Introduction to Geometry
An Introduction to Mathematics 金融学数学基础知识
Answer
FV 15,000 1.5 ln(1.5) 0.405 0.135 e0.135 1.144 r = C0 (1+r)T = 10,000(1+r) 3 = (1+r) 3 = ln(1+r) 3 = 3ln(1+r) = ln (1+r) = 1+r =1+r ≈ 0.144 or 14.4%
THE VALUE OF MONEY: FUTURE VALUE
• Let us assume that one invests an amount of money at time zero (C0) in an account for T years and that the going annual interest rate is r. Then, the value of this amount after the completion of T years shall equal: FV = C0 (1+r)T where FV = Future Value • However, the above assumes annual compounding of the interest, yet the latter could be compounded: Semiannually: FV = C0 [1+(r/2)]2T Quarterly: FV = C0 [1+(r/4)]4T Monthly: FV = C0 [1+(r/12)]12T Daily: FV = C0 [1+(r/360)]360T Continuously: FV = C0 erT
Introduction to Mathematics & Statistics
介绍数学的英语
介绍数学的英语Mathematics is the study of numbers, shapes, patterns, and their relationships. It is a field that deals with logical reasoning and problem-solving using numerical calculations, measurements, and mathematical models. Math is used extensively in various disciplines such as physics, engineering, finance, computer science, and many more.Here are 27 bilingual example sentences related to mathematics:1.数学是一门需要逻辑推理和问题解决的学科。
Mathematics is a discipline that requires logical reasoning and problem-solving.2.数学是一种描述和量化现实世界的语言。
Mathematics is a language that describes and quantifies the real world.3.我们使用数学来解决实际生活中的各种问题。
We use mathematics to solve various problems in everyday life.4.算数是数学的一个重要分支，涉及基本的加减乘除运算。
Arithmetic is an important branch of mathematics that involves basic operations like addition, subtraction, multiplication, and division.5.代数是研究数之间关系和未知量的分支。
几何英语知识点归纳总结
几何英语知识点归纳总结In this article, we will delve into the key concepts and principles of geometry, covering a wide range of topics from basic shapes to advanced theorems. By the end of this article, you will have a comprehensive understanding of geometry and be able to apply its principles to solve a variety of problems.Basic Concepts in Geometry1. Points, Lines, and Planes: The foundation of geometry lies in the ideas of points, lines, and planes. A point is a location in space, represented by a dot. A line is a straight path that extends in both directions infinitely, with no width or thickness. A plane is a flat, two-dimensional surface that extends infinitely in all directions.2. Angles: An angle is formed when two rays share a common endpoint, referred to as the vertex. Angles are measured in degrees, with a full circle representing 360 degrees. There are different types of angles, including acute angles (less than 90 degrees), obtuse angles (greater than 90 degrees), right angles (exactly 90 degrees), straight angles (exactly 180 degrees), and reflex angles (greater than 180 degrees).3. Polygons: A polygon is a closed shape made up of straight line segments. The most common types of polygons are triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), and so on. Polygons can be classified based on the number of sides and angles they have.4. Circles: A circle is a set of all points in a plane that are equidistant from a given center point. The distance from the center to any point on the circle is called the radius, and the distance across the circle through the center is called the diameter. The ratio of the circumference of a circle to its diameter is a constant value known as pi (π), approximately equal to 3.14159.5. Similarity and Congruence: Two geometric figures are similar if they have the same shape but different sizes. They are congruent if they have the same shape and size. These concepts are fundamental in understanding the relationships between different geometric figures.6. Perimeter and Area: The perimeter of a shape is the distance around its boundary, while the area is the measure of the space inside the boundary. Different formulas are used to calculate the perimeter and area of various shapes, such as rectangles, triangles, circles, and so on.Advanced Concepts in Geometry1. Pythagorean Theorem: This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is expressed as a^2 + b^2 = c^2, where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.2. Theorems of Euclidean Geometry: Euclidean geometry, named after the ancient Greek mathematician Euclid, is the study of plane and solid figures based on a set of axioms and theorems. Some of the key theorems include the Parallel Postulate, the Angle Sum Theorem, the Pythagorean Theorem, and the Midpoint Theorem.3. Transformations: Transformations in geometry refer to the ways in which a figure can be moved, reflected, rotated, or scaled without changing its shape or size. Common transformations include translations (sliding), reflections (flipping), rotations (turning), and dilations (resizing).4. Coordinates and Graphs: The coordinate plane is a fundamental tool in geometry, consisting of two perpendicular number lines that intersect at the origin (0,0). Points on the plane are represented by ordered pairs of numbers (x,y) called coordinates. By plotting points on the coordinate plane, geometric figures and relationships can be visualized and analyzed.5. Trigonometry: Trigonometry is the branch of mathematics that deals with the study of angles and the lengths of their sides in triangles. It is an essential tool in understanding the relationships between angles and sides, and is widely used in fields such as engineering, physics, and navigation.6. Three-Dimensional Geometry: In addition to the two-dimensional shapes and figures, geometry also encompasses three-dimensional objects such as prisms, pyramids, spheres, cones, and cylinders. Three-dimensional geometry involves the measurement of volume, surface area, and spatial relationships between solid figures.Applications of GeometryThe principles of geometry have a wide range of applications in various fields of study and professions. Some of the key applications include:1. Architecture: Architects use geometry to design and construct buildings, bridges, and other structures. Understanding principles of symmetry, proportion, and spatial relationships is crucial in creating aesthetically pleasing and structurally sound designs.2. Engineering: Engineers utilize geometry in designing and analyzing mechanical components, electrical circuits, and structural frameworks. Geometric concepts such as vectors, forces, and dimensions play a critical role in the field of engineering.3. Cartography: Cartographers use geometry in creating maps and geographic information systems. By understanding the principles of projections, scales, and angles, accurate representations of geographic features can be produced.4. Art and Design: Artists and designers often incorporate geometric shapes, patterns, and proportions in their work. Understanding geometric principles allows them to create compositions with balance, harmony, and visual appeal.5. Computer Graphics: The field of computer graphics heavily relies on geometric algorithms and principles to create visual representations of virtual environments, objects, and characters in video games, movies, and simulations.ConclusionGeometry is a fundamental branch of mathematics that encompasses the study of shapes, sizes, and properties of space. By understanding the basic concepts of points, lines, angles, polygons, and circles, as well as the advanced principles of transformations, coordinates, trigonometry, and three-dimensional geometry, we are able to make sense of the world around us in a more precise and systematic manner.The principles of geometry have a wide range of applications in various fields of study and professions, including architecture, engineering, cartography, art and design, and computer graphics. By applying the principles of geometry to solve problems and analyze real-world scenarios, we are able to create, innovate, and understand the world in a more profound way. As we continue to advance in our understanding of geometry, we open up new possibilities for exploration, discovery, and creativity in the world of mathematics and beyond.。
几何学简介geometry概要
什么是数学
Mathematical physics Fluid dynamics Numerical analysis Optimization Probability theory Statistics Financial mathematics Mathematical biology Mathematical chemistry Mathematical economics Control theory Mathematical logic Set theory Theory of computation
（希尔伯特（Hilbert）同时也做了相应工作）
《广义相对论》1915年 (狭义相对论, 1905年) 物理几何化，引力对应于曲率
整体微分几何
Gauss-Bonnet-Chen公式
D D
（ - i） k g ds KdA 2 ( D) 其中i为内角，k g为测地曲率， K为Gauss 曲率， ( D)为Euler数
《几何原本》五条公设
1. 2. 3.
4.
5.
过相异两点能且只能作一直线 线段可任意延长 以任意一点为圆心,任意长为半径可作一圆 凡直角都相等 两直线被第三直线所截,若两内角和小于两直 角,则两直线作延长时在此侧相交
第五公设与下列命题等价 •在一平面内,过直线外一点可作且只能作一直线与此直线平行 •三角形内角和为两直角 •存在面积足够大的三角形 •存在矩形 •存在相似而不全等的三角形
曲率角度非欧几何
欧氏几何：曲率为零 双曲几何：曲率为负常数 椭圆几何：曲率为正常数 （上述几何为黎曼几何特例）
平面
伪球面
球面
公理角度非欧几何
希尔伯特（Hilbert）：《几何基础》1899年 （从公理角度对欧氏几何，非欧几何做了严谨阐述）
机器人学基础_第3章_机器人运动学
Kinematics treats motion without regard to the forces that cause it. Within the science of kinematics one studies the position, velocity, acceleration, and all higher order derivatives of the position variables (with respect to time or any other variable). 从几何学 几何学的观点来处 几何学 理手指位置 手指位置P与关节变量 关节变量 手指位置 L1, L2, θ1 和 θ 2的关系称为 运动学(Kinematics)。 运动学
(3.9)
3.1 Representation of Kinematics Equation of Manipulator
17
3.1 Representation of Kinetic Equation of Robot Manipulator
3.1 Representation of Kinematics Equation of Manipulator
12
3.1.1 Kinetic Pose and Oriented Angle Roll, Pitch, Yaw to represent motion pose
geometry函数
geometry函数一、介绍geometry函数是一个用于处理几何图形的函数，它可以实现一系列几何图形的计算和操作。
几何图形是指二维或三维空间中的点、线、面等物体，是数学和物理学中重要的研究对象。
geometry函数可以帮助我们在程序中轻松地处理各种几何图形，包括计算它们的面积、周长、体积等。
二、基本概念在使用geometry函数之前，我们需要了解一些基本概念：1. 点：在二维平面上表示为(x,y)，在三维空间中表示为(x,y,z)。
2. 直线：由两个点确定，在二维平面上通常用斜率截距式表示为y=kx+b，在三维空间中通常用参数方程表示为x=x0+t*a,y=y0+t*b,z=z0+t*c。
3. 圆：由一个圆心和半径确定，在二维平面上通常用标准式表示为(x-a)^2+(y-b)^2=r^2，在三维空间中通常用参数方程表示为x=a+r*cos(t), y=b+r*sin(t), z=c。
4. 矩形：由四个顶点确定，在二维平面上通常用左下角坐标和右上角坐标表示为(x1,y1,x2,y2)，在三维空间中通常用六个面的坐标表示为(x1,y1,z1,x2,y2,z2)。
5. 三角形：由三个点确定，在二维平面上通常用三个顶点坐标表示为(x1,y1,x2,y2,x3,y3)，在三维空间中通常用三个顶点坐标表示为(x1,y1,z1,x2,y2,z2,x3,y3,z3)。
6. 多边形：由多个点确定，在二维平面上通常用顶点坐标数组表示，每个顶点的坐标为(x[i],y[i])，在三维空间中通常用顶点坐标数组表示，每个顶点的坐标为(x[i],y[i],z[i])。
7. 立体图形：包括球体、立方体、圆柱、圆锥等，在三维空间中通常用各自的参数方程表示。
三、函数列表geometry函数包含以下几种类型的函数：1. 点相关函数：包括计算两点之间距离、计算两点之间的中点、判断一个点是否在某条直线上等。
2. 直线相关函数：包括计算两条直线之间的夹角、计算两条直线是否相交、计算一条直线与一个矩形是否相交等。
几何英语知识点总结高中
几何英语知识点总结高中Lines, Angles, and Polygons:- A line is a straight path that extends indefinitely in both directions. It has no endpoints.- A line segment is a part of a line with two endpoints.- A ray is a part of a line that has one endpoint and extends indefinitely in one direction.- An angle is formed by two rays with a common endpoint called the vertex.- A polygon is a closed figure with three or more sides. It can be classified based on the number of sides, such as triangle, quadrilateral, pentagon, hexagon, etc.Properties of Triangles:- The sum of the interior angles of a triangle is 180 degrees.- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.- The longest side of a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle.Pythagorean Theorem:- In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is expressed as a² + b² = c², where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.Quadrilaterals:- A quadrilateral is a four-sided polygon. It can be classified based on the properties of its sides and angles, such as parallelogram, rectangle, square, rhombus, trapezoid, etc. Circles:- A circle is a set of points in a plane that are equidistant from a fixed point called the center.- The circumference of a circle is the distance around the circle, and it is given by the formula C = 2πr, where r is the radius of the circle.- The area of a circle is given by the formula A = πr², where r is the radius of the circle.Similarity and Congruence:- Two figures are said to be similar if they have the same shape but not necessarily the same size.- Two figures are said to be congruent if they have the same shape and size.- The corresponding angles and sides of similar figures are proportional.- Two right-angled triangles are similar if the length of their corresponding sides are in proportion.Coordinate Geometry:- Coordinate geometry is the study of geometric figures using the concepts of algebra and the coordinate plane.- The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by the formula √((x₂ - x₁)² + (y₂ - y₁)²).- The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2).Transformation Geometry:- In transformation geometry, a figure can be transformed through translation, rotation, reflection, or dilation, while maintaining its size and shape.- Translation involves moving a figure from one location to another without changing its size or shape.- Rotation involves turning a figure around a point by a certain angle.- Reflection involves flipping a figure over a line called the line of reflection.- Dilation involves stretching or shrinking a figure by a certain scale factor.Three-Dimensional Figures:- Three-dimensional figures are shapes that have length, width, and height.- Common three-dimensional figures include prisms, pyramids, cylinders, cones, and spheres.- The volume of a three-dimensional figure measures the space it occupies, and it is given by the formula V = Bh, where B is the area of the base and h is the height of the figure. Overall, geometry is a fascinating and important part of mathematics, providing a rich understanding of the shapes and structures that surround us. Understanding the principles and theorems in geometry is not only academically essential but also applicable in everyday life and various professions. Whether it's designing a building, analyzing the trajectory of a projectile, or creating intricate artwork, geometry plays a vital role in shaping the world around us.。
圆的数学定义 英语
圆的数学定义英语Title: The Mathematical Definition of a Circle: A Journey into Geometric PerfectionIn the realm of mathematical geometry, circles hold a unique and captivating place, their elegance and simplicity defying the boundaries of Euclidean space. A circle, by definition, is a two-dimensional shape that forms when all points equidistant from a central point are connected, creating a continuous, circular boundary. This fundamental concept has been studied and revered for centuries, its influence permeating various aspects of mathematics, physics, and art.The circle's mathematical definition begins with its radius, a line segment that connects the center to any point on its circumference. The radius, denoted by 'r,' defines its size and determines the curvature of the circle. The center, often represented by the symbol 'O,' is the point from which all radii emanate. The distance from the center to any point on the circle is the same, a fundamental property that sets it apart from other shapes.The radius is not the only measure in a circle's definition.Circumference, denoted by 'C,' is the total length of the boundary, calculated using the formula 2πr, where π (pi) is a mathematical constant approximately equal to 3.14159. This circumference is the path that a point would take if it were to travel once around the circle. The area, denoted by 'A,' is the region enclosed by the circle, calculated using the formula πr². This area is a measure of the space occupied by the circle.The circle's relationship with other geometric shapes is also crucial. It is the only shape that is both a regular polygon (all sides and angles are equal) and a polygon with the highest number of sides (infinitely many). It is also the parent shape for other shapes like ellipses and ovals, which can be obtained by stretching or compressing a circle.In the realm of algebra, circles play a significant role in equations. The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². This equation, kn own as the circle's equation in standard form, allows us to find the points that lie on the circle's circumference or determine if a given point lies within it.The circle's symmetries are another fascinating aspect. It is a perfect shape, with every point being equidistant from its center, resulting in 180-degree rotational symmetry. This property is reflected in its reflection across any diameter, which bisects the circle and creates another identical circle.In the context of calculus, circles are central to the study of limits and derivatives. The derivative of the circumference with respect to the radius, for instance, is 2π, reflecting the constant rate of change in the circle's circumference as its radius changes.Lastly, circles have profound implications in physics, particularly in the study of motion. In circular motion, an object moves along a circle with a constant speed, its path described by the equation of a circle. This motion is a fundamental part of understanding circular orbits in celestial mechanics.In conclusion, the mathematical definition of a circle is not just a geometric construct, but a fundamental concept that underpins the understanding of the universe. Its simplicity and elegance have inspired countless mathematical discoveries and applications, from the elegant equations of algebra to the principles of physics. As we delve deeper into the realm of mathematics, the circle remains a constant reminder of the harmony and balance inherent in nature's designs.。
清华大学数学科学系本科课程浏览
清华大学数学科学系本科课程浏览课程号课程名课时学分00420033数学模型Mathematical Models 48 3 00420073应用近世代数Applied abstract algebra 48 3 10420213几何与代数(1) Geometry and Algebra(1) 64 4 10420243随机数学方法Stochastic Mathematical Methods 48 3 10420252复变函数引论Introduction to Functions of One Complex Variable 32 2 10420262数理方程引论Introduction to Equations of Mathematical Physics 32 2 10420454高等分析Advanced Analysis 64 4 10420672初等数论与多项式Elementary Number Theory 32 2 10420684几何与代数(1) Geometry and Algebra 64 4 10420692几何与代数(2) Geometry and Algebra(2) 32 2 10420743微积分（I）Calculus（I）48 3 10420746微积分（III）Calculus（III）64 4 10420753微积分（II）Calculus（II）48 3 10420803概率论与数理统计Probability and Statistics 48 3 10420844文科数学Mathematics for Liberal Arts 64 4 10420845大学数学2（社科类） College Mathematics II （For Social Science）48 3 10420854数学实验Mathematical Experiments 48 4 10420874一元微积分Calculus of One Variable 64 4 10420884多元微积分Calculus of Several Variables 64 4 10420892高等微积分B Advanced Calculus B 32 2 10420894高等微积分Advanced Calculus 64 4 10420925数学分析（1）Mathematical Analysis 80 5 10420935数学分析（2）Mathematical Analysis II 80 5 10420944线性代数（1）Linear algebra 64 4 10420946线性代数Linear algebra 32 2 10420963大学数学（1）（社科类）48 3 10420984大学数学(3)(社科类) Collegiate mathematics (3) for social science students 64 4 10420994大学数学(4) Undergraduate Mathematics (4) 64 4 10421692几何与代数(2) Geometry and Algebra(2) 32 2 30420023微分方程（1） Differential Equations （1）48 3 30420033微分方程（2）Differential Equations （2）48 3 30420083复分析Complex analysis 48 3 30420095高等微积分（1）Mathematical analysis (I) 80 5 30420124高等代数与几何(1) Advanced Algebra and Geometry (1) 64 4 30420134高等代数与几何(2) Advanced Algebra and Geometry (2) 64 4 30420224高等微积分（3）Advanced Calculus(3) 64 4 30420334测度与积分Measure and Integration 64 4 30420352概率论介绍A First Course in Probability 32 2 30420364拓扑学Topology 64 4 30420384抽象代数Abstract Algebra 64 4 30420394高等微积分（2）Mathematical analysis (II) 64 4 40420093数理统计Mathematical Statistics 48 3 40420193数理方程与特殊函数Equations in Mathematical Physics and Special Function 48 3 40420534数学规划Mathematical Programming 64 4 40420583概率论（1）Introduction to Stochastics 48 3 40420593数据结构 Data Structures 48 3 40420603集合论Set Theory 48 3 40420614泛函分析（１）Functional Analysis 64 4 40420632数理统计介绍Introduction to Statistics 32 2 40420644微分几何 Differential Geometry#Mathematics。
建筑英语翻译
建筑英语翻译篇一：建筑类英文及翻译外文原文出处：Geotechnical, Geological, and Earthquake Engineering, 1, Volume 10, Seismic Risk Assessment and Retrofitting, Pages 329-342补充垂直支撑对建筑物抗震加固摘要：大量的钢筋混凝土建筑物在整个世界地震活跃地区有共同的缺陷。
弱柱，在一个或多个事故中，由于横向变形而失去垂直承载力。
这篇文章提出一个策略关于补充安装垂直支撑来防止房子的倒塌。
这个策略是使用在一个风险的角度上来研究最近实际可行的性能。
混凝土柱、动力失稳的影响、多样循环冗余的影响降低了建筑系统和组件的强度。
比如用建筑物来说明这个策略的可行性。
1、背景的介绍：建筑受地震震动，有可能达到一定程度上的动力失稳，因为从理论上说侧面上有无限的位移。
许多建筑物，然而，在较低的震动强度下就失去竖向荷载的支撑，这就是横向力不稳定的原因(见图16.1)。
提出了这策略的目的是为了确定建筑物很可能马上在竖向荷载作用下而倒塌，通过补充一些垂直支撑来提高建筑物的安全。
维护竖向荷载支撑的能力，来改变水平力稳定临界失稳的机理，重视可能出现微小的侧向位移(见图16.2)。
在过去的经验表明，世界各地的地震最容易受到破坏的是一些无筋的混凝土框架结构建筑物。
这经常是由于一些无关紧要的漏洞，引起的全部或一大块地方发生破坏，比如整根梁、柱子和板。
去填实上表面来抑制框架的内力，易受影响的底层去吸收大部分的内力和冲力。
这有几种过去被用过的方法可供选择来实施:1、加密上层结构，可以拆卸和更换一些硬度不够强的材料。
2、加密上层结构，可以隔离一些安装接头上的裂缝，从而阻止对框架结构的影响。
3、底楼，或者地板，可以增加结构新墙。
这些措施(项目1、2和3)能有效降低自重，这韧性能满足于一层或多层。
然而，所有这些都有困难和干扰。
在美国，这些不寻常的代价换来的是超过一半更有价值的建筑。
普林斯顿数学指南(英文版)
普林斯顿数学指南(英文版)The Princeton Companion to Mathematics is a comprehensive guide that explores the vast and intricate world of mathematics. This encyclopedic reference work is an indispensable resource for students, educators, and anyonewith an interest in the subject. It covers a wide range of topics, from the fundamentals of arithmetic and geometry tothe more advanced concepts of calculus, number theory, and topology.The book is organized into several sections, eachfocusing on a specific area of mathematics. The first section, "The Foundations of Mathematics," provides an overview of the basic principles and concepts that underlie all mathematical disciplines. This includes an introduction to the history of mathematics, its major branches, and the key figures who have shaped its development over the centuries.The second section, "Algebra and Number Theory," delves into the study of abstract structures, such as groups, rings, and fields, as well as the properties and relationships of numbers. This section also covers the foundations of algebraic geometry, which is concerned with the study of geometric objects defined by polynomial equations.The third section, "Analysis and Calculus," explores the study of continuity, change, and limits, as well as the techniques used to solve problems involving rates of change, such as differentiation and integration. This section also covers the theory of complex analysis, which extends the ideas of real analysis to complex numbers.The fourth section, "Geometry and Topology," focuses on the study of shape, size, and spatial relationships, as well as the properties of abstract spaces that are not necessarily Euclidean. This section includes discussions of classical geometry, such as Euclidean and non-Euclidean geometries, aswell as more modern areas like topology and differential geometry.The fifth section, "Probability and Statistics," deals with the study of randomness and uncertainty, as well as the collection, analysis, and interpretation of data. This section covers the basic principles of probability theory, statistical inference, and statistical modeling.The final section, "Applied Mathematics," highlights the many ways in which mathematics is used to solve real-world problems in fields such as physics, engineering, economics, and computer science. This section includes discussions of optimization, game theory, cryptography, and other areas where mathematical techniques are essential for solving practical problems.Throughout the book, readers will find numerous examples, exercises, and applications that illustrate the concepts and techniques discussed in each section. These include bothhistorical examples, such as the development of calculus or the proof of Fermat's Last Theorem, and contemporary applications, such as the use of chaos theory in meteorology or the role of Fourier analysis in image processing.In addition to its comprehensive coverage of mathematical topics, The Princeton Companion to Mathematics also features biographical sketches of many of the most influential mathematicians in history, from ancient Greek geometers like Euclid and Archimedes to modern giants like Isaac Newton, Carl Friedrich Gauss, and Emmy Noether. These profiles provide insights into the lives and achievements of these remarkable individuals, as well as their contributions to the development of mathematics.Furthermore, the book includes a detailed glossary of mathematical terms and symbols, which can be especially helpful for readers who are new to the subject or encounter unfamiliar concepts. The glossary defines key terms andprovides examples that illustrate their meanings and uses in various contexts.As a reference work, The Princeton Companion to Mathematics is designed to be accessible to a wide range of readers, from beginners to advanced students and professionals. Its clear explanations, extensive examples, and engaging historical narratives make it an ideal resource for anyone seeking to deepen their understanding of mathematics or explore new areas of the subject.In conclusion, The Princeton Companion to Mathematics is a valuable resource for anyone interested in learning more about the fascinating world of mathematics. Its comprehensive coverage of topics, engaging historical narratives, and clear explanations make it an invaluable tool for anyone seeking to enhance their knowledge and appreciation of this vital field of study. Whether you are a student, educator, or simply acurious mind looking to explore the beauty and elegance of mathematics, this guide is an excellent place to start.。
Introduction to Mathematical Programming
The TBA Airlines Problem
• TBA Airlines is a small regional company that specializes in short flights in small airplanes.
• The company has been doing well and has decided to expand its operations. • The basic issue facing management is whether to purchase more small airplanes to add
Maximize Profit = S + 5L ($millions) subject to
Capital Available: 5S + 50L ≤ 100 ($millions) Max Small Planes: S ≤ 2 and S ≥ 0, L ≥ 0.
Graphical Method for Linear Programming
Data for the TBA Airlines Problem
Net annual profit per airplane Purchase cost per airplane Maximum purchase quantity
Small Airplane $1 million 5 million
Explorations in Artificial Intelligence
Prof. Carla P. Gomes gomes@
Module 7 Part 3
Integer Programming
Divisibility
介绍几何图形的作文英语
介绍几何图形的作文英语Geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects in space.It is an important subject that has many practical applications in our daily lives. In this essay, I will introduce several common geometric shapes and their properties.The first shape I would like to introduce is the circle.A circle is a closed curve that is made up of points that are all the same distance from a fixed point called the center. The distance from the center to any point on the circle is called the radius. The circumference of a circleis the distance around the outside edge, and it can be calculated by multiplying the diameter (the distance across the circle through the center) by pi (approximately 3.14). Circles are found in many everyday objects, such as wheels, plates, and coins.Another common geometric shape is the triangle. Atriangle is a three-sided polygon that has three angles. The sum of the angles in a triangle is always 180 degrees. Triangles can be classified based on the length of their sides and the size of their angles. For example, an equilateral triangle has three equal sides and three equal angles of 60 degrees each, while a right triangle has one angle that is 90 degrees. Triangles are used in construction to create stable structures, such as roofs and bridges.Rectangles are another important geometric shape. A rectangle is a four-sided polygon that has two pairs of parallel sides and four right angles. The length of a rectangle is its longer side, while the width is its shorter side. The area of a rectangle can be calculated by multiplying its length by its width, while its perimeter (the distance around the outside edge) is the sum of its four sides. Rectangles are commonly used in architecture and design to create buildings and furniture.Finally, I would like to introduce the sphere. A sphere is a three-dimensional shape that is formed by rotating acircle around its diameter. It is a perfectly round shape that has no edges or corners. The surface area of a sphere can be calculated by multiplying its radius by 4 pi, while its volume can be calculated by multiplying the cube of its radius by 4/3 pi. Spheres are found in many natural objects, such as planets and bubbles.In conclusion, geometry is a fascinating subject that has many practical applications in our daily lives. By understanding the properties of common geometric shapes, we can better appreciate the beauty and complexity of theworld around us.。
计算数学专业英语自我介绍
计算数学专业英语自我介绍Here is an English essay with a title "A Self-Introduction to the Mathematics Major" that is over 1000 words long, without any extra punctuation marks in the body of the text.As a mathematics major, I have always been fascinated by the beauty and complexity of the field. From a young age, I have been drawn to the logical and analytical nature of mathematics, finding joy in solving intricate problems and unraveling the mysteries of the numerical world. My journey through the realm of mathematics has been one of continuous growth and discovery, and I am excited to share a glimpse of my experiences and aspirations.Mathematics, to me, is not just a collection of formulas and theorems, but a language that allows us to understand and describe the world around us. It is a powerful tool that enables us to model and analyze real-world phenomena, from the smallest subatomic particles to the grandest celestial bodies. As a mathematics major, I am captivated by the way in which mathematical principles can be applied to diverse fields, from physics and engineering to economics and computer science.One of the aspects of mathematics that I find most compelling is its inherent logical structure. I am fascinated by the way in which mathematical concepts build upon one another, creating a seamless web of interconnected ideas. The process of deductive reasoning, where one step logically follows from the previous one, allows me to engage in a form of intellectual problem-solving that is both challenging and deeply satisfying.Throughout my studies, I have had the opportunity to delve into a wide range of mathematical disciplines, from abstract algebra and real analysis to numerical methods and partial differential equations. Each course has provided me with a new lens through which to view the world, expanding my understanding of the fundamental principles that govern the universe. As I have progressed through my studies, I have developed a particular interest in the field of computational mathematics, where I have explored the intersection of mathematics and computer science.One of the most rewarding aspects of my mathematics education has been the opportunity to engage in research and collaborative projects. I have had the privilege of working alongside esteemed professors and fellow students, tackling complex problems and contributing to the advancement of mathematical knowledge. Through these experiences, I have honed my critical thinking skills, learned to effectively communicate complex ideas, and developed adeep appreciation for the collaborative nature of scientific inquiry.Beyond the purely academic aspects of my studies, I have also found great fulfillment in sharing my passion for mathematics with others. I have had the opportunity to serve as a tutor and teaching assistant, helping students to overcome their fears and develop a deeper understanding of mathematical concepts. In these roles, I have learned the importance of patience, empathy, and the ability to explain complex ideas in a clear and accessible manner.As I look to the future, I am excited by the myriad of possibilities that await me as a mathematics major. I am particularly interested in the field of data science, where I believe my strong background in mathematics and computational skills can be applied to solve real-world problems and drive innovation. The ability to extract meaningful insights from vast amounts of data is a critical skill in today's data-driven world, and I am eager to contribute my expertise to this rapidly evolving field.Furthermore, I am intrigued by the potential applications of mathematics in the field of artificial intelligence and machine learning. As these technologies continue to shape the future, I believe that a deep understanding of mathematical principles will be essential in the development of more sophisticated and effective algorithms. I am excited by the prospect of using my mathematicalknowledge to push the boundaries of what is possible in these cutting-edge domains.Beyond my academic and professional aspirations, I am also deeply committed to using my mathematical skills to make a positive impact on the world. I believe that mathematics has the power to address some of the most pressing challenges facing our society, from climate change and global health to economic inequality and social justice. I am passionate about using my mathematical expertise to contribute to research and initiatives that seek to improve the lives of people around the world.In conclusion, my journey as a mathematics major has been one of intellectual growth, collaborative discovery, and a deep appreciation for the power of mathematical thinking. As I look to the future, I am filled with a sense of excitement and purpose, driven by the belief that mathematics holds the key to unlocking solutions to some of the most complex problems facing humanity. I am committed to continuing my pursuit of mathematical knowledge, and I am eager to apply my skills in ways that will make a meaningful difference in the world.。
对数学的认识 英语
对数学的认识英语Mathematics is a fundamental discipline that has been integral to human civilization for thousands of years. It is a language of its own, with its own set of rules, principles, and methods that have been developed and refined over time. As a subject, mathematics is often perceived as a daunting and challenging field, but it is also a powerful tool that can be used to solve a wide range of problems and gain a deeper understanding of the world around us.One of the key aspects of mathematics is its ability to provide a logical and systematic way of thinking about and analyzing problems. Through the use of mathematical concepts and techniques, such as algebra, geometry, and calculus, we can break down complex problems into smaller, more manageable parts and develop strategies for solving them. This is particularly important in fields such as engineering, finance, and scientific research, where the ability to analyze and interpret data is essential for making informed decisions and driving innovation.Moreover, mathematics is not just about solving problems; it is also a means of exploring and understanding the underlying patterns and structures that govern the natural world. From the intricatesymmetries of snowflakes to the complex dynamics of planetary motion, mathematics provides a framework for understanding and describing the world around us in a precise and objective manner. This understanding can then be applied to a wide range of practical applications, such as designing more efficient transportation systems, developing new medical treatments, and predicting the behavior of complex systems.One of the key reasons why mathematics is so important is its ability to provide a foundation for logical and critical thinking. By learningto approach problems in a systematic and analytical manner, students of mathematics develop a set of skills that can be applied to a wide range of academic and professional pursuits. These skills include the ability to identify patterns, formulate hypotheses, and test them using empirical evidence. Additionally, the study of mathematics helps to develop problem-solving skills, attention to detail, and the ability to communicate complex ideas clearly and concisely.Despite the importance of mathematics, many students struggle with the subject. This may be due to a variety of factors, such as a lack of confidence, poor teaching methods, or a perception that mathematics is simply too difficult to master. However, with the right approach and support, it is possible for anyone to develop a strong understanding and appreciation of mathematics.One of the keys to success in mathematics is a willingness to persevere and embrace the challenge. Mathematics is often a subject that requires sustained effort and a commitment to continuous learning. Students who are willing to put in the time and effort to practice problem-solving, engage in discussions with their peers, and seek out additional resources and support, are often the ones who are able to succeed in the subject.Additionally, it is important to recognize that mathematics is not just a collection of formulas and algorithms, but a way of thinking and problem-solving that can be applied to a wide range of real-world situations. By focusing on the underlying concepts and principles, rather than just memorizing formulas, students can develop a deeper understanding of the subject and become better equipped to apply their knowledge to practical problems.In conclusion, mathematics is a powerful and versatile discipline that has the potential to transform the way we think about and interact with the world around us. Whether we are working in a highly technical field or simply trying to make sense of the complexities of everyday life, the skills and insights that we gain through the study of mathematics can be invaluable. By embracing the challenge of mathematics and developing a deep appreciation for its importance, we can unlock a world of new possibilities and make a meaningfulcontribution to the advancement of human knowledge and understanding.。
几何性质英语作文
几何性质英语作文The Properties of Geometry。
Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and solids. It is a fundamental part of mathematics and has many practical applications in everyday life. In this essay, we will explore some of the key properties of geometry and how they are used in various fields.One of the most important properties of geometry is the concept of shape and size. Shapes can be classified into various categories such as circles, squares, triangles, and rectangles. Each of these shapes has its own unique properties and can be used to solve a wide range of problems. For example, the area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius. This formula is used in many real-world applications, such as calculating the area of agarden or the surface area of a sphere.Another important property of geometry is the conceptof angles. An angle is formed by two rays that share a common endpoint, called the vertex. Angles can beclassified into different categories such as acute, obtuse, right, and straight angles. The measurement of angles is crucial in various fields such as architecture, engineering, and navigation. For example, architects use angles todesign buildings and structures, while engineers use angles to calculate the forces acting on a structure.Geometry also deals with the properties of lines and planes. A line is a straight path that extends infinitelyin both directions, while a plane is a flat surface that extends infinitely in all directions. These concepts are used in various fields such as art, design, and physics.For example, artists use lines and planes to create perspective in their drawings, while physicists use lines and planes to represent the motion of objects in space.In addition to these properties, geometry also dealswith the properties of solids. Solids can be classifiedinto various categories such as cubes, spheres, cones, and cylinders. Each of these solids has its own uniqueproperties and can be used to solve a wide range of problems. For example, the volume of a cube can becalculated using the formula V = l^3, where V is the volume and l is the length of a side. This formula is used in many real-world applications, such as calculating the volume ofa box or the capacity of a container.In conclusion, geometry is a fundamental part of mathematics that has many practical applications ineveryday life. It deals with the properties andrelationships of points, lines, angles, surfaces, and solids. By understanding the properties of geometry, we can solve a wide range of problems and make informed decisionsin various fields. Whether it's designing a building, navigating a ship, or calculating the volume of a container, geometry plays a crucial role in our lives.。
第2节几何建模(Geometry)
Decompose 将复杂曲面重构成由三角形，四边形曲面组成的简单曲面
Edge
由3条或4条封闭曲线，生成三角形或四边形曲面
Extract 提取实体表面，或按一定参数提取实体内某一面
Fillet
二个面间产生倒角面
Match
当二面交接处有裂纹时，用match消除间隙，以保证连接协调
Ruled
二曲线间产生有理面
简单曲线
Involute
产生渐开线，形成齿廓曲线有用
Revolve
由点，回转轴和转角创建园弧
平面曲线产生：都须输入“Construction Plane List”，即曲线所在平面
2d Normal
在平面内作一条垂线
2d Circle
在平面内画一个园
2d Arc Angles
在平面内，根据园心，半径，起始角，终止角创建一条园弧
Extrude
将曲面沿矢量方向拉伸成实体 注:：Trimmed Surface不能拉伸成-Rep实体
Glide
由基准曲面（Base Surface）沿1或2条路径（Director Curve）滑动， 产生一个实体。注：Trimmed Surface不能作为基准曲面。
Normal
将曲面沿法向偏置一段距离，形成实体
把二条或多条曲线(或边)合并成一条曲线；新曲线一阶导数连续,不会与原曲线一致
把(Chained曲线分解成一组简单曲线
延长曲线 把多条曲线(边)合成一条;新曲线在指定公差内与原曲线一致 将曲线转换成相互连续的分段三次曲线 改变曲线参数方向 修剪曲线到指定位置
示例
■Show /Curve
方式 Attribute Arc Angles Length Range Node
几何geometry
坐标系 coordinate system
• How to build a coordinate system? • Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance , although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the points.
• 经过圆心且其两个端点在圆周上的线段称为这个园的直 径，这条曲线的长度叫做周长。 • A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sides, or one side and an acute angle, are given.
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Computer Science15-499C/15-881,Fall1997Introduction to GeometryInstructor:Michael Erdmann(me+@)Guest Instructor:Yanxi Liu(yanxi+@)Location:Scaife324Time:TR10:30–11:50TA:German(german+@)Office Hours:by appointment1Course OutlineThis course will cover elementary differential and computational geometry.The purpose of the course is to prepare a student for advanced geometrical work in robotics and computer science.Increasingly,cutting-edge results in these areas require a working knowledge of differential geometry,algebraic geometry,algebraic topology, and computational geometry.Much of this work is inaccessible to a student just entering thefield.In this course we will convey the basic tools,definitions,and results of differential geometry and the basic algorithms of computational geometry,so that a student can,either by self-study or through further courses,understand and implement the advanced results in computer science and robotics discovered in the past decade.We will touch upon some of these applications in the course.In particular,we will consider the robot motion planning problem as a core application.The topics are:Motion Planning:Configuration Space,Visibility Graph,Non-Holonomic MotionPlanning,Forces in Cspace.Frame Fields:Curves,Frenet Formulas,Covariant Derivatives,Differential Forms,Connection Forms,Structural Equations.Calculus on Surfaces:Surfaces,Patches,Tangent V ectors,Mappings,Differential Forms,Integration on Surfaces,Manifolds.Shape Operators:Surface Shape Operator,Normal Curvature,Gaussian Curvature.Point and Range Queries:One shot,repeated query,slab method,multidimensional binary tree.Convex Hull Algorithms:Lower bounds,sorting reduction,2D algorithms(Graham’s Scan,Jarvis’March,Quickhull,Mergehull),dynamic convex hull,higher-dimensional algorithms(gift-wrapping,beneath-beyond),3D convexhull.Proximity Algorithms:V oronoi diagram,triangulations,Euclidean minimum spanning tree.Plane Sweep:Polygon union and intersection.Prerequisites for this course are minor,but important.Calculus and linear algebra should be sufficient from the mathematics side,while fundamental data structures and algorithms should be sufficient from the computer science side.The course will consist of a series of intensive lectures.It is expected that a student will spend a few hours each day digesting the material of the lectures and solving some related homework problems. Grades will be based primarily on the homework,and possibly an exam.The course may be taken at either the undergraduate or the graduate levels.Students taking the class for graduate credit will be asked to prepare a project,either a paper or an implementation of an advanced topic.2BibliographyHere are some good texts for background reading.The course texts are the books by O’Neill and Preparata&Shamos.1.W.M.Boothby,An Introduction to Differentiable Manifolds and RiemannianGeometry,Academic Press,New York,1975.2.H.Edelsbrunner,Algorithms in Combinatorial Geometry,Springer-V erlag,Berlin,1987.tombe,Robot Motion Planning,Kluwer Academic Publishers,Boston,1991.4.B.O’Neill,Elementary Differential Geometry,Academic Press,New York,1966.2nd Edition:1997.5.F.P.Preparata and M.I.Shamos,Computational Geometry,Springer-V erlag,NewYork,1985.(Corrected and expanded printing:1988.)6.M.Spivak,Differential Geometry,Publish or Perish,Berkeley,1979.3ScheduleHere is an approximate schedule for the semester.This is subject to change as we seehow quickly we cover the material.AugustLecture Day1T2RDate TopicSep2Dual Spaces,1-Forms4RSep9Mappings6RSep16Frenet Formulas8RSep23Non-holonomic Motions,Connection Forms,Structural Equations 10RSep30Manifolds:configuration space,embeddings,differential forms OctoberLecture Day12ROct7Shape Operator,examples,normal acceleration14ROct14Gauss Theorem,surface area16ROct21Datastructure preliminaries(e.g.,segment tree,DCEL) 18ROct28Range searching,point location20RNovemberLecture Day21TNov6Convex Hull in2D(definitions,theorems,Graham’s scan) 23TNov13Higher Dimensional Convex Hull25TNov20V oronoi Diagram algorithm,Euclidean Minimum Spanning Tree,Triang.27TNov27ThanksgivingDecemberLecture Day28TDec4open。