数学代写|最优化作业代写optimization theory代考|ESE504

2022年10月6日

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数学代写|最优化作业代写optimization theory代考|Bézier Curves

When we desire to represent a set of points $\left(x_i, y_i\right)$ by means of parametric curves, a possibility often used is Bézier curves. This technique consists in determining a couple of cubic Hermite polynomials depending on a parameter $t$ for each couple of consecutive points, a polynomial for $x(t)$ and a polynomial for $y(t)$. Let $\left(x_k, y_k\right)$ and $\left(x_{k+1}, y_{k+1}\right)$ be two such points (Figure 1.19). Any point of the curve is written as $(x(t), y(t))$. Thus, $t=0$ at the beginning point of the curve and $t=1$ at the endpoint, so that $x_k=x(0)$ and $x_{k+1}=x(1)$, and similarly $y_k=y(0)$ and $y_{k+1}=y(1)$.

The derivatives are specified at both extremities, thus $y^{\prime}(t) / x^{\prime}(t)$ for $t=0$ and $t=1$. As two cubic polynomials must be determined, this represents eight unknowns with only six constraints $\left{x_k, y_k, x_{k+1}, y_{k+1}, d y_k / d x_k, d y_{k+1} / d x_{k+1}\right}$. Thus, there exist two degrees of freedom that are fulfilled by specifying two guide-points, each one along a tangent line at one extremity. These guide-points are used to “pull” the curve. Let $\left(x_k+\alpha_k, y_k+\beta_k\right)$ and $\left(x_{k+1}+\alpha_{k+1}, y_{k+1}+\beta_{k+1}\right)$ be the coordinates of these two guidepoints. The Hermite polynomial $x(t)$ must verify $x^{\prime}(0)=\alpha_k$ and $x^{\prime}(1)=\alpha_{k+1}$, and the Hermite polynomial $y(t)$ must also verify $y^{\prime}(0)=\beta_k$ and $y^{\prime}(1)=\beta_{k+1}$. The tangents at the extremities must verify $\beta_k / \alpha_k=d y_k / d x_k$ and $\beta_{k+1} / \alpha_{k+1}=d y_{k+1} / d x_{k+1}$, which leaves a freedom for, either $\alpha$, or $\beta$, hence a displacement of the guide-points along the tangent lines. Both cubic Hermite polynomials are now completely specified and equal to
\begin{aligned} x(t)=& x_k+\alpha_k t+\left[3\left(x_{k+1}-x_k\right)-\left(2 \alpha_k+\alpha_{k+1}\right)\right] t^2+\ & {\left[2\left(x_k-x_{k+1}\right)+\left(\alpha_k+\alpha_{k+1}\right)\right] t^3, \quad t \in[0,1] } \ y(t)=& y_k+\beta_k t+\left[3\left(y_{k+1}-y_k\right)-\left(2 \beta_k+\beta_{k+1}\right)\right] t^2+\ & {\left[2\left(y_k-y_{k+1}\right)+\left(\beta_k+\beta_{k+1}\right)\right] t^3 } \end{aligned} The form of the parametric equations for Bézier curves is very slightly different from the previous Hermite polynomials, as each term $\alpha$ or $\beta$ is multiplied by a factor 3 , but this is not a fundamental change.

数学代写|最优化作业代写optimization theory代考|Numerical Integration

In some simple cases, the calculation of the definite integral
$$\int_a^b f(x) d x$$
is directly possible when the primitive (or antiderivative) function $F(x)$ is known
$$\int f(x) d x=F(x)$$
hence
$$\int_a^b f(x) d x=F(b)-F(a)$$
Most often, this is impossible and the only possible solution is numerical. Frequently, moreover, the function $f(x)$ is only known at a given number of points $x_i, i=$ $0,1, \ldots, n$. In this case, it is possible to search an approximation $g(x)$ of the function $f(x)$ and to proceed to a formal integration.

The interpolation polynomials $P_n(x)$ possess the required approximation properties and are easily integrable. Thus, they will be largely used in numerical integration (also called quadrature).

最优化代写

数学代写|最优化作业代写优化理论代考|Bézier曲线

\begin{aligned} x(t)=& x_k+\alpha_k t+\left[3\left(x_{k+1}-x_k\right)-\left(2 \alpha_k+\alpha_{k+1}\right)\right] t^2+\ & {\left[2\left(x_k-x_{k+1}\right)+\left(\alpha_k+\alpha_{k+1}\right)\right] t^3, \quad t \in[0,1] } \ y(t)=& y_k+\beta_k t+\left[3\left(y_{k+1}-y_k\right)-\left(2 \beta_k+\beta_{k+1}\right)\right] t^2+\ & {\left[2\left(y_k-y_{k+1}\right)+\left(\beta_k+\beta_{k+1}\right)\right] t^3 } \end{aligned} Bézier曲线的参数方程的形式与前面的埃尔米特多项式的形式有非常细微的不同，作为每一项 $\alpha$ 或 $\beta$ 是乘以一个因子3，但这不是一个根本的变化。

数学代写|最优化作业代写优化理论代考|数值积分

$$\int f(x) d x=F(x)$$

$$\int_a^b f(x) d x=F(b)-F(a)$$

$$\int_a^b f(x) d x$$

有限元方法代写

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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。