Introduction to Geometry

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.。

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。