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# 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING THROUGH GEOMETRY

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## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING THROUGH GEOMETRY

(a) One of the earliest examples of mathematical modeling was that of mathematical description of the paths of planets. Looked at from the Earth the paths were not simple curves like circles or ellipses. The next curve known in order of complexity was an epicycloid which is the locus of a point on a circle which rolls on another fixed circle. The path of a planet was not even an epicycloid. However it was found possible to combine suitably a numbe of these epicycloidal curves or epicycles to describe the paths of all the planets. This was highly successful, though quite a complicated model.
(b) Another geometric modeling was involved in use of parabolic mirrors for burning enemy ships by Archimedes by concentrating the Sun’s parallel rays on them. The property that was used is that the line joining a point $P$ on a parabola to the focus $S$ and the line through $P$ parallel to the axis of the parabola make equal angles with the tangent (and the normal) at $P$ so that all parallel rays of the Sun can be reflected to only one point, i.e., to the focus $S$ (Figure 1.6).

(c) A similar geometric modeling is involved in constructing an elliptic sound gallery so that the sound produced at one focus can be heard at the other focus after being reflected back from every point of the ellipse (Figure 1.7).

(d) Based on the observations of Copernicus, Kepler showed that each planet moves in an ellipse with the Sun at one focus. Thus the heliocentric theory of planetory motion completely simplified the description of the paths of the planets. The earlier geocentric theory required a complicated combination of epicycloids. Both the models are correct, but the heliocentric model is much simpler than the geocentric model. However both the models were models for description only. Later Newton showed that the elliptical orbit followed from the universal law of gravitation and thus this model became a model for understanding. Still later in 1957, the elliptic orbits were used as orbits of satellites. At this stage, the model became a model for control. Now the same model can be used for getting optimal orbits for the satellites and as such it can also be used as a model for optimization.

## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING THROUGH ALGEBRA

(a) Finding the Radius of the Earth
This model was used about two thousand years ago. $A$ and $B$ are two points on the surface of the Earth with the same longitude and $d$ miles apart. When the Sun is vertically above $A$ (i.e., it is in the direction $O A$, where $O$ is the center of the Earth; Figure 1.9), the Sun’s rays make an angle of $\theta^{\circ}$ with the vertical at $B$ (i.e., with the line $O B$ ). If $a$ miles is the radius of the Earth, it is easily seen from Figure 1.9 that
$$\frac{d}{2 \pi a}=\frac{\theta}{360} \text { or } a=\frac{360 d}{2 \pi \theta}$$

(b) Motion of Planets
The orbit of each planet is an ellipse with the Sun at one focus. However the ellipticities of the orbits are very small, so that as a first approximation, we can take these orbits as circles with the Sun at the center. Also we know that the planets move under gravitational attraction of the Sun and that for motion in a circle with uniform speed $v$, a central acceleration $v^2 / r$ is required.
If the masses of the Sun and the planet are $S$ and $P$ respectively, we get
$$\frac{G P S}{r^2}=\frac{P v^2}{r} \text { or } v^2=\frac{G S}{r}$$
where $G$ is the constant of gravitation. Further if $T$ is the periodic time of the planet, we have
$$v T=2 \pi r$$
Eliminating $v$ between Eqns. (5) and (6), we get
$$T^2=\frac{4 \pi^2 r^3}{G S}$$
If $T_1, T_2$ are the time periods of two planets with orbital radii $r_1, r_2$, then
$$T_1^2 / T_2^2=r_1^3 / r_2^3$$
so that the squares of the periodic times are proportional to the cubes of the radii of the orbits.
(c) Motions of Satellites
Satellites move under the attraction of the Earth in the same way as the planets move under the attraction of the Sun, so that we get
$$T^2=\frac{4 \pi^2 r^3}{G E}, \frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}=\frac{\left(a+h_1\right)^3}{\left(a+h_2\right)^3}$$

where $E$ is the mass of the Earth, $a$ is the radius of the Earth, and $h_1, h_2$ are the heights of the satellites above the Earth’s surface. Also if $g$ is the acceleration due to gravity at the Earth’s surface, then
$$m g=\frac{G m E}{a^2} \text { or } G E=g a^2$$
From Eqns. (9) and (10), we get
$$T^2=\frac{4 \pi^2(a+h)^3}{g a^2}$$

## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING THROUGH GEOMETRY

(a)数学建模最早的例子之一是对行星运行轨迹的数学描述。从地球上看，这些路径不是简单的曲线，比如圆形或椭圆形。按照复杂程度排序的下一条曲线是外摆线，它是一个圆上的一点在另一个固定圆上滚动的轨迹。行星的轨道甚至不是一条外摆线。然而，人们发现可以适当地结合一些本轮曲线或本轮来描述所有行星的路径。这是一个非常成功的模式，尽管相当复杂。
(b)另一个几何模型是阿基米德利用抛物面镜将太阳的平行光线集中在敌舰上燃烧敌舰。所使用的属性是，将抛物线上的一点$P$与焦点$S$连接起来的线和通过$P$平行于抛物线轴线的线与$P$的切线(和法线)成相等的角度，以便所有平行的太阳光线只能反射到一个点，即焦点$S$(图1.6)。

(c)一个类似的几何建模涉及到构建一个椭圆声廊，这样在一个焦点产生的声音从椭圆的每一点反射回来后，可以在另一个焦点听到(图1.7)。

(d)根据哥白尼的观察，开普勒表明，每颗行星都以太阳为焦点在一个椭圆上运行。这样，行星运动的日心说就完全简化了对行星运行轨迹的描述。早期的地心说需要复杂的外摆线组合。两种模型都是正确的，但日心说模型比地心说模型简单得多。然而，这两个模型都只是用于描述的模型。后来牛顿证明了椭圆轨道遵循万有引力定律，因此这个模型成为了人们理解的模型。在1957年晚些时候，椭圆轨道被用作卫星轨道。在这个阶段，模型变成了用于控制的模型。现在同样的模型可以用来得到卫星的最优轨道因此它也可以用来作为优化的模型。

## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING THROUGH ALGEBRA

(a)计算地球的半径

$$\frac{d}{2 \pi a}=\frac{\theta}{360} \text { or } a=\frac{360 d}{2 \pi \theta}$$

(b)行星运动

$$\frac{G P S}{r^2}=\frac{P v^2}{r} \text { or } v^2=\frac{G S}{r}$$

$$v T=2 \pi r$$

$$T^2=\frac{4 \pi^2 r^3}{G S}$$

$$T_1^2 / T_2^2=r_1^3 / r_2^3$$

(c)卫星运动

$$T^2=\frac{4 \pi^2 r^3}{G E}, \frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}=\frac{\left(a+h_1\right)^3}{\left(a+h_2\right)^3}$$

$$m g=\frac{G m E}{a^2} \text { or } G E=g a^2$$

$$T^2=\frac{4 \pi^2(a+h)^3}{g a^2}$$

## 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。