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# 数学代写|曲线和曲面代写Curves And Surfaces代考|MATH2242 Constructing junction functions and domain patches

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## 数学代写|曲线和曲面代写Curves And Surfaces代考|Constructing junction functions and domain patches

We focus on one curvilinear quadrangle $\Omega_i$, and to make things easier we use new symbols. A part of the mapping scheme from Figure $5.3$ is repeated four times in Figure 5.10, where all four copies share the same mapping $\boldsymbol{\delta}_i$ ( $\boldsymbol{\delta}$ in Theorem 5.1), which we named the domain patch. We are going to construct it now. Each side of the unit square on the left side corresponds to one line segment $\Psi$, which is mapped by $\delta_i$ to the corresponding boundary curve of $\Omega_i$. In the previous steps, we constructed a mapping $\boldsymbol{\beta}$ for each side of the square; two of those mappings are represented by the bicubic patches determined by the domain net, while the other two are the auxiliary domain patches represented by the curves dividing $\Omega$ and by the cross-boundary derivatives. The auxiliary domain patches corresponding to the line segments $u=0$ and $v=0$ will be denoted here by $\boldsymbol{q}_0$ and $\boldsymbol{r}_0$, while the symbols $\boldsymbol{q}_1$ and $\boldsymbol{r}_1$ will be used for the other two, corresponding to $u=1$ and $v=1$.

## 数学代写|曲线和曲面代写Curves And Surfaces代考|Constructing basis function patches

Consider a Sabin net in $\mathbb{R}^3$ whose projection on $\mathbb{R}^2$, obtained by rejecting the $z$-coordinate of all vertices, is the domain net. The surface made of bicubic patches represented by this Sabin net is the graph of a scalar function of class $C^2$, defined in the area $A \backslash \Omega$. We are going to extend any such function to obtain a function $\phi_j$ of class $C^1$ or $C^2$ in the entire area $A$. To do this, for each curvilinear quadrangle $\Omega_i$ which is an image of the unit square under the mapping $\boldsymbol{\delta}i$, we construct a scalar function (a bivariate polynomial, denoted by $\mu$ in Figure $5.3$, and by $\mu{i j}$ if an indication of the area $\Omega_i$ is needed) whose domain is the unit square. The extension of the function from $A \backslash \Omega$ to the entire area $A$ is in $\Omega_i$ the composition $\mu_{i j} \circ \delta_i^{-1}$.

Actually, we are going to find bases of two linear vector spaces whose elements are functions in $A$. The elements $\hat{\phi}_j$ of a basis of the first space, denoted by $V_1$, are functions taking non-zero values at the boundary of the area $\Omega$; any such function is related to a Sabin net of radius 2 having only one vertex with the $z$-coordinate not equal to 0 . The Sabin net of radius 2 with the extraordinary vertex incident with $k$ edges (corresponding to a $k$-sided hole in the surface) has $6 k+1$ vertices. Therefore, we need $6 k+1$ functions, which form a basis of the space $V_1$; each of them corresponds to a Sabin net having one vertex with the coordinate $z=1$ and the other vertices in the $x y$ plane. The orthogonal projection of all these Sabin nets on this plane is the domain net.

The second space, $V_0$, is made of functions taking non-zero values only in the area $\Omega$. This space is needed to construct regular final patches and to optimise their shape. Its dimension depends on the partition of the full angle determined by the halflines tangent to the curves dividing $\Omega$ at the central point.

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