📚 AS Mathematics: Parametric Equations Mastery Guide | AS 数学：参数方程考点精讲
Parametric equations are a powerful tool in AS Mathematics that describe curves using an independent parameter, typically t. Unlike Cartesian equations that directly relate x and y, parametric equations express both coordinates as functions of t: x = f(t), y = g(t). Mastering this topic enables you to tackle a variety of problems, from sketching unfamiliar curves to computing tangents and normals efficiently. This guide covers all essential concepts you need to excel in your exam.
参数方程是 AS 数学中一个强有力的工具，它通过一个独立参数（通常为 t）来描述曲线。与直接关联 x 和 y 的笛卡尔方程不同，参数方程将两个坐标分别表示为 t 的函数：x = f(t)，y = g(t)。掌握这一专题能够帮助你高效解决各种问题，从绘制陌生曲线到计算切线与法线。本指南涵盖所有你需要精通的关键概念，助你在考试中脱颖而出。
1. What Are Parametric Equations? | 参数方程概念
A parametric equation defines a curve by giving x and y as separate functions of a third variable, the parameter. Often the parameter is denoted by t (but can be θ, u, etc.). For example, x = 2t, y = t² describes a parabola. As t varies, the point (x, y) traces the curve. The parameter often represents time or an angle, making parametric forms ideal for modelling real-world motion.
参数方程通过将 x 和 y 分别定义为第三个变量（参数）的函数来描述曲线。参数通常用 t 表示（也可能用 θ、u 等）。例如，x = 2t, y = t² 就描述了一条抛物线。随着 t 变化，点 (x, y) 会描绘出整条曲线。由于参数常常代表时间或角度，参数方程非常适合模拟现实世界中的运动。
A classic example is the unit circle: x = cos θ, y = sin θ, where θ is the angle measured from the positive x‑axis. Another common form is x = a cos t, y = b sin t, which gives an ellipse with semi‑axes a and b. Recognising these standard forms speeds up graph sketching and elimination tasks.
一个经典例子是单位圆：x = cos θ, y = sin θ，其中 θ 是从正 x 轴测量的角度。另一种常见形式是 x = a cos t, y = b sin t，它给出长短半轴分别为 a 和 b 的椭圆。识别这些标准形式能加速图形绘制和消参过程。
2. Eliminating the Parameter | 消去参数
To find the Cartesian equation (an equation purely in x and y), we eliminate the parameter t. The simplest method is direct substitution: solve one parametric equation for t in terms of x (or y) and substitute into the other. For instance, if x = t + 3 and y = 2t − 1, then t = x − 3 gives y = 2(x − 3) − 1 = 2x − 7. This yields a straight line.
要找出笛卡尔方程（仅含 x 和 y 的方程），我们需要消去参数 t。最简单的方法是直接代入：从其中一个参数方程解出 t 用 x 表示，再代入另一个方程。例如，若 x = t + 3, y = 2t − 1，则 t = x − 3，代入得 y = 2(x − 3) − 1 = 2x − 7。这便得到了一条直线方程。
When trigonometric functions are involved, use standard identities. For example, given x = 3 sec θ, y = 4 tan θ, we apply sec² θ − tan² θ = 1. Substituting sec θ = x/3 and tan θ = y/4 yields (x/3)² − (y/4)² = 1, a hyperbola. Also, with x = a cos θ, y = b sin θ, using cos² θ + sin² θ = 1 leads to (x/a)² + (y/b)² = 1, an ellipse.
当含有三角函数时，可利用标准恒等式。例如，已知 x = 3 sec θ, y = 4 tan θ，利用 sec² θ − tan² θ = 1，代入 sec θ = x/3, tan θ = y/4，得到 (x/3)² − (y/4)² = 1，即双曲线。另外，对于 x = a cos θ, y = b sin θ，运用 cos² θ + sin² θ = 1 可得 (x/a)² + (y/b)² = 1，为椭圆。
Always check the domain of t: if t has a restricted range, the Cartesian equation may represent only part of the full curve. For example, x = √(t), y = t − 4 (t ≥ 0) transforms to x² = t, so y = x² − 4, but with x ≥ 0 – only the right branch exists.
务必检查参数的定义域：若 t 有取值范围，笛卡尔方程可能只表示完整曲线的一部分。例如，x = √(t), y = t − 4 (t ≥ 0) 转化为 x² = t，于是 y = x² − 4，但 x ≥ 0，因此只有右半支存在。
3. Domain and Range from Parametric Equations | 定义域与值域
Because x and y are defined via the parameter, their allowed values (domain and range) are determined by the range of t. For x = f(t), y = g(t), compute the set of all possible x‑values as t runs over its domain to obtain the domain; similarly, find all possible y‑values for the range. Graphical methods or evaluating min/max of f(t) and g(t) are useful.
由于 x 和 y 是通过参数定义的，它们允许的取值范围（定义域和值域）由 t 的范围决定。对于 x = f(t), y = g(t)，计算当 t 在其定义域内变化时所有可能 x 值的集合，即可得到定义域；类似地，找出所有可能的 y 值得到值域。图形法或求 f(t)、g(t) 的最值都很实用。
Example: x = 2 sin t, y = cos t, 0 ≤ t ≤ π. Here x ranges from 2 sin 0 = 0 to 2 sin(π/2) = 2 and back to 2 sin π = 0, so domain is [0, 2]. y = cos t decreases from 1 to −1, so range is [−1, 1]. The curve is only the upper half (or a portion) of the ellipse ¼ x² + y² = 1.
例如：x = 2 sin t, y = cos t, 0 ≤ t ≤ π。此时 x 从 2 sin 0 = 0 到 2 sin(π/2) = 2，再回到 2 sin π = 0，因此定义域为 [0, 2]。y = cos t 从 1 递减到 −1，故值域为 [−1, 1]。该曲线仅是椭圆 ¼ x² + y² = 1 的一部分（上半部一部分）。
In exam questions, you may be asked to state the domain of the Cartesian equation after elimination – always refer back to the original parametric restrictions. Ignoring the parameter range leads to an incomplete description of the curve.
在考试题中，你可能会被要求在消参后给出笛卡尔方程的定义域——务必回顾原始的参变数限制。忽略参变数的范围会导致对曲线描述不完整。
4. Differentiation: First Derivative | 一阶导数
To find the gradient dy/dx from parametric equations, use the chain rule: dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This formula gives dy/dx still expressed in terms of t. If you need the gradient at a specific point, first find the corresponding t‑value.
要从参数方程求梯度 dy/dx，利用链式法则：dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。该公式给出的 dy/dx 仍是用 t 表示的。若需求某具体点处的梯度，须先找出对应的 t 值。
dy/dx = (dy/dt) ÷ (dx/dt)
Example: x = t², y = t³. Then dx/dt = 2t, dy/dt = 3t², so dy/dx = (3t²)/(2t) = (3/2)t, for t ≠ 0. At t = 2, the gradient is 3, and the point is (4, 8). This straightforward differentiation step is essential for tangent and normal problems.
例如：x = t², y = t³。则 dx/dt = 2t, dy/dt = 3t²，因此 dy/dx = (3t²)/(2t) = (3/2)t，t ≠ 0。当 t = 2 时，梯度为 3，对应点为 (4, 8)。这一直接微分步骤对于切线与法线问题至关重要。
Be careful when dx/dt = 0 at some t: the gradient may be vertical or undefined, indicating a cusp or a point where the tangent is parallel to the y‑axis.
注意当某些 t 使得 dx/dt = 0 时：梯度可能为垂直或未定义，表明存在尖点（cusp）或切线平行于 y 轴的情况。
5. Equation of Tangent and Normal | 切线与法线方程
Once you have the gradient m = dy/dx at a known point (x₁, y₁) obtained from a parameter value t, the tangent line equation is y − y₁ = m (x − x₁). The normal line (perpendicular) has gradient −1/m, giving y − y₁ = (−1/m)(x − x₁). Always write the final equation in the required form: y = mx + c, ax + by + c = 0, etc.
求出梯度 m = dy/dx 以及由参数值 t 确定的已知点 (x₁, y₁) 后，切线方程即为 y − y₁ = m (x − x₁)。法线（垂直于切线）的梯度为 −1/m，方程为 y − y₁ = (−1/m)(x − x₁)。最终方程应按题目要求化为 y = mx + c 或 ax + by + c = 0 等形式。
Worked example: curve defined by x = 2t + 1, y = t² − t. At t = 1: x₁ = 3, y₁ = 0. dx/dt = 2, dy/dt = 2t − 1 ⇒ at t = 1, dy/dx = (1)/2 = 1/2. Tangent: y − 0 = ½ (x − 3) → 2y = x − 3. Normal: gradient = −2, equation: y = −2(x − 3) → y = −2x + 6.
例题：曲线由 x = 2t + 1, y = t² − t 定义。在 t = 1 时：x₁ = 3, y₁ = 0。dx/dt = 2, dy/dt = 2t − 1 ⇒ t = 1 时 dy/dx = 1/2。切线：y − 0 = ½ (x − 3) → 2y = x − 3。法线：梯度 = −2，方程：y = −2(x − 3) → y = −2x + 6。
Always double‑check that the point indeed lies on the curve. The calculations become straightforward once you find dy/dx and the coordinates – just don’t forget to convert the parameter to the specific values.
务必再次确认该点确实在曲线上。一旦找到 dy/dx 和坐标，计算就变得简单直接——只须记住将参数转换为具体数值。
6. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative d²y/dx² tells us about the curvature of the graph. From parametric equations, we use: d²y/dx² = d(dy/dx)/dx = [d(dy/dx)/dt] / (dx/dt). In other words, differentiate dy/dx with respect to t, then divide by dx/dt. The result is expressed in terms of t.
二阶导数 d²y/dx² 告诉我们图像的弯曲情况。从参数方程出发，我们使用：d²y/dx² = d(dy/dx)/dx = [d(dy/dx)/dt] / (dx/dt)。换言之，先对 dy/dx 关于 t 求导，再除以 dx/dt。结果仍用 t 表达。
d²y/dx² = (d/dt[dy/dx]) / (dx/dt)
Continuing the earlier example x = t², y = t³, we had dy/dx = 3t/2. Differentiate w.r.t t: d/dt(3t/2) = 3/2. dx/dt = 2t. Thus d²y/dx² = (3/2) / (2t) = 3/(4t). For t = 2, d²y/dx² = 3/8 > 0, so the curve is concave upwards at that point.
继续前面的例子 x = t², y = t³，我们有 dy/dx = 3t/2。对 t 求导：d/dt(3t/2) = 3/2。dx/dt = 2t。因此 d²y/dx² = (3/2) / (2t) = 3/(4t)。当 t = 2 时，d²y/dx² = 3/8 > 0，故曲线在该点处凹向上。
A negative second derivative indicates concave downwards. In exam questions, you may be asked to verify a point of inflection – where d²y/dx² = 0 and changes sign. Remember to check that dx/dt ≠ 0 at that t, otherwise the formula breaks.
负的二阶导数表示凹向下。在考题中，可能要求验证拐点——即 d²y/dx² = 0 且变号的位置。记住要检查在该 t 处 dx/dt ≠ 0，否则公式不成立。
7. Turning Points via Parametric Equations | 利用参数方程求驻点
Stationary (turning) points occur where dy/dx = 0, i.e. when dy/dt = 0 and dx/dt ≠ 0. To classify them, you can either use the second derivative d²y/dx² or check the sign change of dy/dx around the t‑value. The nature (maximum, minimum, or horizontal inflection) is determined just as in Cartesian analysis.
驻点（极值点）发生在 dy/dx = 0 处，即 dy/dt = 0 且 dx/dt ≠ 0。对其进行分类时，可利用二阶导数 d²y/dx²，也可检查 dy/dx 在 t 值附近的符号变化。驻点的性质（极大值、极小值或水平拐点）与笛卡尔分析中的判定方法相同。
Example: x = t³ − 3t, y = t². Then dx/dt = 3t² − 3, dy/dt = 2t. Set dy/dx = 0 → 2t / (3t² − 3) = 0 → t = 0. At t = 0, x = 0, y = 0. dx/dt = −3 ≠ 0, so (0,0) is a stationary point. d²y/dx² calculation determines it is a minimum if d²y/dx² > 0, etc. (Here d²y/dx² = −2/9 < 0? Let's compute: dy/dx = 2t/(3t²−3), d/dt = quotient rule → we can check sign changes.)
例如：x = t³ − 3t, y = t²。则 dx/dt = 3t² − 3, dy/dt = 2t。令 dy/dx = 0 → 2t / (3t² − 3) = 0 → t = 0。在 t = 0 时，x = 0, y = 0。dx/dt = −3 ≠ 0，故 (0,0) 为驻点。再通过计算 d²y/dx² 或符号变化判定其性质。
Remember that if both dx/dt and dy/dt are zero at the same t, the point could be a cusp or an isolated point – further investigation using higher derivatives or graph behaviour is needed. AS exams tend to avoid such degenerate cases, but it’s good to be aware.
如果 dx/dt 和 dy/dt 在同一 t 处同时为零，该点可能是尖点或孤立点——需要借助高阶导数或图形行为进一步分析。AS 考试通常避免此类退化情形，但了解它很有好处。
8. Intersections with Axes | 与坐标轴的交点
To find where a parametric curve crosses the x‑axis, set y = g(t) = 0 and solve for t; then compute x = f(t) to get the coordinates. Similarly, for y‑axis intersections, set x = f(t) = 0, solve for t, and find y. Always list all possible values of t within the given range.
要求参数曲线与 x 轴的交点，设 y = g(t) = 0 解出 t，再计算 x = f(t) 得到坐标。类似地，与 y 轴的交点则设 x = f(t) = 0 解出 t，并求出 y。务必列出给定范围内所有可能的 t 值。
Example: Curve given by x = 2 cos t, y = 3 sin t, 0 ≤ t ≤ 2π. For x‑axis intersections, y = 0 → 3 sin t = 0 → t = 0, π, 2π. Corresponding points: (2, 0), (−2, 0), (2, 0). For y‑axis, x = 0 → cos t = 0 → t = π/2, 3π/2 → points (0, 3) and (0, −3). This method is straightforward but always substitute back to confirm.
例：曲线由 x = 2 cos t, y = 3 sin t, 0 ≤ t ≤ 2π 给出。与 x 轴相交时 y = 0 → 3 sin t = 0 → t = 0, π, 2π。对应点：(2, 0), (−2, 0), (2, 0)。与 y 轴相交时 x = 0 → cos t = 0 → t = π/2, 3π/2 → 点 (0, 3) 和 (0, −3)。方法简单，但务必代回确认。
These intersection points are particularly helpful when sketching the parametric curve; they provide key anchor points before you determine the general shape.
这些交点在绘制参数曲线草图时格外有用；它们能在确定大致形状前提供关键的锚点。
9. Common Mistakes to Avoid | 常见错误
- Forgetting to check dx/dt ≠ 0 before using the derivative formula. If dx/dt = 0, the gradient is vertical or undefined – handle as a special case.
- Eliminating the parameter too early: sometimes it’s easier to find dy/dx directly in parametric form, especially for tangents.
- Mixing up the order when eliminating: substituting incorrectly and losing the correct relationship.
- Ignoring parameter restrictions, which leads to an incorrect Cartesian domain (e.g., missing that x must be non‑negative).
- When computing second derivative, forgetting to divide by dx/dt again – the formula is not just d²y/dt² / dx²/dt².
- 在使用导数公式前忘记检查 dx/dt ≠ 0。若 dx/dt = 0
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