数学代写|线性代数代写linear algebra代考|MATH1014

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

Computer graphics are images displayed or animated on a computer screen. Applications of computer graphics are widespread and growing rapidly. For instance, computeraided design (CAD) is an integral part of many engineering processes, such as the aircraft design process described in the chapter introduction. The entertainment industry has made the most spectacular use of computer graphics – from the special effects in Amazing Spider-Man 2 to PlayStation 4 and Xbox One.

Most interactive computer software for business and industry makes use of computer graphics in the screen displays and for other functions, such as graphical display of data, desktop publishing, and slide production for commercial and educational presentations. Consequently, anyone studying a computer language invariably spends time learning how to use at least two-dimensional (2D) graphics.

This section examines some of the basic mathematics used to manipulate and display graphical images such as a wire-frame model of an airplane. Such an image (or picture) consists of a number of points, connecting lines or curves, and information about how to fill in closed regions bounded by the lines and curves. Often, curved lines are approximated by short straight-line segments, and a figure is defined mathematically by a list of points.

Among the simplest 2D graphics symbols are letters used for labels on the screen. Some letters are stored as wire-frame objects; others that have curved portions are stored with additional mathematical formulas for the curves.

EXAMPLE 1 The capital letter $\mathrm{N}$ in Figure 1 is determined by eight points, or vertices. The coordinates of the points can be stored in a data matrix, $D$.
$x$-coordinate $\left[\begin{array}{cccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \ 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 \ 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]=D$
In addition to $D$, it is necessary to specify which vertices are connected by lines, but we omit this detail.

The main reason graphical objects are described by collections of straight-line segments is that the standard transformations in computer graphics map line segments onto other line segments. (For instance, see Exercise 27 in Section 1.8.) Once the vertices that describe an object have been transformed, their images can be connected with the appropriate straight lines to produce the complete image of the original object.

数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

By analogy with the $2 \mathrm{D}$ case, we say that $(x, y, z, 1)$ are homogeneous coordinates for the point $(x, y, z)$ in $\mathbb{R}^3$. In general, $(X, Y, Z, H)$ are homogeneous coordinates for $(x, y, z)$ if $H \neq 0$ and
$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$
Each nonzero scalar multiple of $(x, y, z, 1)$ gives a set of homogeneous coordinates for $(x, y, z)$. For instance, both $(10,-6,14,2)$ and $(-15,9,-21,-3)$ are homogeneous coordinates for $(5,-3,7)$.

The next example illustrates the transformations used in molecular modeling to move a drug into a protein molecule.
EXAMPLE 7 Give $4 \times 4$ matrices for the following transformations:
a. Rotation about the $y$-axis through an angle of $30^{\circ}$. (By convention, a positive angle is the counterclockwise direction when looking toward the origin from the positive half of the axis of rotation-in this case, the $y$-axis.)
b. Translation by the vector $\mathbf{p}=(-6,4,5)$.
SOLUTION
a. First, construct the $3 \times 3$ matrix for the rotation. The vector $\mathbf{e}_1$ rotates down toward the negative $z$-axis, stopping at $\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. The vector $\mathbf{e}_2$ on the $y$-axis does not move, but $\mathbf{e}_3$ on the $z$-axis rotates down toward the positive $x$-axis, stopping at $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. See Figure 5. From Section $1.9$, the standard matrix for this rotation is
$$\left[\begin{array}{ccc} \sqrt{3} / 2 & 0 & .5 \ 0 & 1 & 0 \ -.5 & 0 & \sqrt{3} / 2 \end{array}\right]$$
So the rotation matrix for homogeneous coordinates is
$$A=\left[\begin{array}{cccc} \sqrt{3} / 2 & 0 & .5 & 0 \ 0 & 1 & 0 & 0 \ -.5 & 0 & \sqrt{3} / 2 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
b. We want $(x, y, z, 1)$ to map to $(x-6, y+4, z+5,1)$. The matrix that does this is
$$\left[\begin{array}{rrrr} 1 & 0 & 0 & -6 \ 0 & 1 & 0 & 4 \ 0 & 0 & 1 & 5 \ 0 & 0 & 0 & 1 \end{array}\right]$$

线性代数代考

数学代写|线性代数代写linear algebra代考|APPLICATIONS TO COMPUTER GRAPHICS

$\left[\begin{array}{llllllllllllllllllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 0 & .5 & .5 & 6 & 6 & 5.5 & 5.5 & 0 & 0 & 0 & 6.42 & 0 & 8 & 8 & 1.58 & 8\end{array}\right]$ 此外 $D$ ，有必要指定哪些顶点由线连接，但我们省略了这个细节。

数学代写|线性代数代写linear algebra代考|Homogeneous 3D Coordinates

$$x=\frac{X}{H}, \quad y=\frac{Y}{H}, \quad \text { and } \quad z=\frac{Z}{H}$$

a。旋转关于 $y$-轴通过一个角度 $30^{\circ}$. (按照惯例，正角是从旋转轴的正半边看原点时的逆时针方向一一在 这种情况下， $y$-轴。)
$\mathrm{b}$ 。向量翻译 $\mathbf{p}=(-6,4,5)$.

$\left(\cos 30^{\circ}, 0,-\sin 30^{\circ}\right)=(\sqrt{3} / 2,0,-.5)$. 载体 $\mathbf{e}_2$ 在 $y$-轴不移动，但 $\mathbf{e}_3$ 在 $z$-axis 向下旋转到正 $x$ 轴，停在 $\left(\sin 30^{\circ}, 0, \cos 30^{\circ}\right)=(.5,0, \sqrt{3} / 2)$. 参见图 5。从部分 $1.9$ ，这个旋转的标准矩阵是

b. 我们想要 $(x, y, z, 1)$ 映射到 $(x-6, y+4, z+5,1)$. 这样做的矩阵是

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。