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数学代写|曲线和曲面代写Curves And Surfaces代考|M4190 JOINING A SPLINE PATCH TO A TRIMMED SPLINE PATCH

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数学代写|曲线和曲面代写Curves And Surfaces代考|JOINING A SPLINE PATCH TO A TRIMMED SPLINE PATCH

Consider a planar spline curve $c$ located in the rectangular domain of a spline patch $p$. Suppose that a part of the domain of this patch bounded by the curve $c$ is rejected, thus producing a trimmed patch whose boundary, or its part, is a spline curve having the parametrisation $\boldsymbol{p} \circ \boldsymbol{c}$. A specific design may require constructing a spline patch $p^*$ adherent to this curve in such a way that the junction of the two patches is smooth.

If the patch $\boldsymbol{p}$ is bicubic and the degree of the curve $\boldsymbol{c}$ is 3 , then the parametrisation $\boldsymbol{p} \circ \boldsymbol{c}$ has the degree $(3+3) \cdot 3=18$. Splines of that high or even higher degrees are troublesome, making the smoothness of the junction a goal very hard to score. Even the positional continuity would require that the degree of the patch $p^$ with respect to one parameter be at least 18 . Moreover, the sequence of knots of the spline patch $p^$ for this parameter consists of the knots of the spline curve $c$ and knots corresponding to intersections of this curve with the lines $u=u_i$ and $v=v_j$, where $u_i$ and $v_j$ are knots of the spline patch $p$; the latter have to be found by solving nonlinear algebraic equations.

For the reasons given above, it makes sense to give up even the positional continuity of the junction. Instead of the patch $p^$ of an impractically high degree, whose junction with the trimmed patch $\boldsymbol{p}$ is of class $G^1$ or $G^2$, it is possible to construct a bicubic B-spline patch $\hat{\boldsymbol{p}}^$ whose boundary approximates the boundary of the trimmed patch within a given tolerance. However, to develop such a construction we need to include in our theoretical considerations the patch $p^$ and to recognise the conditions which have to be satisfied by the patch $p$ and by the trimming curve $c$. If these conditions are satisfied, there exists a patch $p^$ whose junction with $p$ at the boundary curve obtained by trimming is of class $G^1$ or $G^2$. The patch $\hat{p}^$ being the result of the construction approximates $p^$ and its junction with $p$ may be said to be of class “quasi $G^1$ ” or “quasi $G^{2 “}$.

The idea of the construction is to obtain the boundary curve and one or two cross-boundary derivatives of the patch $\hat{\boldsymbol{p}}^$, and then to construct this patch by solving an interpolation problem. The cross-boundary derivatives of the patch $\hat{p}^$ are constructed using the partial derivatives of $\boldsymbol{p}$ at the points of the curve $c$ and junction functions.

数学代写|曲线和曲面代写Curves And Surfaces代考|HAHN’S SCHEME OF FILLING POLYGONAL HOLES

In [1988] Hahn outlined a general method of filling polygonal holes in surfaces made of tensor product patches. This method, with various modifications, was implemented in numerous constructions developed later. An outline of this outline given below will serve us as a reference point in the analysis of compatibility conditions made in this chapter and as a framework for constructions described in the next chapter.

A given surface with a hole is made of smooth regular tensor product patches having common boundary curves. Each pair of patches having a common curve may be reparametrised so that their rectangular domains have a common edge and the parametrisation of the surface made of the two patches over the union of the two rectangles is of class $C^n$. The boundary of the hole consists of $k$ smooth curves (made of boundary curves of the patches making the surface). The goal is to construct $k$ tensor product patches which would fill the hole; the junctions between the new patches and the given ones and between any two new patches having a common boundary are supposed to be of class $G^n$.

The first step of the construction is to find the cross-boundary derivatives of the given patches surrounding the hole, up to the order $n$ (Fig. 4.1a). The second step is to choose the common corner of the patches to be constructed, i.e., the “central point” of the filling surface, and vectors which will be the first-order derivatives of the boundary curves of the final patches at this point (Fig. 4.1b). These vectors, which must be coplanar (they determine the tangent plane of the surface at the central point), are related to what we call a partition of the full angle, which later in this chapter will be the subject of extensive study. Then in the third step (Fig. 4.1c) derivatives up to the order $2 n$ of one of the patches filling the hole at the central point are fixed. By reparametrisation of this patch (and using the generalised Fàa di Bruno’s formula, see Section A.11), it is possible to obtain the partial derivatives up to the order $2 n$ of all the other patches at the central

point (Fig. 4.1d). Then, by solving Hermite interpolation problems, we can construct the curves between the central point and the points in the middle of the edges of the hole (Fig. 4.1e); these curves will be the common curves of the patches. The next step is to construct auxiliary patches along these curves; the auxiliary patches determine planes tangent to the final patches along the curves (Fig. 4.1f) and, if a higher order geometric continuity is the goal, also the normal curvatures and attributes of the surface determined by higher order cross-boundary derivatives.

Hahn suggested constructing directly cross-boundary derivatives of one of the patches adjacent to each of the common curves and using them to construct the cross-boundary derivatives of the other patch using junction functions. In this way the final patches are constructed in a non-symmetrical way (Fig. 4.1g). Having the cross-boundary derivatives along all four boundary curves, we can obtain the final patches (Fig. 4.1h) as Coons patches (Section A.9)-bicubically blended, if the surface is of class $G^1$, biquintically if $G^2$, etc.

数学代写|曲线和曲面代写Curves And Surfaces代考|HAHN’S SCHEME OF FILLING POLYGONAL HOLES

Hahn 建议直接构造与每条公共曲线相邻的一个块的跨界导数，并使用它们使用连接函数构造另一个块的跨界导数。通过这种方 式，最終的补丁以非对称方式构建（图 4.1g）。沿着所有四个边界曲线具有跨边界导数，如果表面属于类，我们可以获得最终的 补丁 (图 4.1h) 作为 Coons 补丁 (第 A.9 节) -双三次混合 $G^1$ ，如果 $G^2$ ，ETC。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。