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# 计算机代写|计算机图形学代考Computer Graphics代考|CSCI371 Background

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## 计算机代写|计算机图形学代考Computer Graphics代考|Background

Modern algebraic notation has evolved over thousands of years where different civilisations developed ways of annotating mathematical and logical problems. The word ‘algebra’ comes from the Arabic ‘al-jabr w’al-muqabal’ meaning ‘restoration and reduction’. In retrospect, it does seem strange that centuries passed before the ‘equals’ sign $(=)$ was invented, and concepts such as ‘zero’ (CE 876) were introduced, especially as they now seem so important. But we are not at the end of this evolution, because new forms of annotation and manipulation will continue to emerge as new mathematical objects are invented.

One fundamental concept of algebra is the idea of giving a name to an unknown quantity. For example, $m$ is often used to represent the slope of a $2 \mathrm{D}$ line, and $c$ is the line’s $y$-coordinate where it intersects the $y$-axis. René Descartes formalised the idea of using letters from the beginning of the alphabet $(a, b, c, \ldots)$ to represent arbitrary quantities, and letters at the end of the alphabet $(p, q, r, s, t, \ldots, x, y, z)$ to represent quantities such as pressure $(p)$, time $(t)$ and coordinates $(x, y, z)$.

With the aid of the basic arithmetic operators: $+,-, \times$, / we can develop expressions that describe the behaviour of a physical process or a logical computation. For example, the expression $a x+b y-d$ equals zero for a straight line. The variables $x$ and $y$ are the coordinates of any point on the line and the values of $a, b$ and $d$ determine the position and orientation of the line. The $=$ sign permits the line equation to be expressed as a self-evident statement:
$$0=a x+b y-d .$$
Such a statement implies that the expressions on the left- and right-hand sides of the $=$ sign are ‘equal’ or ‘balanced’, and in order to maintain equality or balance,whatever is done to one side, must also be done to the other. For example, adding $d$ to both sides, the straight-line equation becomes
$$d=a x+b y .$$
Similarly, we could double or treble both expressions, divide them by 4 , or add 6 , without disturbing the underlying relationship. When we are first taught algebra, we are often given the task of rearranging a statement to make different variables the subject. For example, (3.1) can be rearranged such that $x$ is the subject:
\begin{aligned} y &=\frac{x+4}{2-\frac{1}{z}} \ y\left(2-\frac{1}{z}\right) &=x+4 \ x &=y\left(2-\frac{1}{z}\right)-4 \end{aligned}

## 计算机代写|计算机图形学代考Computer Graphics代考|Solving the Roots of a Quadratic Equation

Problem solving is greatly simplified if one has solved it before, and having a good memory is always an advantage. In mathematics, we keep coming across problems that have been encountered before, apart from different numbers. For example, $(a+b)(a-b)$ always equals $a^2-b^2$, therefore factorising the following is a trivial exercise:
\begin{aligned} &a^2-16=(a+4)(a-4) \ &x^2-49=(x+7)(x-7) \ &x^2-2=(x+\sqrt{2})(x-\sqrt{2}) . \end{aligned}
A perfect square has the form:
$$a^2+2 a b+b^2=(a+b)^2 .$$
Consequently, factorising the following is also a trivial exercise:
\begin{aligned} a^2+4 a b+4 b^2 &=(a+2 b)^2 \ x^2+14 x+49 &=(x+7)^2 \ x^2-20 x+100 &=(x-10)^2 \end{aligned}

## 计算机代写|计算机图形学代考Computer Graphics代考|Background

$$0=a x+b y-d .$$

$$d=a x+b y .$$

$$y=\frac{x+4}{2-\frac{1}{z}} y\left(2-\frac{1}{z}\right) \quad=x+4 x=y\left(2-\frac{1}{z}\right)-4$$

## 计算机代写|计算机图形学代考Computer Graphics代考|Solving the Roots of a Quadratic Equation

$$a^2-16=(a+4)(a-4) \quad x^2-49=(x+7)(x-7) x^2-2=(x+\sqrt{2})(x-\sqrt{2}) .$$

$$a^2+2 a b+b^2=(a+b)^2 .$$

$$a^2+4 a b+4 b^2=(a+2 b)^2 x^2+14 x+49 \quad=(x+7)^2 x^2-20 x+100=(x-10)^2$$

## 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。