family of curves

简明释义

曲线族

英英释义

例句

1.Mathematicians study a family of curves 曲线族 to understand the solutions to differential equations.

数学家研究曲线族 family of curves 以理解微分方程的解。

2.The family of curves 曲线族 generated from the simulation helps predict the system's behavior under various conditions.

从模拟生成的曲线族 family of curves 有助于预测系统在不同条件下的行为。

3.The family of curves 曲线族 plotted on the graph shows how temperature affects pressure in a gas.

图表上绘制的曲线族 family of curves 显示了温度如何影响气体的压力。

4.In fluid dynamics, we often analyze a family of curves 曲线族 that illustrates flow rates at different temperatures.

在流体动力学中，我们经常分析一个曲线族 family of curves，它展示了不同温度下的流速。

5.In engineering, a family of curves 曲线族 can represent different stress-strain relationships for various materials.

在工程中，曲线族 family of curves 可以表示不同材料的应力-应变关系。

作文

In mathematics, particularly in the field of calculus and geometry, the term family of curves refers to a collection of curves that share common characteristics or are defined by a common equation but differ in certain parameters. This concept is crucial for understanding how various mathematical functions can be represented graphically and how they relate to one another. A classic example of a family of curves is the set of parabolas defined by the equation y = ax², where 'a' can take any real number value. As 'a' changes, the shape and position of the parabola change, yet they all belong to the same family due to their shared quadratic nature.Understanding the family of curves is essential not only in pure mathematics but also in applied fields such as physics and engineering. For instance, in physics, the trajectories of projectiles can be modeled using parabolic equations, which form a family of curves based on different initial velocities and angles. Each curve in this family illustrates a different possible path that a projectile might take under the influence of gravity, demonstrating the practical implications of these mathematical concepts.Moreover, the family of curves concept extends beyond parabolas. It includes other types of curves, such as ellipses, hyperbolas, and trigonometric functions. Each type of curve can be expressed with a general equation that allows for the manipulation of parameters to generate various specific curves within that family. For example, the equation of an ellipse can be written as (x²/a²) + (y²/b²) = 1, where 'a' and 'b' determine the size and shape of the ellipse. By varying 'a' and 'b', one can create a family of curves that represent different ellipses, showcasing how changing parameters affects the overall shape while maintaining the fundamental properties of ellipses.The study of family of curves is not merely an academic exercise; it has real-world applications in various domains. In economics, for example, supply and demand curves can be seen as a family of curves that illustrate how price changes affect the quantity supplied and demanded. Each curve may represent a different market condition or consumer preference, highlighting the dynamic nature of economic models.Furthermore, in computer graphics and animation, understanding family of curves is vital for rendering shapes and motions. Designers often use Bézier curves, which are defined by control points, to create smooth and scalable curves. The manipulation of these control points generates a family of curves that can be used to model everything from simple shapes to complex animations, demonstrating the versatility of this mathematical concept.In conclusion, the term family of curves encompasses a wide range of mathematical phenomena that are interconnected through shared equations and parameters. Whether in theoretical mathematics or practical applications, the understanding of family of curves provides valuable insights into how different functions behave and interact. By studying these families, we gain a deeper appreciation for the beauty and complexity of mathematics, as well as its relevance in our everyday lives.

在数学中，特别是在微积分和几何领域，术语family of curves指的是一组共享共同特征或由共同方程定义但在某些参数上有所不同的曲线。这个概念对于理解各种数学函数如何图形化表示以及它们之间的关系至关重要。一个经典的family of curves的例子是由方程y = ax²定义的一组抛物线，其中'a'可以取任何实数值。当'a'变化时，抛物线的形状和位置也会变化，但由于它们共享二次性质，所有这些抛物线都属于同一家族。理解family of curves不仅在纯数学中是必不可少的，而且在物理学和工程等应用领域也是如此。例如，在物理学中，抛射物的轨迹可以用抛物线方程建模，这些方程基于不同的初始速度和角度形成一个family of curves。这个家族中的每条曲线展示了抛射物在重力影响下可能采取的不同路径，体现了这些数学概念的实际意义。此外，family of curves的概念超越了抛物线。它还包括其他类型的曲线，例如椭圆、双曲线和三角函数。每种曲线都可以用一个通用方程来表示，允许通过操纵参数来生成该家族中的各种特定曲线。例如，椭圆的方程可以写成(x²/a²) + (y²/b²) = 1，其中'a'和'b'决定椭圆的大小和形状。通过改变'a'和'b'，可以创建一个family of curves，表示不同的椭圆，展示了改变参数如何影响整体形状，同时保持椭圆的基本属性。对family of curves的研究不仅仅是学术练习；它在各个领域都有现实世界的应用。在经济学中，例如，供求曲线可以视为一种family of curves，它展示了价格变化如何影响供给和需求的数量。每条曲线可能代表不同的市场条件或消费者偏好，突显了经济模型的动态性。此外，在计算机图形学和动画中，理解family of curves对于渲染形状和运动至关重要。设计师通常使用Bézier曲线，这些曲线由控制点定义，用于创建平滑且可缩放的曲线。控制点的操控生成了一组family of curves，可用于从简单形状到复杂动画的建模，展示了这一数学概念的多样性。总之，术语family of curves涵盖了一系列相互关联的数学现象，这些现象通过共享的方程和参数相连。无论是在理论数学还是实际应用中，对family of curves的理解都提供了对不同函数行为和相互作用的宝贵见解。通过研究这些家族，我们对数学的美丽和复杂性有了更深的欣赏，以及它在我们日常生活中的相关性。