m)P_{i,j}^{(3)}\\&-8a_2u_mP_{i,j}^{(4)}+4(a_1v_n+a_2u_m)P_{i,j}^{(5)}-8a_1v_nP_{i,j}^{(6)}+4a_2(1-u_m)P_{i,j}^{(7)}+8a_2u_mP_{i,j}^{(8)}\\ \end{aligned} \end{aligned}$$

((A.3))

For $S_{i,j+1}$ patch, ${\bar{P}}_{i,j+1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i,j+1}^{(0)}=&{}\!\!\!{\bar{P}}_{i,j}^{(0)}+{\bar{P}}_{i,j}^{(1)}+{\bar{P}}_{i,j}^{(2)}+{\bar{P}}_{i,j}^{(3)} \\ {\bar{P}}_{i,j+1}^{(1)}=&{}\!\!\!4P_{i,j+1}^{(2)}-P_{i,j+1}^{(3)}-3{\bar{P}}_{i,j+1}^{(0)}+\frac{1}{2}{\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i,j+1}^{(2)}=&{}\!\!\!-4P_{i,j+1}^{(2)}+2P_{i,j+1}^{(3)}+2{\bar{P}}_{i,j+1}^{(0)}-\frac{3}{2}{\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i,j+1}^{(3)}=&{}\!\!\!\frac{\sum _{m=1}^M\sum _{n=1}^N a_1(6u_m-3)F_{i,j+1}^{m,n}}{\sum _{m=1}^M\sum _{n=1}^N a_1^2(6u_m-3)^2}\\ {\bar{P}}_{i,j+1}^{(4)}=&{}\!\!\!{\bar{P}}_{i,j}^{(4)}+{\bar{P}}_{i,j}^{(7)}+{\bar{P}}_{i,j}^{(8)}, {\bar{P}}_{i,j+1}^{(5)}={\bar{P}}_{i,j}^{(5)}+{\bar{P}}_{i,j}^{(9)}, {\bar{P}}_{i,j+1}^{(6)}={\bar{P}}_{i,j}^{(6)}\\ {\bar{P}}_{i,j+1}^{(7)}=&{}\!\!\!-3P_{i,j+1}^{(5)}-P_{i,j+1}^{(3)}+4P_{i,j+1}^{(4)}+4P_{i,j+1}^{(6)}-4P_{i,j+1}^{(2)}-3{\bar{P}}_{i,j+1}^{(4)}-2{\bar{P}}_{i,j+1}^{(5)}-\frac{3}{2}{\bar{P}}_{i,j+1}^{(6)}\\ {\bar{P}}_{i,j+1}^{(8)}=&{}\!\!\!2P_{i,j+1}^{(5)}-2P_{i,j+1}^{(3)}-4P_{i,j+1}^{(6)}+4P_{i,j+1}^{(2)}+2{\bar{P}}_{i,j+1}^{(4)}+2{\bar{P}}_{i,j+1}^{(5)}+2{\bar{P}}_{i,j+1}^{(6)}\\ {\bar{P}}_{i,j+1}^{(9)}=&{}\!\!\!2P_{i,j+1}^{(5)}+2P_{i,j+1}^{(3)}-4P_{i,j+1}^{(4)}-{\bar{P}}_{i,j+1}^{(5)}-\frac{3}{2}{\bar{P}}_{i,j+1}^{(6)}\\ \end{array}\right. } \end{aligned}$$

((A.4))

where

$$\begin{aligned} \begin{aligned} F_{i,j+1}^{m,n}=&2a_1(-4P_{i,j+1}^{(2)}+2P_{i,j+1}^{(3)}+2{\bar{P}}_{i,j+1}^{(0)}+v_n2{\bar{P}}_{i,j+1}^{(8)})\\&+2a_2({\bar{P}}_{i,j+1}^{(5)}+3v_n{\bar{P}}_{i,j+1}^{(6)}+u_m{\bar{P}}_{i,j+1}^{(9)})\\ \end{aligned} \end{aligned}$$

((A.5))

For $S_{i+1,j}$ patch, ${\bar{P}}_{i+1,j}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i+1,j}^{(0)}=&{}\!\!\!{\bar{P}}_{i,j}^{(0)}+{\bar{P}}_{i,j}^{(4)}+{\bar{P}}_{i,j}^{(5)}+{\bar{P}}_{i,j}^{(6)} \\ {\bar{P}}_{i+1,j}^{(1)}=&{}\!\!\!{\bar{P}}_{i,j}^{(1)}+{\bar{P}}_{i,j}^{(7)}+{\bar{P}}_{i,j}^{(9)}, {\bar{P}}_{i+1,j}^{(2)}={\bar{P}}_{i,j}^{(2)}+{\bar{P}}_{i,j}^{(8)}, {\bar{P}}_{i+1,j}^{(3)}={\bar{P}}_{i,j}^{(3)}\\ {\bar{P}}_{i+1,j}^{(4)}=&{}\!\!\!4P_{i+1,j}^{(8)}-3{\bar{P}}_{i+1,j}^{(0)}-P_{i+1,j}^{(7)}+\frac{1}{2}{\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j}^{(5)}=&{}\!\!\!-4P_{i+1,j}^{(8)}+2{\bar{P}}_{i+1,j}^{(0)}+2P_{i+1,j}^{(7)}-\frac{3}{2}{\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j}^{(6)}=&{}\!\!\!\frac{\sum _{m=1}^M\sum _{n=1}^N a_2(6v_n-3)F_{i+1,j}^{m,n}}{\sum _{m=1}^M\sum _{n=1}^N a_2^2(6v_n-3)^2}\\ {\bar{P}}_{i+1,j}^{(7)}=&{}\!\!\!-3P_{i+1,j}^{(5)}-P_{i+1,j}^{(7)}+4P_{i+1,j}^{(6)}+4P_{i+1,j}^{(4)}\\ &{}\!\!\!-4P_{i+1,j}^{(8)}-3{\bar{P}}_{i+1,j}^{(1)}-2{\bar{P}}_{i+1,j}^{(2)}-\frac{3}{2}{\bar{P}}_{i+1,j}^{(3)}\\ {\bar{P}}_{i+1,j}^{(8)}=&{}\!\!\!2P_{i+1,j}^{(5)}+2P_{i+1,j}^{(7)}-4P_{i+1,j}^{(6)}-{\bar{P}}_{i+1,j}^{(2)}-\frac{3}{2}{\bar{P}}_{i+1,j}^{(3)}\\ {\bar{P}}_{i+1,j}^{(9)}=&{}\!\!\!2P_{i+1,j}^{(5)}-2P_{i+1,j}^{(7)}-4P_{i+1,j}^{(4)}+4P_{i+1,j}^{(8)}\\ &{}\!\!\!+2{\bar{P}}_{i+1,j}^{(1)}+2{\bar{P}}_{i+1,j}^{(2)}+2{\bar{P}}_{i+1,j}^{(3)}\\ \end{array}\right. } \end{aligned}$$

((A.6))

where

$$\begin{aligned} \begin{aligned} F_{i,j+1}^{m,n}=&2a_1({\bar{P}}_{i+1,j}^{(2)}+3u_m{\bar{P}}_{i+1,j}^{(3)}+v_n{\bar{P}}_{i+1,j}^{(8)})\\&+2a_2(u_m{\bar{P}}_{i+1,j}^{(9)}-4P_{i+1,j}^{(8)}+2{\bar{P}}_{i+1,j}^{(0)}+2P_{i+1,j}^{(7)})\\ \end{aligned} \end{aligned}$$

((A.7))

For $S_{i,j-1}$ patch, ${\bar{P}}_{i,j-1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i,j-1}^{(0)}=&{}P_{i,j-1}^{(1)},{\bar{P}}_{i,j-1}^{(1)}=-3P_{i,j-1}^{(1)}+4P_{i,j-1}^{(2)}-{\bar{P}}_{i,j}^{(0)}+\frac{1}{2}{\bar{P}}_{i,j-1}^{(3)}\\ {\bar{P}}_{i,j-1}^{(2)}=&{}2P_{i,j-1}^{(1)}-4P_{i,j-1}^{(2)}+2{\bar{P}}_{i,j}^{(0)}-\frac{3}{2}{\bar{P}}_{i,j-1}^{(3)}\\ {\bar{P}}_{i,j-1}^{(3)}=&{}\frac{\sum _{m=1}^M\sum _{n=1}^N a_1(6u_m-3)F_{i,j-1}^{m,n}}{\sum _{m=1}^M\sum _{n=1}^N a_1^2(6u_m-3)^2}\\ {\bar{P}}_{i,j-1}^{(4)}=&{}4P_{i,j-1}^{(8)}-3P_{i,j-1}^{(1)}-P_{i,j-1}^{(7)}+\frac{1}{2}{\bar{P}}_{i,j}^{(6)}\\ {\bar{P}}_{i,j-1}^{(5)}=&{}-4P_{i,j-1}^{(8)}+2P_{i,j-1}^{(1)}+2P_{i,j-1}^{(7)}-\frac{3}{2}{\bar{P}}_{i,j}^{(6)}, {\bar{P}}_{i,j-1}^{(6)}={\bar{P}}_{i,j}^{(6)}\\ {\bar{P}}_{i,j-1}^{(7)}=&{}4P_{i,j-1}^{(6)}-4P_{i,j-1}^{(2)}+5P_{i,j-1}^{(1)}-4P_{i,j-1}^{(8)}\\ &{}-P_{i,j-1}^{(7)}-2{\bar{P}}_{i,j}^{(5)}-\frac{5}{2}{\bar{P}}_{i,j}^{(6)}-{\bar{P}}_{i,j}^{(4)}\\ {\bar{P}}_{i,j-1}^{(8)}=&{}-4P_{i,j-1}^{(6)}+4P_{i,j-1}^{(2)}-2P_{i,j-1}^{(1)}+2P_{i,j-1}^{(7)}+2{\bar{P}}_{i,j}^{(5)}+2{\bar{P}}_{i,j}^{(6)}+2{\bar{P}}_{i,j}^{(4)}\\ {\bar{P}}_{i,j-1}^{(9)}=&{}4P_{i,j-1}^{(8)}-2P_{i,j-1}^{(7)}-2P_{i,j-1}^{(1)}+{\bar{P}}_{i,j}^{(5)}+\frac{3}{2}{\bar{P}}_{i,j}^{(6)}\\ \end{array}\right. } \end{aligned}$$

((A.8))

where

$$\begin{aligned} \begin{aligned} F_{i,j-1}^{m,n}=&2a_1(2P_{i,j-1}^{(1)}-4P_{i,j-1}^{(2)}+2{\bar{P}}_{i,j}^{(0)}+v_n{\bar{P}}_{i,j-1}^{(8)})\\&+2a_2({\bar{P}}_{i,j-1}^{(5)}-3v_n{\bar{P}}_{i,j-1}^{(6)}+u_m{\bar{P}}_{i,j-1}^{(9)})\\ \end{aligned} \end{aligned}$$

((A.9))

For $S_{i-1,j}$ patch, ${\bar{P}}_{i-1,j}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i-1,j}^{(0)}=&{}P_{i-1,j}^{(1)}, {\bar{P}}_{i-1,j}^{(1)}=4P_{i-1,j}^{(2)}-3P_{i-1,j}^{(1)}-P_{i-1,j}^{(3)}+\frac{1}{2}{\bar{P}}_{i,j}^{(3)}\\ {\bar{P}}_{i-1,j}^{(2)}=&{}-4P_{i-1,j}^{(2)}+2P_{i-1,j}^{(1)}+2P_{i-1,j}^{(3)}-\frac{3}{2}{\bar{P}}_{i,j}^{(3)}, {\bar{P}}_{i-1,j}^{(3)}={\bar{P}}_{i,j}^{(3)}\\ {\bar{P}}_{i-1,j}^{(4)}=&{}4P_{i-1,j}^{(8)}-3P_{i-1,j}^{(1)}+\frac{1}{2}{\bar{P}}_{i-1,j}^{(6)}-{\bar{P}}_{i,j}^{(0)}\\ {\bar{P}}_{i-1,j}^{(5)}=&{}-4P_{i-1,j}^{(8)}+2P_{i-1,j}^{(1)}-\frac{3}{2}{\bar{P}}_{i-1,j}^{(6)}+2{\bar{P}}_{i,j}^{(0)}\\ {\bar{P}}_{i-1,j}^{(6)}=&{}\frac{\sum _{m=1}^M\sum _{n=1}^N a_2(6v_n-3)F_{i-1,j}^{m,n}}{\sum _{m=1}^M\sum _{n=1}^N a_2^2(6v_n-3)^2}\\ {\bar{P}}_{i-1,j}^{(7)}=&{}4P_{i-1,j}^{(4)}-4P_{i-1,j}^{(8)}+5P_{i-1,j}^{(1)}-4P_{i-1,j}^{(2)}\\ &{}-P_{i-1,j}^{(3)}-2{\bar{P}}_{i,j}^{(2)}-\frac{5}{2}{\bar{P}}_{i,j}^{(3)}-{\bar{P}}_{i,j}^{(1)}\\ {\bar{P}}_{i-1,j}^{(8)}=&{}-2P_{i-1,j}^{(1)}+4P_{i-1,j}^{(2)}-2P_{i-1,j}^{(3)}+{\bar{P}}_{i,j}^{(2)}+\frac{3}{2}{\bar{P}}_{i,j}^{(3)}\\ {\bar{P}}_{i-1,j}^{(9)}=&{}-4P_{i-1,j}^{(4)}+4P_{i-1,j}^{(8)}-2P_{i-1,j}^{(1)}+2P_{i-1,j}^{(3)}\\ &{}+2{\bar{P}}_{i,j}^{(2)}+2{\bar{P}}_{i,j}^{(3)}+2{\bar{P}}_{i,j}^{(1)}\\ \end{array}\right. } \end{aligned}$$

((A.10))

where

$$\begin{aligned} \begin{aligned} F_{i-1,j}^{m,n}=&2a_1({\bar{P}}_{i-1,j}^{(2)}+3u_m{\bar{P}}_{i-1,j}^{(3)}+v_n{\bar{P}}_{i-1,j}^{(8)})\\&+2a_2(u_m{\bar{P}}_{i-1,j}^{(9)}-4P_{i-1,j}^{(8)}+2P_{i-1,j}^{(1)}+2{\bar{P}}_{i,j}^{(0)})\\ \end{aligned} \end{aligned}$$

((A.11))

For $S_{i+1,j+1}$ patch, ${\bar{P}}_{i+1,j+1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i+1,j+1}^{(0)}\!=\!&{}{\bar{P}}_{i+1,j}^{(0)}+{\bar{P}}_{i+1,j}^{(1)}+{\bar{P}}_{i+1,j}^{(2)}+{\bar{P}}_{i+1,j}^{(3)}\\ {\bar{P}}_{i+1,j+1}^{(1)}=&{}{\bar{P}}_{i,j+1}^{(1)}+{\bar{P}}_{i,j+1}^{(7)}+{\bar{P}}_{i,j+1}^{(9)}, {\bar{P}}_{i+1,j+1}^{(2)}={\bar{P}}_{i,j+1}^{(2)}+{\bar{P}}_{i,j+1}^{(8)}, {\bar{P}}_{i+1,j+1}^{(3)}={\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i+1,j+1}^{(4)}=&{}{\bar{P}}_{i+1,j}^{(4)}+{\bar{P}}_{i+1,j}^{(7)}+{\bar{P}}_{i+1,j}^{(8)}, {\bar{P}}_{i+1,j+1}^{(5)}={\bar{P}}_{i+1,j}^{(5)}+{\bar{P}}_{i+1,j}^{(9)}, {\bar{P}}_{i+1,j+1}^{(6)}={\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j+1}^{(7)}=&{}-3P_{i+1,j+1}^{(5)}\!+\!4P_{i+1,j+1}^{(6)}\!+\!4P_{i+1,j+1}^{(4)}\!-\!5{\bar{P}}_{i+1,j+1}^{(0)}-3{\bar{P}}_{i+1,j+1}^{(1)}\!\\ &{}-\!2{\bar{P}}_{i+1,j+1}^{(2)}\!-\!\frac{3}{2}{\bar{P}}_{i+1,j+1}^{(3)}\!-\!3{\bar{P}}_{i+1,j+1}^{(4)}-2{\bar{P}}_{i+1,j+1}^{(5)}-\frac{3}{2}{\bar{P}}_{i+1,j+1}^{(6)}\\ {\bar{P}}_{i+1,j+1}^{(8)}=&{}2P_{i+1,j+1}^{(5)}-4P_{i+1,j+1}^{(6)}+2{\bar{P}}_{i+1,j+1}^{(0)}-{\bar{P}}_{i+1,j+1}^{(2)}\\ &{}-\frac{3}{2}{\bar{P}}_{i+1,j+1}^{(3)}\!+\!2{\bar{P}}_{i+1,j+1}^{(4)}\!+\!2{\bar{P}}_{i+1,j+1}^{(5)}\!+\!2{\bar{P}}_{i+1,j+1}^{(6)}\\ {\bar{P}}_{i+1,j+1}^{(9)}=&{}2P_{i+1,j+1}^{(5)}\!-\!4P_{i+1,j+1}^{(4)}\!+\!2{\bar{P}}_{i+1,j+1}^{(0)}\!+\!2{\bar{P}}_{i+1,j+1}^{(1)}\\ &{}+2{\bar{P}}_{i+1,j+1}^{(2)}\!+\!2{\bar{P}}_{i+1,j+1}^{(3)}\!-\!{\bar{P}}_{i+1,j+1}^{(5)}\!-\!\frac{3}{2}{\bar{P}}_{i+1,j+1}^{(6)}\\ \end{array}\right. } \end{aligned}$$

((A.12))

For $S_{i-1,j-1}$ patch, ${\bar{P}}_{i-1,j-1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i-1,j-1}^{(0)}=&{}P_{i-1,j-1}^{(1)}, {\bar{P}}_{i-1,j-1}^{(1)}=4P_{i-1,j-1}^{(2)}-3P_{i-1,j-1}^{(1)}+\frac{1}{2}{\bar{P}}_{i,j-1}^{(3)}-{\bar{P}}_{i-1,j}^{(0)}\\ {\bar{P}}_{i-1,j-1}^{(2)}=&{}-4P_{i-1,j-1}^{(2)}+2P_{i-1,j-1}^{(1)}-\frac{3}{2}{\bar{P}}_{i,j-1}^{(3)}+2{\bar{P}}_{i-1,j}^{(0)}\\ {\bar{P}}_{i-1,j-1}^{(3)}=&{}{\bar{P}}_{i,j-1}^{(3)}, {\bar{P}}_{i-1,j-1}^{(4)}=4P_{i-1,j-1}^{(8)}-3P_{i-1,j-1}^{(1)}+\frac{1}{2}{\bar{P}}_{i-1,j}^{(6)}-{\bar{P}}_{i,j-1}^{(0)}\\ {\bar{P}}_{i-1,j-1}^{(5)}=&{}-4P_{i-1,j-1}^{(8)}+2P_{i-1,j-1}^{(1)}-\frac{3}{2}{\bar{P}}_{i-1,j}^{(6)}+2{\bar{P}}_{i,j-1}^{(0)}, {\bar{P}}_{i-1,j-1}^{(6)}={\bar{P}}_{i-1,j}^{(6)}\\ {\bar{P}}_{i-1,j-1}^{(7)}=&{}5P_{i-1,j-1}^{(1)}-4P_{i-1,j-1}^{(2)}-4P_{i-1,j-1}^{(8)}+2{\bar{P}}_{i-1,j}^{(0)}\\ &{}+{\bar{P}}_{i-1,j}^{(4)}-\frac{1}{2}{\bar{P}}_{i-1,j}^{(6)}+{\bar{P}}_{i,j-1}^{(0)}-{\bar{P}}_{i,j-1}^{(2)}-\frac{3}{2}{\bar{P}}_{i,j-1}^{(3)}\\ {\bar{P}}_{i-1,j-1}^{(8)}=&{}-2P_{i-1,j-1}^{(1)}+4P_{i-1,j-1}^{(2)}-2{\bar{P}}_{i-1,j}^{(0)}+{\bar{P}}_{i,j-1}^{(2)}+\frac{3}{2}{\bar{P}}_{i,j-1}^{(3)}\\ {\bar{P}}_{i-1,j-1}^{(9)}=&{}-2P_{i-1,j-1}^{(1)}+4P_{i-1,j-1}^{(8)}-2{\bar{P}}_{i,j-1}^{(0)}+{\bar{P}}_{i-1,j}^{(5)}+\frac{3}{2}{\bar{P}}_{i-1,j}^{(6)}\\ \end{array}\right. } \end{aligned}$$

((A.13))

For $S_{i-1,j+1}$ patch, ${\bar{P}}_{i-1,j+1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i-1,j+1}^{(0)}=&{}{\bar{P}}_{i-1,j}^{(0)}+{\bar{P}}_{i-1,j}^{(1)}+{\bar{P}}_{i-1,j}^{(2)}+{\bar{P}}_{i-1,j}^{(3)}\\ {\bar{P}}_{i-1,j+1}^{(1)}=&{}4P_{i-1,j+1}^{(2)}-P_{i-1,j+1}^{(3)}-3{\bar{P}}_{i-1,j+1}^{(0)}+\frac{1}{2}{\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i-1,j+1}^{(2)}=&{}-4P_{i-1,j+1}^{(2)}+2P_{i-1,j+1}^{(3)}+2{\bar{P}}_{i-1,j+1}^{(0)}-\frac{3}{2}{\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i-1,j+1}^{(3)}=&{}{\bar{P}}_{i,j+1}^{(3)}, {\bar{P}}_{i-1,j+1}^{(4)}={\bar{P}}_{i-1,j}^{(4)}+{\bar{P}}_{i-1,j}^{(7)}+{\bar{P}}_{i-1,j}^{(8)}\\ {\bar{P}}_{i-1,j+1}^{(5)}=&{}{\bar{P}}_{i-1,j}^{(5)}+{\bar{P}}_{i-1,j}^{(9)}, {\bar{P}}_{i-1,j+1}^{(6)}={\bar{P}}_{i-1,j}^{(6)}\\ {\bar{P}}_{i-1,j+1}^{(7)}=&{}4P_{i-1,j+1}^{(4)}-P_{i-1,j+1}^{(3)}-4P_{i-1,j+1}^{(2)}+{\bar{P}}_{i-1,j+1}^{(0)}-2{\bar{P}}_{i-1,j+1}^{(4)}\\ &{}-{\bar{P}}_{i-1,j+1}^{(5)}-\frac{1}{2}{\bar{P}}_{i-1,j+1}^{(6)}-2{\bar{P}}_{i,j+1}^{(2)}-\frac{5}{2}{\bar{P}}_{i,j+1}^{(3)}-{\bar{P}}_{i,j+1}^{(1)}\\ {\bar{P}}_{i-1,j+1}^{(8)}=&{}-2P_{i-1,j+1}^{(3)}+4P_{i-1,j+1}^{(2)}-2{\bar{P}}_{i-1,j+1}^{(0)}+{\bar{P}}_{i,j+1}^{(2)}+\frac{3}{2}{\bar{P}}_{i,j+1}^{(3)}\\ {\bar{P}}_{i-1,j+1}^{(9)}=&{}-4P_{i-1,j+1}^{(4)}+2P_{i-1,j+1}^{(3)}+2{\bar{P}}_{i-1,j+1}^{(0)}+2{\bar{P}}_{i-1,j+1}^{(4)}\\ &{}+{\bar{P}}_{i-1,j+1}^{(5)}+\frac{1}{2}{\bar{P}}_{i-1,j+1}^{(6)}+2{\bar{P}}_{i,j+1}^{(1)}+2{\bar{P}}_{i,j+1}^{(2)}+2{\bar{P}}_{i,j+1}^{(3)}\\ \end{array}\right. } \end{aligned}$$

((A.14))

For $S_{i+1,j-1}$ patch, ${\bar{P}}_{i+1,j-1}^{(k)}$ ($k=0, 1, 2, 3,...,9$) are explicitly determined by the following analytical formulae:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{P}}_{i+1,j-1}^{(0)}=&{}{\bar{P}}_{i,j-1}^{(0)}+{\bar{P}}_{i,j-1}^{(4)}+{\bar{P}}_{i,j-1}^{(5)}+{\bar{P}}_{i,j-1}^{(6)}\\ {\bar{P}}_{i+1,j-1}^{(1)}=&{}{\bar{P}}_{i,j-1}^{(1)}+{\bar{P}}_{i,j-1}^{(7)}+{\bar{P}}_{i,j-1}^{(9)}, {\bar{P}}_{i+1,j-1}^{(2)}={\bar{P}}_{i,j-1}^{(2)}+{\bar{P}}_{i,j-1}^{(8)}\\ {\bar{P}}_{i+1,j-1}^{(3)}=&{}{\bar{P}}_{i,j-1}^{(3)}, {\bar{P}}_{i+1,j-1}^{(4)}=4P_{i+1,j-1}^{(8)}-P_{i+1,j-1}^{(7)}-3{\bar{P}}_{i+1,j-1}^{(0)}+\frac{1}{2}{\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j-1}^{(5)}=&{}-4P_{i+1,j-1}^{(8)}+2P_{i+1,j-1}^{(7)}+2{\bar{P}}_{i+1,j-1}^{(0)}-\frac{3}{2}{\bar{P}}_{i+1,j}^{(6)}, {\bar{P}}_{i+1,j-1}^{(6)}={\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j-1}^{(7)}=&{}-4P_{i+1,j-1}^{(8)}-P_{i+1,j-1}^{(7)}+4P_{i+1,j-1}^{(6)}+{\bar{P}}_{i+1,j-1}^{(0)}-2{\bar{P}}_{i+1,j-1}^{(1)}\\ {} &{}-{\bar{P}}_{i+1,j-1}^{(2)}-\frac{1}{2}{\bar{P}}_{i+1,j-1}^{(3)}-{\bar{P}}_{i+1,j}^{(4)}-2{\bar{P}}_{i+1,j}^{(5)}-\frac{5}{2}{\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j-1}^{(8)}=&{}2P_{i+1,j-1}^{(7)}-4P_{i+1,j-1}^{(6)}+2{\bar{P}}_{i+1,j-1}^{(0)}+2{\bar{P}}_{i+1,j-1}^{(1)}\\ {} &{}+{\bar{P}}_{i+1,j-1}^{(2)}+\frac{1}{2}{\bar{P}}_{i+1,j-1}^{(3)}+2{\bar{P}}_{i+1,j}^{(4)}+2{\bar{P}}_{i+1,j}^{(5)}+2{\bar{P}}_{i+1,j}^{(6)}\\ {\bar{P}}_{i+1,j-1}^{(9)}=&{}4P_{i+1,j-1}^{(8)}-2P_{i+1,j-1}^{(7)}-2{\bar{P}}_{i+1,j-1}^{(0)}\\ &{}+{\bar{P}}_{i+1,j}^{(5)}+\frac{3}{2}{\bar{P}}_{i+1,j}^{(6)}\\ \end{array}\right. } \end{aligned}$$

((A.15))