parametric equation

简明释义

参数方程

英英释义

例句

1.In robotics, a parametric equation 参数方程 can define the path that a robotic arm should follow.

在机器人技术中，参数方程 参数方程 可以定义机器人手臂应该遵循的路径。

2.To find the intersection points of two curves, we can set their parametric equations 参数方程 equal to each other.

为了找到两条曲线的交点，我们可以将它们的 参数方程 参数方程 设为相等。

3.The motion of a pendulum can be described using a parametric equation 参数方程 involving sine and cosine functions.

摆的运动可以用包含正弦和余弦函数的 参数方程 参数方程 来描述。

4.Engineers use a parametric equation 参数方程 to describe the trajectory of a projectile.

工程师使用 参数方程 参数方程 来描述抛射物的轨迹。

5.In computer graphics, a parametric equation 参数方程 is often used to model curves and surfaces.

在计算机图形学中，参数方程 参数方程 常用于建模曲线和表面。

作文

In mathematics, a parametric equation is a way of expressing a set of quantities as explicit functions of an independent variable, known as a parameter. This concept is particularly useful in geometry and physics, where curves and surfaces can be described more flexibly than using traditional equations. For instance, consider the circle. While we can represent a circle using the standard equation x² + y² = r², this form does not easily allow us to express the coordinates of points on the circle as functions of a single variable. Instead, we can use a parametric equation to define the coordinates (x, y) in terms of an angle θ: x = r cos(θ) and y = r sin(θ). Here, θ serves as the parameter that varies, allowing us to trace out the entire circle as θ ranges from 0 to 2π.The beauty of parametric equations lies in their ability to describe complex shapes and motions. For example, in physics, the trajectory of a projectile can be represented using parametric equations. If we let t represent time, we can express the horizontal and vertical positions of the projectile as functions of time: x(t) = v₀ cos(θ) t and y(t) = v₀ sin(θ) t - (1/2)gt², where v₀ is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity. By analyzing these parametric equations, we can gain insight into the motion of the projectile, such as its maximum height and range.Moreover, parametric equations are not limited to two dimensions. In three-dimensional space, we can extend the idea further. For example, a helix can be described using parametric equations in terms of a single parameter t: x(t) = r cos(t), y(t) = r sin(t), and z(t) = ct, where r is the radius of the helix and c determines the vertical spacing between loops. This flexibility allows mathematicians and scientists to model real-world phenomena more accurately.In computer graphics, parametric equations are crucial for rendering curves and surfaces. Designers often use these equations to create smooth transitions and shapes, which are essential in animation and visual effects. For instance, Bézier curves, widely used in graphic design, are defined by parametric equations that allow for easy manipulation of the curve's shape by adjusting control points.In conclusion, parametric equations provide a powerful tool for representing mathematical relationships in a variety of fields. They enable us to describe curves, surfaces, and trajectories in a flexible and intuitive manner. As we continue to explore the applications of parametric equations in mathematics, physics, and computer graphics, it becomes evident that they are not just abstract concepts but essential elements that enhance our understanding of the world around us. By mastering parametric equations, we equip ourselves with the means to tackle complex problems and innovate in various disciplines.

在数学中，参数方程是一种将一组量表示为独立变量（称为参数）的显式函数的方式。这个概念在几何和物理中尤其有用，因为它比传统方程更灵活地描述曲线和表面。例如，考虑圆。虽然我们可以使用标准方程x² + y² = r²来表示一个圆，但这种形式并不容易让我们将圆上点的坐标表示为一个变量的函数。相反，我们可以使用参数方程来定义坐标(x, y)以角度θ为函数：x = r cos(θ)和y = r sin(θ)。在这里，θ作为参数变化，允许我们在θ从0到2π的范围内描绘整个圆。参数方程的美在于它们能够描述复杂的形状和运动。例如，在物理学中，抛射体的轨迹可以使用参数方程来表示。如果我们让t代表时间，我们可以将抛射体的水平和垂直位置表示为时间的函数：x(t) = v₀ cos(θ) t和y(t) = v₀ sin(θ) t - (1/2)gt²，其中v₀是初始速度，θ是发射角，g是重力加速度。通过分析这些参数方程，我们可以深入了解抛射体的运动，例如其最大高度和射程。此外，参数方程不仅限于二维空间。在三维空间中，我们可以进一步扩展这个概念。例如，螺旋线可以使用关于单个参数t的参数方程来描述：x(t) = r cos(t)，y(t) = r sin(t)，z(t) = ct，其中r是螺旋的半径，c决定了环之间的垂直间距。这种灵活性使得数学家和科学家能够更准确地建模现实世界现象。在计算机图形学中，参数方程对渲染曲线和表面至关重要。设计师常常使用这些方程来创建平滑的过渡和形状，这在动画和视觉效果中是必不可少的。例如，广泛用于图形设计的贝塞尔曲线是由参数方程定义的，通过调整控制点可以轻松操纵曲线的形状。总之，参数方程为在多个领域表示数学关系提供了强大的工具。它们使我们能够以灵活和直观的方式描述曲线、表面和轨迹。随着我们继续探索参数方程在数学、物理和计算机图形学中的应用，显而易见的是，它们不仅仅是抽象概念，而是增强我们对周围世界理解的基本元素。通过掌握参数方程，我们为自己配备了应对复杂问题和在各个学科中创新的手段。