spherical coordinates

简明释义

球面座标

英英释义

例句

1.Astronomers often use spherical coordinates (球坐标) to describe the positions of stars and galaxies in the universe.

天文学家通常使用球坐标（球坐标）来描述宇宙中恒星和星系的位置。

2.When working with three-dimensional models, using spherical coordinates (球坐标) can make it easier to define points on a sphere.

在处理三维模型时，使用球坐标（球坐标）可以更容易地定义球面上的点。

3.The formula for the volume of a sphere is derived using spherical coordinates (球坐标).

球的体积公式是通过使用球坐标（球坐标）推导出来的。

4.To calculate the surface area of a sphere, we apply the principles of spherical coordinates (球坐标).

为了计算球的表面积，我们应用球坐标（球坐标）的原理。

5.In physics, we often convert Cartesian coordinates to spherical coordinates (球坐标) to simplify the calculations of problems involving symmetry.

在物理学中，我们经常将笛卡尔坐标转换为球坐标（球坐标）以简化涉及对称性的问题的计算。

作文

Understanding the concept of spherical coordinates is essential for anyone studying mathematics, physics, or engineering. Unlike the traditional Cartesian coordinate system, which uses two-dimensional (2D) or three-dimensional (3D) axes to define a point in space, spherical coordinates offer a different approach by using a radius and angles. This system is particularly useful when dealing with problems that have inherent symmetry, such as those involving spheres or circular patterns.In the spherical coordinates system, a point in three-dimensional space is defined by three values: the radial distance from the origin, the polar angle (often denoted as theta), and the azimuthal angle (often denoted as phi). The radial distance represents how far the point is from the origin, while the polar angle measures the angle down from the positive z-axis. The azimuthal angle describes the angle around the z-axis from the positive x-axis. This method of representation is especially advantageous in scenarios where the geometry of the problem is spherical in nature.For instance, consider the case of calculating the gravitational field around a planet. Using spherical coordinates simplifies the mathematics involved because the gravitational field has spherical symmetry. Instead of working with complicated integrals in Cartesian coordinates, one can use spherical coordinates to easily express the equations governing the gravitational force. The transformation from Cartesian to spherical coordinates can also help visualize the problem better, making it easier to understand the underlying physics.Moreover, spherical coordinates are not just limited to theoretical applications; they are widely used in various fields, including computer graphics, robotics, and even in medical imaging technologies like MRI scans. In computer graphics, spherical coordinates are utilized to create realistic models of objects by defining their shapes and surfaces more intuitively. For example, when rendering a sphere, it is much simpler to specify points using spherical coordinates than Cartesian coordinates, as the latter would require complex calculations to ensure the points lie on the surface of the sphere.In robotics, spherical coordinates can be employed to navigate and control robotic arms or drones. By defining positions in spherical coordinates, engineers can simplify the programming of movements and ensure precise control over the robot’s actions, especially in environments where obstacles may be present.Furthermore, in the field of astronomy, spherical coordinates play a crucial role in mapping the positions of celestial bodies. Astronomers often use spherical coordinates to describe the location of stars and galaxies in the universe. The vastness of space makes it impractical to utilize Cartesian coordinates, as the distances involved can be astronomical. Therefore, spherical coordinates provide a more natural way to express these locations.In conclusion, mastering spherical coordinates is vital for students and professionals alike in various scientific fields. The ability to switch between different coordinate systems enhances problem-solving skills and offers new perspectives on mathematical and physical phenomena. Whether it’s in the classroom, laboratory, or industry, understanding spherical coordinates empowers individuals to tackle complex challenges with greater ease and efficiency. As such, it is a fundamental concept that should be embraced and studied thoroughly by anyone aiming to excel in science and engineering disciplines.

理解spherical coordinates（球坐标）的概念对于任何学习数学、物理或工程的人来说都是至关重要的。与传统的笛卡尔坐标系统不同，后者使用二维（2D）或三维（3D）坐标轴来定义空间中的点，spherical coordinates（球坐标）则通过半径和角度提供了一种不同的方法。这种系统在处理具有固有对称性的问题时特别有用，例如涉及球体或圆形图案的问题。在spherical coordinates（球坐标）系统中，三维空间中的一个点由三个值定义：从原点的径向距离、极角（通常用theta表示）和方位角（通常用phi表示）。径向距离表示点离原点的远近，而极角测量从正z轴向下的角度。方位角描述从正x轴绕z轴的角度。这种表示方法在问题的几何形状呈球形时尤其有利。例如，考虑计算行星周围的引力场的情况。使用spherical coordinates（球坐标）简化了相关的数学运算，因为引力场具有球对称性。与其在笛卡尔坐标系中进行复杂的积分运算，不如使用spherical coordinates（球坐标）轻松表达引力公式。将笛卡尔坐标转换为spherical coordinates（球坐标）也可以帮助更好地可视化问题，使理解基础物理变得更加容易。此外，spherical coordinates（球坐标）不仅限于理论应用；它们广泛用于多个领域，包括计算机图形学、机器人技术，甚至医学成像技术如MRI扫描。在计算机图形学中，spherical coordinates（球坐标）用于通过更直观地定义物体的形状和表面来创建逼真的模型。例如，在渲染一个球体时，使用spherical coordinates（球坐标）指定点要比笛卡尔坐标简单得多，因为后者需要复杂的计算以确保点位于球体的表面上。在机器人技术中，spherical coordinates（球坐标）可以用于导航和控制机器人手臂或无人机。通过在spherical coordinates（球坐标）中定义位置，工程师可以简化运动编程，并确保对机器人的动作进行精确控制，尤其是在可能存在障碍物的环境中。此外，在天文学领域，spherical coordinates（球坐标）在绘制天体的位置时发挥着关键作用。天文学家常常使用spherical coordinates（球坐标）来描述恒星和星系在宇宙中的位置。空间的浩瀚使得使用笛卡尔坐标变得不切实际，因为涉及的距离可能是天文数字。因此，spherical coordinates（球坐标）提供了一种更自然的方式来表达这些位置。总之，掌握spherical coordinates（球坐标）对各科学领域的学生和专业人士来说至关重要。能够在不同坐标系统之间切换增强了解决问题的能力，并为数学和物理现象提供新的视角。无论是在课堂、实验室还是工业界，理解spherical coordinates（球坐标）使个人能够更轻松高效地应对复杂挑战。因此，这是一项基本概念，任何希望在科学和工程学科中取得优异成绩的人都应当深入学习和掌握。