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数学代写|曲线和曲面代写Curves And Surfaces代考|Removing knots and algebra of splines

If a spline curve of degree $n$ is represented in B-spline basis defined for a knot sequence $\hat{u}0, \ldots, \hat{u}{N+1}$, with a knot $\hat{u}{k+1}$ of multiplicity $r+1$, such that $\hat{u}_k \leq \hat{u}{k+1}<\hat{u}{k+2}$ and $n{k+1}$, then this knot is removable, i.e., there exists a representation of this curve with the knot sequence $\left(u_0, \ldots, u_N\right)=\left(\hat{u}0, \ldots, \hat{u}_k, \hat{u}{k+2}, \ldots, \hat{u}_{N+1}\right)$.

Algorithm A.9 is based on a linear dependency between the points $\boldsymbol{d}0, \ldots, \boldsymbol{d}{N-n-1}$ and the points $\hat{\boldsymbol{d}}0, \ldots, \hat{\boldsymbol{d}}{N-n}$ obtained by knot insertion:
\begin{aligned} &\boldsymbol{d}i=\hat{\boldsymbol{d}}_i \quad \text { for } i \leq k-n, \ &\frac{u{i+n}-\hat{u}{k+1}}{u{i+n}-u_i} \boldsymbol{d}{i-1}+\frac{\hat{u}{k+1}-u_i}{u_{i+n}-u_i} \boldsymbol{d}i=\hat{\boldsymbol{d}}_i \quad \text { for } i=k-n+1, \ldots, k-r, \ &\boldsymbol{d}{i-1}=\hat{\boldsymbol{d}}i \text { for } i>k-r . \ & \end{aligned} Given the control points $\hat{\boldsymbol{d}}_i$ representing an arbitrary B-spline curve we can write these equations with the intention of finding unknown control points $\boldsymbol{d}_i$. The number of equations is the number of unknown control points plus 1 . This system of equations is consistent if the knot $\hat{u}_k$ is removable and the curve has a shorter representation. To obtain a good accuracy, it is best to compute the points $\boldsymbol{d}{k-n}, \ldots, \boldsymbol{d}_{k-r}$ by solving a linear least-squares problem.

数学代写|曲线和曲面代写Curves And Surfaces代考|Convergence of repeated knot insertion

If an infinite sequence of numbers $v_1, v_2, \ldots$ is dense in the interval $\left[u_n, u_{N-n}\right)$, then the process of inserting these numbers as new knots produces an infinite sequence of representations of a given curve $\boldsymbol{s}$, with the control polylines made of increasing numbers of shorter and shorter line segments. These polylines converge to the curve. It may be proved (see Cohen and Schumaker [1985]) that the distance between the polyline and the curve is estimated by the expression $C h^2$, where the constant $C$ depends on the curve and $h$ is the maximal distance between consecutive knots of its representation. This is a fast convergence; often it suffices to insert relatively few knots in order to obtain a polyline being a very good approximation of the spline curve. This fact is exploited by the methods of generating surfaces via mesh refinement discussed in Section A.6.

数学代写|曲线和曲面代写Curves And Surfaces代考|THE DNIDED DIFFERENCES ALGORITHM

$$p_0(x)=1, p_1(x) \quad=x-u_0, p_2(x)=\left(x-u_0\right)\left(x-u_1\right), \quad \vdots p_n(x)=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) .$$

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。

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数学代写|曲线和曲面代写Curves And Surfaces代考|LAGRANGE AND HERMITE INTERPOLATION

Let $u_0, \ldots, u_n$ be given numbers. If each of them is different from the others, we can specify arbitrary numbers $f_0, \ldots, f_n$ and look for a function $h$ such that $h\left(u_i\right)=f_i$ for $i=0, \ldots, n$. This is called a Lagrange interpolation problem. The existence and uniqueness of its solution depends on additional conditions to be satisfied by the function $h$. In the linear space $\mathbb{R}[\cdot]_n$ the solution is unique; in other words, there exists a unique polynomial $h$ of degree at most $n$, satisfying the interpolation conditions considered above.

We obtain a more general problem by allowing some (or all) interpolation knots $u_0, \ldots, u_n$ to coincide. Obviously, we cannot specify two different function values for the same argument. But if a number $u_i$ appears $r$ times in the sequence $u_0, \ldots, u_n$ (we say that the knot $u_i$ has the multiplicity $r$ ), then we can demand that the function to be found at the knot $u_i$ and its derivatives up to the order $r-1$ take prescribed values. This interpolation problem, bearing the name of Charles Hermite, also has a unique solution in the space of real polynomials of degree at most $n$. Examples are shown in Figure A.1.

Functions of interpolation may be searched also in other function spaces, e.g. of splines or trigonometric polynomials; the existence of solutions depends on algebraic properties of those spaces. Having a Lagrange interpolation problem and a basis $\left{g_0, \ldots, g_n\right}$ of a function space, we can write the following system of linear equations:
$$\left[\begin{array}{ccc} g_0\left(u_0\right) & \ldots & g_n\left(u_0\right) \ \vdots & & \vdots \ g_0\left(u_n\right) & \ldots & g_n\left(u_n\right) \end{array}\right]\left[\begin{array}{c} a_0 \ \vdots \ a_n \end{array}\right]=\left[\begin{array}{c} f_0 \ \vdots \ f_n \end{array}\right]$$

数学代写|曲线和曲面代写Curves And Surfaces代考|THE DIVIDED DIFFERENCES ALGORITHM

The matrix of the system (A.1) written for the power basis $\left{1, x, \ldots, x^n\right}$ is full; it may be obtained with $O\left(n^2\right)$ operations and then the system may be solved with $O\left(n^3\right)$ operations, e.g. with the Gaussian elimination. This computational cost may be reduced by using a different basis. Let $p_0, \ldots, p_n$ be polynomials defined as follows:
\begin{aligned} p_0(x) &=1, \ p_1(x) &=x-u_0, \ p_2(x) &=\left(x-u_0\right)\left(x-u_1\right), \ & \vdots \ p_n(x) &=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) . \end{aligned}

数学代写|曲线和曲面代写Curves And Surfaces代考|THE DNIDED DIFFERENCES ALGORITHM

$$p_0(x)=1, p_1(x) \quad=x-u_0, p_2(x)=\left(x-u_0\right)\left(x-u_1\right), \quad \vdots p_n(x)=\left(x-u_0\right)\left(x-u_1\right) \ldots\left(x-u_{n-1}\right) .$$

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。

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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.

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。

Posted on Categories:Curves And Surfaces, 数学代写, 曲线和曲面

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avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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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。

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数学代写|曲线和曲面代写Curves And Surfaces代考|GEOMETRIC CONTINUITY AT A COMMON BOUNDARY

Consider two patches represented by smooth regular parametrisations, $p(s, t)$ and $p^(u, v)$. We assume that the patches have a common boundary curve being a constant parameter curve of the first patch, and corresponding to $t=t_0$, and a constant parameter curve of the second patch, corresponding to $v=v_0$. Let $I$ denote the line segment $v=v_0$, bounding the domain of the parametrisation $p^$. We assume that the two parametrisations of the common curve, obtained by restricting the parametrisations $\boldsymbol{p}$ and $\boldsymbol{p}^$ and denoted by $\overline{\boldsymbol{p}}$ and $\underline{p}^$, are identical: $\overline{\boldsymbol{p}}(s)=$ $\underline{p}^(u)$ for $s=u$. This assumption guarantees the positional continuity of the junction of the two patches: $$\bar{p}=\underline{p}^ .$$
The derivation of equations of geometric continuity for a junction of two patches is similar to that of curve arcs. Using a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, whose coordinates are described by scalar functions $s$ and $t$ (see Figure 3.1), we obtain a new parametrisaton of the patch $p$ :
$$q(u, v)=p(s(u, v), t(u, v)) .$$
We assume that the partial derivatives of $\boldsymbol{q}$ up to the order $n$ at each point of the line segment $I$ are equal to the corresponding derivatives of the parametrisation $p^*$.

We can obtain the derivatives of the parametrisation $\boldsymbol{q}$ using the generalised Fàa di Bruno’s formula (A.55) for functions of two variables. Then, we restrict them to the line segment $I$. We can notice that if the derivatives with respect to $v$ of $\boldsymbol{q}$ and $p^$ of any order $k$ are equal at each point of the line segment $I$, then also $$\left.\frac{\partial^{m+k}}{\partial u^m \partial v^k} \boldsymbol{q}\right|{v=v_0}=\left.\frac{\partial^{m+k}}{\partial u^m \partial v^k} \boldsymbol{p}^\right|{v=v_0}$$

for all $m$ such that these derivatives exist. Therefore, we can focus our attention on the partial derivatives with respect to $v$ – the parameter changing across the boundary-which is why they are called the cross-boundary derivatives of the patches.

The cross-boundary derivatives of the parametrisation $q$ are related to those of $p$ in the following way:
\begin{aligned} \overline{\boldsymbol{q}}v &=\bar{s}_v \overline{\boldsymbol{p}}_s+\bar{t}_v \overline{\boldsymbol{p}}_t, \ \overline{\boldsymbol{q}}{v v} &=\bar{s}{v v} \overline{\boldsymbol{p}}_s+\bar{t}{v v} \overline{\boldsymbol{p}}t+\bar{s}_v^2 \overline{\boldsymbol{p}}{s s}+2 \bar{s}v \bar{t}_v \overline{\boldsymbol{p}}{s t}+\bar{t}v^2 \overline{\boldsymbol{p}}{t t} \end{aligned}
etc. The general (and rather impractical) formula, which is a special case of (A.55), is
$\frac{\partial^j}{\partial v^j} \overline{\boldsymbol{q}}=\sum_{k=1}^j \sum_{h=0}^k a_{j k h} \frac{\partial^k}{\partial s^h \partial t^{k-h}} \overline{\boldsymbol{p}}$,
$a_{j k h}=\left(\begin{array}{l}k \ h\end{array}\right) \sum_{\substack{m_1+\cdots+m_k=j \ m_1, \ldots, m_k>0}} \frac{j !}{k ! m_{1} ! \ldots m_{k} !} \bar{s}{v^{m_1}} \ldots \bar{s}{v^{m_h}} \bar{t}{v^{m{h+1}} \ldots} \ldots \bar{t}_{v^{m_k}}$.

数学代写|曲线和曲面代写Curves And Surfaces代考|INTERPRETATION

We take a closer look at the junctions of patches of class $G^n$ for $n=1$ and $n=2$, bearing in mind that the patches have a common curve $\overline{\boldsymbol{p}}=\underline{p}^$. From the assumption that $\overline{\boldsymbol{p}}(s)=\underline{p}^(u)$ if $s=u$ it follows that all partial derivatives of the parametrisations $\boldsymbol{p}$ and $\boldsymbol{p}^*$ with respect to $s$ and $u$ at the junction points agree.

Case $n=1$. At any point of the common curve the cross-boundary derivative of $p^*$ is a linear combination of the first-order partial derivatives of $\boldsymbol{p}$. The partial derivatives of both patches at any point of their common curve determine the same plane (see Figure 3.2). Geometrically $G^1$ continuity is the continuity of tangent plane of the surface made of the two patches. One can also talk about the continuity of the normal vector, which is equivalent.
The same geometric interpretation applies to the equations for the homogeneous representations of the patches. The homogeneous patches reside in the four-dimensional space. At any junction point the triples of vectors, $\overline{\boldsymbol{P}}(s), \overline{\boldsymbol{P}}_s(s), \overline{\boldsymbol{P}}_t(s)$ and $\underline{\boldsymbol{P}}^(u), \underline{\boldsymbol{P}}_u^(u), \underline{\boldsymbol{P}}_v^*(u)$ span the same three-dimensional linear subspace (i.e., hyperplane) $\Pi(u)$ of $\mathbb{R}^4$. The common tangent plane $\pi(u)$ of the rational patches is represented by this hyperplane. ${ }^1$

Case $n=2$. The first and second fundamental forms (see Section A.10.2) are expressed by the derivatives of the first and second order of the surface’s parametrisation. The forms may be used to find the curvature of curves obtained by intersecting the surface with planes. If there exists a regular parametrisation of class $C^2$ of the surface, e.g. described piecewise by $q$ and $p^*$, then the curvature of the intersection of the surface with any plane not tangent to the surface is continuous.

On the other hand, having a surface whose all planar sections (with non-tangent planes) are curves with the curvature continuous, it is possible to find local regular parametrisations of class $C^2$ of this surface. ${ }^2$ Thus geometric continuity of the second order is equivalent to the curvature continuity of the surface.

数学代写|曲线和曲面代写Curves And Surfaces代考|GEOMETRIC CONTINUITY AT A COMMON BOUNDARY

$$\bar{p}=\underline{p}$$

$$q(u, v)=p(s(u, v), t(u, v)) .$$

$a_{j k h}=(k h) \sum_{m_1+\cdots+m_k=j m_1, \ldots, m_k>0} \frac{j !}{k ! m_{1} ! \ldots m_k} \bar{s} v^{m_1} \ldots \bar{s} v^{m_h} \bar{t} v^{m h+1} \ldots \ldots \bar{t}_{v^m k}$.

数学代写曲线和曲面代写Curves And Surfaces代考|INTERPRETATION

${ }^2$ 因此二阶几何连续性等价于曲面的曲率连续性。

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。

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数学代写|曲线和曲面代写Curves And Surfaces代考|Further properties and examples

A consequence of Property (iii) in the definition of $\beta$-spline functions is the affine invariance of the representation: if $A$ is an affine transformation, then the image $A(s)$ of a curve given by Formula (2.16) is represented by the control points $A\left(\boldsymbol{d}_i\right)$.

The convex hull property occurs if all $\beta$-spline functions are nonnegative, which depends on the connection parameters. In particular, if $\beta_{l, 1}=1$ and $\beta_{l, j}=0$ for $j>1$, then the $\beta$-spline functions are B-spline functions, which are nonnegative. Small enough perturbations of these particular connection parameters do not destroy this property, but it does not hold in general. If the functions $P_{k-n}^n, \ldots, P_k^n$ are nonnegative then the arc $\left{s(t): t \in\left[u_k, u_{k+1}\right]\right}$ is contained in the convex hull of the control points $\boldsymbol{d}_{k-n}, \ldots, \boldsymbol{d}_k$.

Figure $2.9$ shows two planar $\beta$-spline curves obtained with the knots and parameters that were used to define the cubic and quartic $\beta$-spline functions shown in Figure 2.8.

Figure $2.10$ shows two curves in the three-dimensional space: a cubic curve of class $G^2$ and a quartic curve of class $G^3$, with the same knots and connection parameters as the planar curves in the previous examples. The picture shows their curvature and torsion. To draw the picture, the Frenet frame has been found at a number of points of the curves and then the unit normal vector $\boldsymbol{n}$ was multiplied by the curvature $\kappa$ and a constant factor, and the binormal vector $\boldsymbol{b}$ was multiplied by the torsion $\tau$ and a constant factor. The curvature of both the curves is continuous. We can see points of discontinuity of the torsion of the cubic curve, while the torsion of the quartic curve is continuous.

The cubic curves in Figure $2.11$ have the same equidistant knots and the same (up to translations) control polygons; each of them was obtained with a different pair of global shape parameters, $\beta_1$ and $\beta_2$, which, for cubic curves, are named bias and tension. By looking at these curves one can see how the parameters influence the shape of the curve.

数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

There is another very natural way of associating a $S^1$-valued curve (and consequently a degree) with a closed regular plane curve.

Definition 2.4.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$, and let $\mathbf{t}:[a, b] \rightarrow S^1$ be its tangent versor, given by
$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$
The rotation index $\rho(\sigma)$ of $\sigma$ is the degree of the map $\mathbf{t}$; it counts the number of full turns made by the tangent versor to $\sigma$.

Corollary 2.1.18 provides us with a simple formula to compute the rotation index:

Proposition 2.4.2. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$ with oriented curvature $\tilde{\kappa}:[a, b] \rightarrow \mathbb{R}$. Then
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$
Proof. By Corollary 2.1.18,
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \bar{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

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。

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数学代写|曲线和曲面代写Curves And Surfaces代考|The Jordan curve theorem

In this section we shall complete the proof of the Jordan curve theorem for regular curves, by showing that the complement of the support of a simple

closed regular plane curve of class $C^2$ has at least two components. To get there we need a new ingredient, which we shall construct by using the degree introduced in Section 2.1.

Given a continuous closed plane curve, there are (at least) two ways to associate with it a curve with values in $S^1$, and consequently a degree. In this section we are interested in the first way, while in next section we shall use the second one.

Definition 2.3.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a continuous closed plane curve. Given a point $p \notin \sigma([a, b])$ we may define $\phi_p:[a, b] \rightarrow S^1$ by setting
$$\phi_p(t)=\frac{\sigma(t)-p}{|\sigma(t)-p|} .$$
The winding number $\iota_p(\sigma)$ of $\sigma$ with respect to $p$ is, by definition, the degree of $\phi_p$; it measures the number of times $\sigma$ goes around the point $p$.

Fig. $2.4$ shows the winding number of a curve with respect to several points, computed as we shall see in Example 2.3.5.

数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

There is another very natural way of associating a $S^1$-valued curve (and consequently a degree) with a closed regular plane curve.

Definition 2.4.1. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$, and let $\mathbf{t}:[a, b] \rightarrow S^1$ be its tangent versor, given by
$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$
The rotation index $\rho(\sigma)$ of $\sigma$ is the degree of the map $\mathbf{t}$; it counts the number of full turns made by the tangent versor to $\sigma$.

Corollary 2.1.18 provides us with a simple formula to compute the rotation index:

Proposition 2.4.2. Let $\sigma:[a, b] \rightarrow \mathbb{R}^2$ be a closed regular plane curve of class $C^1$ with oriented curvature $\tilde{\kappa}:[a, b] \rightarrow \mathbb{R}$. Then
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$
Proof. By Corollary 2.1.18,
$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

数学代写|曲线和曲面代写Curves And Surfaces代考|The Jordan curve theorem

$$\phi_p(t)=\frac{\sigma(t)-p}{|\sigma(t)-p|} .$$

数学代写|曲线和曲面代写Curves And Surfaces代考|The turning tangents theorem

$$\mathbf{t}(t)=\frac{\sigma^{\prime}(t)}{\left|\sigma^{\prime}(t)\right|} .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \tilde{\kappa}\left|\sigma^{\prime}\right| \mathrm{d} t=\frac{1}{2 \pi} \int_a^b \frac{\operatorname{det}\left(\sigma^{\prime}, \sigma^{\prime \prime}\right)}{\left|\sigma^{\prime}\right|^2} \mathrm{~d} t .$$

$$\rho(\sigma)=\frac{1}{2 \pi} \int_a^b \operatorname{det}\left(\mathbf{t}, \mathbf{t}^{\prime}\right) \mathrm{d} t$$

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。

Posted on Categories:Curves And Surfaces, 数学代写, 曲线和曲面

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数学代写|曲线和曲面代写Curves And Surfaces代考|Whitney’s Theorem

The goal of this section is to give a proof of Whitney’s Theorem 1.1.7. Let us start with some preliminary results.

Lemma 1.5.1. There exists a function $\alpha: \mathbb{R} \rightarrow[0,1)$ which is monotonic, of class $C^{\infty}$ and such that $\alpha(t)=0$ if and only if $t \leq 0$.
Proof. Set
$$\alpha(t)= \begin{cases}\mathrm{e}^{-1 / t} & \text { if } t>0 \ 0 & \text { if } t \leq 0\end{cases}$$
see Fig. 1.9.(a). Clearly, $\alpha$ takes values in $[0,1)$, is monotonic, is zero only in $\mathbb{R}^{-}$, and is of class $C^{\infty}$ in $\mathbb{R}^*$; we have only to check that it is of class $C^{\infty}$ in the origin too. To verify this, it suffices to prove that the right and left limits of all derivatives in the origin coincide, that is, that
$$\lim _{t \rightarrow 0^{+}} \alpha^{(n)}(t)=0$$
for all $n \geq 0$. Assume we have proved the existence, for all $n \in \mathbb{N}$, of a polynomial $p_n$ of degree $2 n$ such that
$$\forall t>0 \quad \alpha^{(n)}(t)=\mathrm{e}^{-1 / t} p_n(1 / t)$$

数学代写|曲线和曲面代写Curves And Surfaces代考|Classification of 1-submanifolds

As promised at the end of Section 1.1, we want to discuss now another possible approach to the problem of defining what a curve is. As we shall see, even if in the case of curves this approach will turn out to be too restrictive, for surfaces it will be the correct way to follow (as you shall learn in Section 3.1).
The idea consists in concentrating on the support. The support of a curve has to be a subset of $\mathbb{R}^n$ that looks (at least locally) like an interval of the real line. What we have seen studying curves suggests that a way to give concrete form to the concept of “looking like” consists in using homeomorphisms with the image that are regular curves of class at least $C^1$ too. So we introduce:
Definition 1.6.1. A 1-submanifold of class $C^k$ in $\mathbb{R}^n$ (with $k \in \mathbb{N}^* \cup{\infty}$ and $n \geq 2$ ) is a connected subset $C \subset \mathbb{R}^n$ such that for all $p \in C$ there exist a neighborhood $U \subset \mathbb{R}^n$ of $p$, an open interval $I \subseteq \mathbb{R}$, and a map $\sigma: I \rightarrow \mathbb{R}^n$ (called local parametrization) of class $C^k$, such that:
(i) $\sigma(I)=C \cap U$;
(ii) $\sigma$ is a homeomorphism with its image;
(iii) $\sigma^{\prime}(t) \neq O$ for all $t \in I$.
If $\sigma(I)=C$, we shall say that $\sigma$ is a global parametrization. A periodic parametrization is a map $\sigma: \mathbb{R} \rightarrow \mathbb{R}^n$ of class $C^k$ which is periodic of period $\ell>0$, with $\sigma(\mathbb{R})=C$, and such that for all $t_0 \in \mathbb{R}$ the restriction $\left.\sigma\right|_{\left(t_0, t_0+\ell\right)}$ is a local parametrization of $C$ having image $C \backslash\left{\sigma\left(t_0\right)\right}$.

数学代写|曲线和曲面代写Curves And Surfaces代考|Whitney’s Theorem

$$\alpha(t)= \begin{cases}\mathrm{e}^{-1 / t} & \text { if } t>00 \quad \text { if } t \leq 0\end{cases}$$

数学代写曲线和曲面代写Curves And Surfaces代考|Classification of 1submanifolds

，开区间 $I \subseteq \mathbb{R}$, 和一张地图 $\sigma: I \rightarrow \mathbb{R}^n$ 类的（称为局部参数化) $C^k$ ，这样:
(i) $\sigma(I)=C \cap U$
(二) $\sigma$ 是与其象的同胚;
(E) $\sigma^{\prime}(t) \neq O$ 对所有人 $t \in I$.

$\sigma(\mathbb{R})=C$ ，这样对于所有人 $t_0 \in \mathbb{R}$ 限制 $\left.\sigma\right|{(t 0, t 0+\ell)}$ 是一个局部参数化 $C$ 有形象 $\backslash$ left 的分隔符缺失或无法识别

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。

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数学代写|曲线和曲面代写Curves And Surfaces代考|Local theory of curves

Elementary geometry gives a fairly accurate and well-established notion of what is a straight line, whereas is somewhat vague about curves in general. Intuitively, the difference between a straight line and a curve is that the former is, well, straight while the latter is curved. But is it possible to measure how curved a curve is, that is, how far it is from being straight? And what, exactly, is a curve? The main goal of this chapter is to answer these questions. After comparing in the first two sections advantages and disadvantages of several ways of giving a formal definition of a curve, in the third section we shall show how Differential Calculus enables us to accurately measure the curvature of a curve. For curves in space, we shall also measure the torsion of a curve, that is, how far a curve is from being contained in a plane, and we shall show how curvature and torsion completely describe a curve in space. Finally, in the supplementary material, we shall present (in Section 1.4) the local canonical shape of a curve; we shall prove a result (Whitney’s Theorem 1.1.7, in Section 1.5) useful to understand what cannot be the precise definition of a curve; we shall study (in Section 1.6) a particularly well-behaved type of curves, foreshadowing the definition of regular surface we shall see in Chapter 3 ; and we shall discuss (in Section 1.7) how to deal with curves in $\mathbb{R}^n$ when $n \geq 4$.

数学代写|曲线和曲面代写Curves And Surfaces代考|How to define a curve

What is a curve (in a plane, in space, in $\mathbb{R}^n$ )? Since we are in a mathematical textbook, rather than in a book about military history of Prussian light cavalry, the only acceptable answer to such a question is a precise definition, identifying exactly the objects that deserve being called curves and those that do not. In order to get there, we start by compiling a list of objects that we consider without a doubt to be curves, and a list of objects that we consider without a doubt not to be curves; then we try to extract properties possessed by the former objects and not by the latter ones.

Example 1.1.1. Obviously, we have to start from straight lines. A line in a plane can be described in at least three different ways:

• as the graph of a first degree polynomial: $y=m x+q$ or $x=m y+q$;
• as the vanishing locus of a first degree polynomial: $a x+b y+c=0$;
• as the image of a map $f: \mathbb{R} \rightarrow \mathbb{R}^2$ having the form $f(t)=(\alpha t+\beta, \gamma t+\delta)$.
A word of caution: in the last two cases, the coefficients of the polynomial (or of the map) are not uniquely determined by the line; different polynomials (or maps) may well describe the same subset of the plane.

Example 1.1.2. If $I \subseteq \mathbb{R}$ is an interval and $f: I \rightarrow \mathbb{R}$ is a (at least) continuous function, then its graph
$$\Gamma_f={(t, f(t)) \mid t \in I} \subset \mathbb{R}^2$$
surely corresponds to our intuitive idea of what a curve should be. Note that we have
$$\Gamma_f={(x, y) \in I \times \mathbb{R} \mid y-f(x)=0},$$
that is a graph can always be described as a vanishing locus too. Moreover, it also is the image of the map $\sigma: I \rightarrow \mathbb{R}^2$ given by $\sigma(t)=(t, f(t))$.

数学代写|曲线和曲面代写Curves And Surfaces代考|How to define a curve

• 作为一次㝖项式的图: $y=m x+q$ 或者 $x=m y+q$;
• 作为一次多项式的消失轨迹: $a x+b y+c=0$;
• 作为地图的图像 $f: \mathbb{R} \rightarrow \mathbb{R}^2$ 有形式 $f(t)=(\alpha t+\beta, \gamma t+\delta)$.
需要注意的是: 在最后两种情况下，多项式 (或映射) 的条数不是由线唯一确定的; 不同的多项式（或映射）可以很好地描 述平面的同一子集。
示例 1.1.2。如果 $I \subseteq \mathbb{R}$ 是 个区间并且 $f: I \rightarrow \mathbb{R}$ 是 个 (至少) 连紏函数，那么它的图
$$\Gamma_f=(t, f(t)) \mid t \in I \subset \mathbb{R}^2$$
肯定符合我们对曲线应该是什么的直观想法。请注意，我们有
$$\Gamma_f=(x, y) \in I \times \mathbb{R} \mid y-f(x)=0,$$
那是一个图也总是可以描述为一个消失的轨迹。而且，它也是地图的图像 $\sigma: I \rightarrow \mathbb{R}^2$ 由 $\sigma(t)=(t, f(t))$.

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。