Parametric Equations | 参数方程

📚 Parametric Equations | 参数方程

Parametric equations describe curves by expressing x and y as functions of a third variable, usually t or θ. In Edexcel A Level Mathematics, you will sketch parametric curves, eliminate the parameter to find Cartesian equations, differentiate to find tangents and normals, and integrate to find areas.

参数方程通过将 x 和 y 表示为第三个变量（通常是 t 或 θ）的函数来描述曲线。在 Edexcel A Level 数学中，你将学习绘制参数曲线、消去参数得到笛卡尔方程、求导以确定切线和法线，以及积分求面积。

1. What Are Parametric Equations? | 什么是参数方程？

A parametric curve is defined by two equations x = f(t) and y = g(t), where t is the parameter. Each value of t gives one point (x, y) on the curve.

参数曲线由两个方程 x = f(t) 和 y = g(t) 定义，其中 t 是参数。每个 t 值对应曲线上的一个点 (x, y)。

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

For example, x = t² and y = 2t define a parabola. As t increases, the point traces out the curve in a specific direction.

例如，x = t² 和 y = 2t 定义一条抛物线。随着 t 增大，点沿特定方向描绘出曲线。

2. The Domain of the Parameter | 参数的定义域

The parameter t is often restricted to an interval such as t ∈ [0, 5] or θ ∈ [0, 2π]. This restriction gives a particular section of the curve rather than the whole locus.

参数 t 通常被限制在一个区间内，例如 t ∈ [0, 5] 或 θ ∈ [0, 2π]。这种限制给出曲线的特定部分，而不是整个轨迹。

Always check the given domain before sketching or eliminating the parameter. For x = t², y = 2t with t ≥ 0, the graph is only the upper half of the Cartesian equation y² = 4x.

在绘制草图或消去参数之前，务必检查给定的定义域。对于 t ≥ 0 的 x = t², y = 2t，其图像仅是笛卡尔方程 y² = 4x 的上半部分。

3. Sketching Parametric Curves | 参数曲线草图

To sketch a parametric curve, make a table of t, x, and y values, plot the points, and join them in increasing order of t. Add an arrow to show the direction of increasing t.

要绘制参数曲线，先列出 t、x 和 y 的数值表，描点，并按 t 增大的顺序连接。用箭头标出 t 增大的方向。

Choose key t values within the domain.
选择定义域内的关键 t 值。
Plot (x, y) points exactly as coordinate pairs.
将 (x, y) 点作为坐标对精确描出。
Label direction from smallest t to largest t.
从最小 t 到最大 t 标出方向。

If the question gives a Cartesian equation to compare with, check that your parametric sketch matches its shape and orientation.

如果题目给出了用于比较的笛卡尔方程，请检查你的参数草图是否与其形状和方向一致。

4. Eliminating the Parameter | 消去参数

Eliminating the parameter means removing t to obtain a Cartesian equation involving only x and y. There are two main methods: substitution and using trigonometric identities.

消去参数意味着去掉 t，得到只含 x 和 y 的笛卡尔方程。主要有两种方法：代入法和使用三角恒等式。

Method 1: Solve one parametric equation for t and substitute into the other. For x = t² and y = 2t, write t = y/2, so x = (y/2)² = y²/4.

方法一：从其中一个参数方程解出 t，并代入另一个方程。对于 x = t² 和 y = 2t，写出 t = y/2，因此 x = (y/2)² = y²/4。

Method 2: Use identities such as cos²θ + sin²θ = 1. For x = a cos θ and y = b sin θ, write cos θ = x/a and sin θ = y/b, then (x/a)² + (y/b)² = 1.

方法二：使用诸如 cos²θ + sin²θ = 1 的恒等式。对于 x = a cos θ 和 y = b sin θ，写出 cos θ = x/a 和 sin θ = y/b，然后得到 (x/a)² + (y/b)² = 1。

After eliminating the parameter, state any restrictions on x or y that come from the range of t or θ.

消去参数后，说明由 t 或 θ 的范围带来的 x 或 y 的任何限制。

5. Finding dy/dx | 求一阶导数 dy/dx

By the chain rule, the gradient of a parametric curve is found using dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0.

根据链式法则，参数曲线的斜率由 dy/dx = (dy/dt) ÷ (dx/dt) 求得，前提是 dx/dt ≠ 0。

dy/dx = (dy/dt) ÷ (dx/dt)

For x = t² and y = 2t, dx/dt = 2t and dy/dt = 2, so dy/dx = 2/(2t) = 1/t.

对于 x = t² 和 y = 2t，dx/dt = 2t，dy/dt = 2，因此 dy/dx = 2/(2t) = 1/t。

At a point where dx/dt = 0 but dy/dt ≠ 0, the tangent is vertical. At a point where dy/dt = 0 but dx/dt ≠ 0, the tangent is horizontal.

在 dx/dt = 0 而 dy/dt ≠ 0 的点处，切线是垂直的。在 dy/dt = 0 而 dx/dt ≠ 0 的点处，切线是水平的。

6. Tangents and Normals | 切线与法线

To find the tangent or normal at a point, first identify the value of t at that point. Then calculate dy/dx at this t to get the gradient m.

要求某点处的切线或法线，首先确定该点对应的 t 值。然后计算该 t 值处的 dy/dx，得到斜率 m。

The tangent equation is y − y₁ = m(x − x₁). The normal gradient is −1/m, so the normal equation is y − y₁ = (−1/m)(x − x₁).

切线方程为 y − y₁ = m(x − x₁)。法线斜率为 −1/m，因此法线方程为 y − y₁ = (−1/m)(x − x₁)。

Example: For x = t², y = 2t at t = 1, the point is (1, 2) and m = 1. The tangent is y − 2 = 1(x − 1), or y = x + 1.

示例：对于 x = t², y = 2t，在 t = 1 处，点为 (1, 2)，斜率 m = 1。切线为 y − 2 = 1(x − 1)，即 y = x + 1。

7. Second Derivative | 二阶导数

The second derivative d²y/dx² measures the curvature of a parametric curve. It is not simply d²y/dt² divided by d²x/dt².

二阶导数 d²y/dx² 衡量参数曲线的弯曲程度。它不是简单地将 d²y/dt² 除以 d²x/dt²。

d²y/dx² = d/dt(dy/dx) ÷ dx/dt

First find dy/dx as a function of t, differentiate it with respect to t, then divide by dx/dt.

首先将 dy/dx 表示为 t 的函数，对其关于 t

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Parametric Equations | 参数方程