A-Level Mathematics: Parametric Equations – Basic Concepts and Representation | A-Level数学：参数方程的基本概念与表示

📚 A-Level Mathematics: Parametric Equations – Basic Concepts and Representation | A-Level数学：参数方程的基本概念与表示

In A-Level Mathematics, a parametric equation defines a set of related quantities as explicit functions of an independent variable called a parameter. Instead of expressing y directly in terms of x, both x and y are written in terms of a third variable, often t, which represents time, angle, or another underlying quantity.

在A-Level数学中，参数方程将一组相关的量定义为独立变量（称为参数）的显函数。它不是直接用x表示y，而是将x和y都写成第三个变量（通常是t）的函数，t可以代表时间、角度或其他潜在的量。

1. What Is a Parametric Equation? | 什么是参数方程？

A parametric equation is a pair of functions that express coordinates (x, y) in terms of a parameter, typically t. The general form is:

参数方程是一对函数，用参数（通常为t）表示坐标(x, y)。一般形式为：

x = f(t), y = g(t)

As t varies over a given interval, the point (x, y) traces out a curve in the plane. This representation is particularly useful for describing motion, since t often corresponds to time.

当t在给定区间内变化时，点(x, y)在平面上描绘出一条曲线。这种表示特别适用于描述运动，因为t通常对应时间。

2. The Role of the Parameter | 参数的作用

The parameter does not appear in the final Cartesian equation, but it controls how the curve is traced. It can encode direction, speed, and the domain of the curve. For example, the same circle can be traced clockwise or anticlockwise by changing the sign of the parametric functions.

参数不会出现在最终的笛卡尔方程中，但它控制曲线的绘制方式。它可以编码方向、速度以及曲线的定义域。例如，同一个圆可以通过改变参数函数的符号来按顺时针或逆时针方向绘制。

A single Cartesian equation can often be represented by many different parametric forms. Choosing a convenient parameter simplifies calculations in calculus and geometry.

一个笛卡尔方程通常可以由许多不同的参数形式表示。选择合适的参数可以简化微积分和几何中的计算。

3. Parametric Equations of a Straight Line | 直线的参数方程

A straight line passing through a point (x₀, y₀) with direction vector (a, b) can be written as:

经过点(x₀, y₀)且方向向量为(a, b)的直线可以写成：

x = x₀ + at, y = y₀ + bt

Here t is any real number. When t = 0, the point is (x₀, y₀); when t = 1, it is (x₀ + a, y₀ + b). The parameter determines the position along the line.

这里t为任意实数。当t = 0时，点为(x₀, y₀)；当t = 1时，点为(x₀ + a, y₀ + b)。参数决定直线上点的位置。

4. Parametric Equations of a Circle | 圆的参数方程

A circle with centre (h, k) and radius r can be parametrised using the angle θ:

圆心为(h, k)、半径为r的圆可以用角度θ参数化：

x = h + r cos θ, y = k + r sin θ

As θ increases from 0 to 2π, the point moves once around the circle anticlockwise. This form is closely linked to the unit circle and trigonometric identities.

当θ从0增加到2π时，点绕圆逆时针移动一周。这种形式与单位圆和三角恒等式密切相关。

5. Parametric Equations of an Ellipse | 椭圆的参数方程

An ellipse with centre at the origin and semi-axes a and b can be expressed as:

中心在原点、半轴为a和b的椭圆可以表示为：

x = a cos t, y = b sin t

Here t is not the angle from the x-axis, but an eccentric angle. When a = b, the ellipse becomes a circle. This parameterisation is useful for integrating along the ellipse.

这里t不是与x轴的夹角，而是离心角。当a = b时，椭圆变成圆。这种参数化在沿椭圆积分时非常有用。

6. Parametric Equations of a Parabola | 抛物线的参数方程

The standard parabola y² = 4ax can be written using parameter t as:

标准抛物线y² = 4ax可以用参数t写成：

x = at², y = 2at

This is a natural parameterisation because it avoids square roots and allows rational expressions. Many properties of the parabola, such as tangents and normals, are easier to derive using t.

这是一种自然的参数化，因为它避免了平方根并允许有理表达式。抛物线的许多性质，如切线和法线，使用t推导更容易。

7. Eliminating the Parameter | 消去参数

To convert a parametric equation to Cartesian form, we eliminate t. For example, from x = at² and y = 2at, we can solve t = y/(2a), then substitute into x:

要将参数方程转换为笛卡尔形式，我们需要消去t。例如，由x = at²和y = 2at，可以先解出t = y/(2a)，然后代入x：

x = a (y/(2a))² = y²/(4a) → y² = 4ax

When eliminating, be careful about restrictions on x and y. The parameter may only take certain values, which limits the domain of the resulting Cartesian equation.

消参时要注意x和y的限制。参数可能只取某些值，这会限制所得笛卡尔方程的定义域。

8. Domain and Range in Parametric Curves | 参数曲线的定义域与值域

The parameter interval determines which part of the curve is drawn. For example, x = cos t, y = sin t with 0 ≤ t ≤ π gives only the upper semicircle, not the full circle.

参数区间决定绘制曲线的哪一部分。例如，x = cos t, y = sin t且0 ≤ t ≤ π只给出上半圆，而不是整个圆。

Similarly, x = t², y = t³ with t ∈ ℝ gives a curve where x ≥ 0, even though the Cartesian form y² = x³ might suggest all x. Always state the domain of the parameter.

类似地，x = t², y = t³且t ∈ ℝ给出的曲线满足x ≥ 0，尽管笛卡尔形式y² = x³可能暗示所有x。务必说明参数的定义域。

9. Differentiation of Parametric Equations | 参数方程的微分

If x = f(t) and y = g(t), the derivative dy/dx is found using the chain rule:

如果x = f(t)且y = g(t)，导数dy/dx可以通过链式法则求得：

dy/dx = (dy/dt) ÷ (dx/dt) = g'(t) / f'(t)

This formula works when dx/dt ≠ 0. It gives the gradient of the tangent to the curve at the point corresponding to parameter t.

这个公式在dx/dt ≠ 0时成立。它给出曲线上对应于参数t的点处的切线斜率。

10. Second Derivatives and Arc Length | 二阶导数与弧长

The second derivative d²y/dx² can be obtained by differentiating dy/dx with respect to t and dividing by dx/dt:

二阶导数d²y/dx²可以通过对dy/dx关于t求导，再除以dx/dt得到：

d²y/dx² = [d/dt (dy/dx)] / (dx/dt)

For arc length, if the curve is defined parametrically, the length from t = a to t = b is given by the integral of √[(dx/dt)² + (dy/dt)²] dt over [a, b]. This shows the power of parametric forms in geometry.

对于弧长，如果曲线是参数定义的，从t = a到t = b的长度由√[(dx/dt)² + (dy/dt)²]在[a, b]上的积分给出。这显示了参数形式在几何中的强大作用。

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A-Level Mathematics: Parametric Equations – Basic Concepts and Representation | A-Level数学：参数方程的基本概念与表示