Parametric Equations: Key Points Mastery | 参数方程 考点精讲

📚 Parametric Equations: Key Points Mastery | 参数方程 考点精讲

Parametric equations are a powerful way of describing curves by expressing both x and y in terms of a third variable, usually t or θ. In WJEC A-Level Mathematics, you are expected to master eliminating the parameter, differentiating parametric functions, finding equations of tangents and normals, and using integration to compute areas and volumes. This article consolidates all essential techniques with clear English explanations followed by matched Chinese translations.

参数方程通过引入第三个变量（通常为 t 或 θ）分别定义 x 与 y，是描述曲线的重要方法。在 WJEC A-Level 数学中，你需要掌握消参、参数函数求导、求切线与法线方程，以及用积分计算面积和体积。本文梳理全部核心技巧，并提供中英对照的详细讲解。

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

A curve can be defined by expressing x and y as functions of a parameter: x = f(t), y = g(t). As t varies, the point (x, y) traces out the curve. This is particularly useful for describing motion, where t often represents time, or for curves that are not functions in the usual y = f(x) sense.

曲线可以通过参数方程定义为 x = f(t), y = g(t)。随着 t 的变化，点 (x, y) 描出曲线。这在描述运动时尤为有用（此时 t 常代表时间），也适合那些无法用普通函数 y = f(x) 表示的曲线。

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2. Eliminating the Parameter | 消去参数

To convert parametric equations into a Cartesian equation y = f(x) (or a relation between x and y), solve for t from one equation and substitute into the other. For example, if x = t + 2 and y = t², then t = x – 2 gives y = (x – 2)². Always consider any domain restrictions that arise from the original parametric range of t.

要将参数方程转化为笛卡尔方程 y = f(x)（或 x 与 y 的关系式），可以从一个方程解出 t，代入另一个。例如 x = t + 2, y = t²，则 t = x – 2，得到 y = (x – 2)²。务必注意原参数 t 的取值范围可能带来的定义域限制。

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3. Eliminating with Trigonometric Identities | 利用三角恒等式消参

When the parameter appears inside trigonometric functions, such as x = a cos θ, y = b sin θ, use the identity cos²θ + sin²θ = 1. Rearranging: (x/a)² + (y/b)² = 1 gives the standard ellipse equation. For x = a sec θ, y = b tan θ, recall that 1 + tan²θ = sec²θ to obtain (x/a)² – (y/b)² = 1, a hyperbola.

当参数出现在三角函数中时，例如 x = a cos θ, y = b sin θ，利用恒等式 cos²θ + sin²θ = 1。整理得 (x/a)² + (y/b)² = 1，即标准椭圆方程。对于 x = a sec θ, y = b tan θ，利用 1 + tan²θ = sec²θ，可得 (x/a)² – (y/b)² = 1，即双曲线。

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4. Differentiation: First Derivative | 一阶导数

For parametric equations x = f(t), y = g(t), the derivative dy/dx is given by the chain rule: dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0. Compute dy/dt and dx/dt separately, then divide. The result is expressed in terms of t, but you can substitute back to x if a Cartesian form is required.

对于参数方程 x = f(t), y = g(t)，由链式法则得 dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。分别求出 dy/dt 和 dx/dt，再相除。结果用 t 表示，但若需要笛卡尔形式，可将 t 代回 x 的关系。

dy/dx = (dy/dt) / (dx/dt)

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5. Equations of Tangents | 切线方程

To find the equation of the tangent to a parametric curve at a specific point (given t = t₀), first evaluate the coordinates x₀ = f(t₀), y₀ = g(t₀) and the gradient m = dy/dx at t = t₀. Then use the point-gradient form: y – y₀ = m (x – x₀). Simplify to the required form, usually ax + by + c = 0 or y = mx + c.

要找到参数曲线上给定 t = t₀ 处的切线方程，先计算点坐标 x₀ = f(t₀), y₀ = g(t₀)，以及该点的梯度 m = dy/dx|_{t=t₀}。然后运用点斜式：y – y₀ = m (x – x₀)。化简为所需形式，通常为 ax + by + c = 0 或 y = mx + c。

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6. Equations of Normals | 法线方程

The normal is perpendicular to the tangent. Its gradient is the negative reciprocal of the tangent’s gradient: m_normal = -1 / m, provided m ≠ 0. Use the same point (x₀, y₀) and write y – y₀ = m_normal (x – x₀). When m = 0, the tangent is horizontal and the normal is vertical (x = x₀). When the tangent is vertical, the normal is horizontal (y = y₀).

法线与切线垂直，其梯度为切线梯度的负倒数：m_normal = -1 / m（前提 m ≠ 0）。使用同一点 (x₀, y₀) 写出 y – y₀ = m_normal (x – x₀)。当 m = 0 时，切线水平，法线为垂直 x = x₀；当切线垂直时，法线为水平 y = y₀。

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7. Second Derivative | 二阶导数

The second derivative d²y/dx² measures the rate of change of the gradient. For parametric curves, use the formula: d²y/dx² = d(dy/dx)/dt ÷ dx/dt. First find dy/dx as a function of t, then differentiate it with respect to t, and finally divide by dx/dt. The result tells you about concavity and helps identify the nature of stationary points.

二阶导数 d²y/dx² 表示梯度变化率。对于参数曲线，使用公式：d²y/dx² = d(dy/dx)/dt ÷ dx/dt。先求出 dy/dx 关于 t 的表达式，对其求 t 导，再除以 dx/dt。结果可判断曲线的凹性，并帮助确定驻点的性质。

d²y/dx² = [d/dt (dy/dx)] / (dx/dt)

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8. Points of Intersection with Axes | 与坐标轴的交点

To find where a parametric curve crosses the x-axis, set y(t) = 0 and solve for t, then find the corresponding x-coordinate. Similarly, for intersections with the y-axis, set x(t) = 0 and solve for t. Check that the resulting t lies within the allowed parameter domain. These steps are essential for accurate curve sketching.

求参数曲线与 x 轴的交点，令 y(t) = 0，解出 t，再代入求 x 坐标。同理，求与 y 轴交点则令 x(t) = 0 解 t。确保所得 t 在参数允许范围内。这些步骤对于精确绘制曲线图像至关重要。

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9. Area Under a Parametric Curve | 参数曲线下的面积

The area bounded by a parametric curve and the x-axis between t = a and t = b is given by ∫ y dx = ∫ y(t) (dx/dt) dt, where the limits of t correspond to the x-values. Always ensure you integrate in the order of increasing x; if the curve goes ‘backwards’, you may need to reverse limits or split the region. This is a direct application of substitution in definite integrals.

由参数曲线与 x 轴在 t = a 到 t = b 所围面积公式为 ∫ y dx = ∫ y(t) (dx/dt) dt，其中 t 的上下限对应相应的 x 值。务必确保积分沿 x 增加方向进行；若曲线“反向”运动，可能需要调换上下限或分段积分。这是定积分换元法的直接应用。

Area = ∫_t=α^β y(t) (dx/dt) dt

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10. Volume of Revolution (Parametric) | 旋转体体积（参数形式）

When a parametric curve is rotated 360° about the x-axis, the volume generated between t = a and t = b is V = π ∫ y² dx = π ∫ y(t)² (dx/dt) dt. Again, the t-limits must match the x-interval. The same principle extends to rotation about the y-axis using V = π ∫ x² dy, substituting dy = (dy/dt) dt. Watch out for gaps or overlapping regions when the curve crosses the axis.

当参数曲线绕 x 轴旋转 360° 时，在 t = a 到 t = b 之间生成的体积为 V = π ∫ y² dx = π ∫ y(t)² (dx/dt) dt。同样，t 的积分限需与 x 的区间对应。该原理也适用于绕 y 轴旋转：V = π ∫ x² dy，其中 dy = (dy/dt) dt。注意曲线穿过轴时可能出现的空隙或重叠区域。

V = π ∫_t=α^β y² (dx/dt) dt

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