surface of revolution

简明释义

回转面

英英释义

例句

1.The shape of a wine glass is a classic example of a surface of revolution 旋转曲面.

酒杯的形状是一个经典的旋转曲面的例子。

2.The surface of revolution 旋转曲面 can be generated by rotating a curve around an axis.

通过围绕一个轴旋转曲线，可以生成旋转曲面。

3.In calculus, we often calculate the volume of a solid formed by a surface of revolution 旋转曲面 using integration.

在微积分中，我们常常通过积分计算由旋转曲面形成的固体的体积。

4.When designing a vase, one must consider the surface of revolution 旋转曲面 to achieve a pleasing aesthetic.

在设计花瓶时，必须考虑旋转曲面以达到令人愉悦的美感。

5.Engineers use the concept of surface of revolution 旋转曲面 to design components like pipes and tanks.

工程师使用旋转曲面的概念来设计管道和储罐等组件。

作文

In the field of mathematics, particularly in geometry, the concept of a surface of revolution plays a crucial role in understanding three-dimensional shapes. A surface of revolution is generated by rotating a two-dimensional shape around an axis. This process creates a surface that has symmetrical properties, making it essential for various applications in engineering, architecture, and art. For instance, when we take a simple shape like a circle and rotate it around a straight line, we produce a three-dimensional object known as a sphere. Similarly, if we rotate a rectangle around one of its edges, we get a cylinder. These examples illustrate the significance of surface of revolution in creating complex shapes from simpler ones.The mathematical formulation of a surface of revolution can be described using parametric equations. For a function f(x) defined on an interval [a, b], the surface formed by revolving this function around the x-axis can be represented parametrically as follows:x = t,y = f(t) * cos(θ),z = f(t) * sin(θ),where t varies from a to b and θ varies from 0 to 2π. This representation helps us visualize how the rotation of a curve generates a surface in three-dimensional space. Understanding these equations allows mathematicians and engineers to calculate areas and volumes of these surfaces, which is vital for many practical applications.In engineering, the design of components often involves surface of revolution shapes. For example, the design of turbine blades or propellers relies heavily on the properties of surfaces generated by revolution. The aerodynamic efficiency of these components can be significantly improved by optimizing their shapes, which are often modeled as surface of revolution. By analyzing the flow of air or fluid around these shapes, engineers can enhance performance and reduce drag, leading to more efficient machines.Architecture also benefits from the concept of surface of revolution. Many modern buildings incorporate curves and rounded shapes that are aesthetically pleasing and structurally sound. The iconic Guggenheim Museum in Bilbao, designed by Frank Gehry, showcases the beauty of surface of revolution in its flowing forms. These architectural designs not only capture the eye but also demonstrate the principles of physics and engineering that govern their stability and strength.In the realm of art, artists have long used the idea of surface of revolution to create sculptures and installations that challenge our perception of space. By manipulating forms that exist in three dimensions, artists can evoke emotions and provoke thought. The use of curves and rotational symmetry in art reflects the natural world, where many objects, such as shells and flowers, exhibit similar properties.In conclusion, the concept of surface of revolution extends far beyond the confines of mathematics. It permeates various disciplines, including engineering, architecture, and art. By understanding how two-dimensional shapes can generate complex three-dimensional surfaces, we unlock the potential for innovation and creativity in designing functional and aesthetic structures. The study of surface of revolution not only enhances our mathematical knowledge but also enriches our appreciation for the world around us, where geometry and beauty intertwine seamlessly.

在数学领域，特别是几何学中，旋转曲面的概念在理解三维形状方面起着至关重要的作用。旋转曲面是通过围绕一个轴旋转一个二维形状生成的。这个过程创造出具有对称性质的表面，使其在工程、建筑和艺术等多个应用中至关重要。例如，当我们取一个简单的形状，如圆，并围绕一条直线旋转时，我们会产生一个称为球体的三维物体。同样，如果我们围绕矩形的一条边旋转，就会得到一个圆柱体。这些例子说明了旋转曲面在从简单形状创建复杂形状中的重要性。旋转曲面的数学公式可以使用参数方程来描述。对于定义在区间[a, b]上的函数f(x)，围绕x轴旋转该函数所形成的曲面可以用如下参数表示：x = t,y = f(t) * cos(θ),z = f(t) * sin(θ),其中t在a到b之间变化，θ从0变化到2π。这个表示帮助我们可视化曲线的旋转如何在三维空间中生成一个表面。理解这些方程使数学家和工程师能够计算这些表面的面积和体积，这对于许多实际应用至关重要。在工程中，组件的设计通常涉及旋转曲面形状。例如，涡轮叶片或螺旋桨的设计在很大程度上依赖于旋转生成的表面的特性。通过优化这些形状，工程师可以显著提高这些组件的空气动力效率，这些形状通常被建模为旋转曲面。通过分析空气或流体围绕这些形状的流动，工程师可以提高性能并减少阻力，从而使机器更加高效。建筑也从旋转曲面的概念中受益。许多现代建筑采用曲线和圆润的形状，这些形状既美观又结构稳固。弗兰克·盖里的毕尔巴鄂古根海姆博物馆展示了旋转曲面在其流畅形态中的美。这些建筑设计不仅吸引眼球，还展示了支配其稳定性和强度的物理和工程原理。在艺术领域，艺术家们长期以来一直利用旋转曲面的思想创作雕塑和装置，挑战我们对空间的感知。通过操控存在于三维中的形状，艺术家可以唤起情感并激发思考。艺术中曲线和旋转对称性的使用反映了自然界，其中许多物体，如贝壳和花朵，也展现出类似的特性。总之，旋转曲面的概念远远超出了数学的范围。它渗透到工程、建筑和艺术等多个学科中。通过理解二维形状如何生成复杂的三维表面，我们解锁了在设计功能性和美观结构方面的创新和创造力的潜力。对旋转曲面的研究不仅增强了我们的数学知识，还丰富了我们对周围世界的欣赏，在这里，几何与美丽无缝交织。