Posted on Categories:Non-Euclidean Geometry, 数学代写, 非欧几何

数学代写非欧几何代写Non-Euclidean Geometry代考|MAT470 The Existence of Similar Figures

avatest™

avatest™帮您通过考试

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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

数学代写非欧几何代写Non-Euclidean Geometry代考|The Existence of Similar Figures

The following statement is also equivalent to the Fifth Postulate and may be substituted for it, leading to the same consequences.
There exists a pair of similar triangles, i.e., triangles which are not congruent, but bave the three angles of one equal, respectively, to the three angles of the otber.

To show that this is equivalent to the Fifth Postulate, we need only show how to deduce the latter from it, since every student of Euclid knows that the use of the Postulate leads to a geometry in which similar figures exist.

Given two triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ (Fig. 7) with angles $A, B$ and $C$ equal, respectively, to angles $A^{\prime}, B^{\prime}$ and $C^{\prime}$. Let $A B$ be greater than $A^{\prime} B^{\prime}$. On $A B$ construct $A D$ equal to $A^{\prime} B^{\prime}$ and on $A C$ construct $A E$ equal to $A^{\prime} C^{\prime}$. Draw $D E$. Then triangles $A D E$ and $A^{\prime} B^{\prime} C^{\prime}$ are congruent. The reader can easily show that $A E$ is less than $A C$, for the assumption that $A E$ is greater than or equal to $A C$ leads to a contradiction. It will not be difficult now to prove that the quadrilateral $B C E D$ has the sum of its four angles equal to four right angles.

Very shortly we shall prove, ${ }^8$ without the use of the Fifth Postulate or its equivalent, that (a) the sum of the angles of a triangle can never be greater than two right angles, provided the straight line is assumed to be infinite, and (b) if one triangle has the sum of its angles equal to two right angles, then the sum of the angles of every triangle is equal to two right angles. By the use of these facts, our proof is easily completed.

By drawing $B E$, two triangles, $B D E$ and $B C E$, are formed. The angle-sum for nelther is greater than two right angles; if the anglesum for either were less than two right angles, that for the other would have to be greater. We conclude that the sum of the angles for each triangle is equal to two right angles and that the same is then true for every triangle.

数学代写非欧几何代写Non-Euclidean Geometry代考|Equidistant Straight Lines

Another noteworthy substitute is the following:
There exists a pair of straight lines everywbere equally distant from one anotber.

Once the Fifth Postulate is adopted, this statement follows, for then all parallels have this property of being everywhere equally distant. If the above statement is postulated, we can easily deduce the Fifth Postulate by first proving that there exists a triangle with the sum of its angles equal to two right angles.

Let $A B$ and $C D$ (Fig. 8) be the two lines everywhere equally distant. From any two points $O$ and $Q$ on $C D$ draw $O P$ and $Q R$ perpendicular to $A B$, and from any point $S$ on $A B$ draw $S T$ perpendicular to $C D$. By hypothesis $O P, Q R$ and $S T$ are equal. Since right triangles $O P S$ and $O T S$ are congruent,
Similarly
$\angle P S O=\angle T O S$.
$\angle R S Q=\angle T Q S$.
It follows that the sum of the angles of triangle $O S Q$ is equal to two right angles.

∠磷小号○=∠吨○小号.
∠R小号问=∠吨问小号.

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。