📚 Parametric Equations: Key Exam Topics for IB & OCR Maths | 参数方程考点精讲
Parametric equations offer a flexible way to represent curves by introducing a third variable, known as the parameter. In IB and OCR mathematics, this topic bridges algebra, calculus, and geometry, appearing frequently in Paper 1 and Paper 2 questions. Mastering the conversion between parametric and Cartesian forms, differentiation, and integral applications is essential for top marks.
参数方程通过引入第三个变量（参数），提供了一种灵活描述曲线的方式。在 IB 和 OCR 数学考试中，该主题联结了代数、微积分和几何，常常出现在卷一和卷二试题中。掌握参数形式与笛卡尔形式的互化、求导以及积分应用，是取得高分的关键。
1. What Are Parametric Equations? | 什么是参数方程？
In a parametric representation, both coordinates x and y are expressed as functions of an independent parameter, usually t or θ. For example, x = 2cos t, y = 2sin t (0 ≤ t < 2π) defines a circle of radius 2 centred at the origin.
在参数表示中，x 和 y 坐标都表示为某个独立参数（通常为 t 或 θ）的函数。例如，x = 2cos t, y = 2sin t (0 ≤ t < 2π) 定义了一个圆心在原点、半径为 2 的圆。
Unlike the standard y = f(x) form, a parametric curve can loop, cross itself, or move back and forth. This makes it ideal for modelling motion, where t represents time, giving a clear description of the path and direction of a particle.
与标准的 y = f(x) 形式不同，参数曲线可以回环、自交或来回移动。这使其非常适合用于运动建模，其中 t 表示时间，能清晰描述粒子的轨迹和方向。
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Parametric equations can represent curves that fail the vertical line test, such as circles, ellipses, and cycloids.
参数方程可以表示不满足垂直线检验的曲线，例如圆、椭圆和摆线。
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They allow us to track the position of a moving object over time.
它们允许我们追踪运动物体随时间变化的位置。
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Many standard curves have neat parametric forms that simplify differentiation and integration.
许多标准曲线具有简洁的参数形式，可以简化求导和积分。
2. Eliminating the Parameter: Obtaining the Cartesian Equation | 消去参数：得到笛卡尔方程
To find the Cartesian equation linking x and y directly, we eliminate the parameter. The strategy depends on whether the parameter appears algebraically or through trigonometric functions.
要找出直接联系 x 和 y 的笛卡尔方程，我们需要消去参数。策略取决于参数是以代数形式还是以三角函数形式出现。
Algebraic elimination: Solve one equation for t (or a simple expression involving t) and substitute into the other. Example: x = 2t + 1, y = t² − 3 → t = (x − 1)/2 → y = ((x − 1)/2)² − 3 = (x − 1)²/4 − 3.
代数消元法：从其中一个方程解出 t（或含 t 的简单表达式），代入另一个方程。例如：x = 2t + 1, y = t² − 3 → t = (x − 1)/2 → y = ((x − 1)/2)² − 3 = (x − 1)²/4 − 3。
Trigonometric elimination: Use identities such as sin²t + cos²t = 1, sec²t − tan²t = 1, or double-angle formulas. Example: x = sin t, y = cos 2t → cos 2t = 1 − 2sin²t → y = 1 − 2x².
三角消元法：使用恒等式，如 sin²t + cos²t = 1、sec²t − tan²t = 1，或倍角公式。例如：x = sin t, y = cos 2t → cos 2t = 1 − 2sin²t → y = 1 − 2x²。
When eliminating, always note the restricted domain of the Cartesian equation imposed by the original parametric range. For x = √t, y = t + 1 with t ≥ 0, the Cartesian equation is y = x² + 1 but only for x ≥ 0.
消去参数时，务必留意原参数范围对笛卡尔方程定义域的限制。例如 x = √t, y = t + 1, t ≥ 0，则笛卡尔方程为 y = x² + 1，但仅当 x ≥ 0 时成立。
3. Sketching Parametric Curves | 描绘参数曲线
To sketch a parametric curve, create a table of t, x, and y values for key values of the parameter. Plot these points and join them smoothly, adding arrows to show the direction of increasing t. If possible, eliminate the parameter to recognise the shape of the curve, then restrict the sketch accordingly.
要绘制参数曲线，可列出关键参数值对应的 t、x、y 表格，描点并平滑连接，用箭头标出 t 增大的方向。若有可能，先消去参数识别曲线形状，再据此限制图形范围。
For example, x = t² − 1, y = t − 2 for −2 ≤ t ≤ 3 produces a parabolic arc. A few calculated points reveal the orientation: as t increases, the curve moves upward and to the right.
例如，x = t² − 1, y = t − 2 (−2 ≤ t ≤ 3) 产生一段抛物线弧。计算几个点就能看出走向：随着 t 增大，曲线向右上方移动。
Look for symmetry: if x(−t) = x(t) and y(−t) = −y(t), the curve is symmetric about the x-axis. Use derivatives (dy/dx) to identify turning points and direction.
注意对称性：如果 x(−t) = x(t) 且 y(−t) = −y(t)，则曲线关于 x 轴对称。可利用导数 dy/dx 识别转向点和方向。
Common parametric curves: circles (x = r cos t, y = r sin t), ellipses (x = a cos t, y = b sin t), parabolas (x = at², y = 2at), and cycloids (x = a(t − sin t), y = a(1 − cos t)).
常见参数曲线：圆（x = r cos t, y = r sin t）、椭圆（x = a cos t, y = b sin t）、抛物线（x = at², y = 2at）和摆线（x = a(t − sin t), y = a(1 − cos t)）。
4. First Derivative: dy/dx | 一阶导数：dy/dx
The gradient of a parametric curve is found using the chain rule:
dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
参数曲线的梯度通过链式法则求得：
dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。
This works because dy/dx = (dy/dt) × (dt/dx), and dt/dx = 1/(dx/dt). Compute the derivatives with respect to t first, then divide. For example, if x = t³, y = t²: dx/dt = 3t², dy/dt = 2t → dy/dx = (2t)/(3t²) = 2/(3t).
这一方法成立的原因是 dy/dx = (dy/dt) × (dt/dx)，而 dt/dx = 1/(dx/dt)。先求关于 t 的导数，再相除。例如，若 x = t³, y = t²：dx/dt = 3t², dy/dt = 2t → dy/dx = (2t)/(3t²) = 2/(3t)。
If dx/dt = 0 and dy/dt ≠ 0, the tangent is vertical. If both are zero, further investigation is needed (usually a cusp or self-intersection).
若 dx/dt = 0 而 dy/dt ≠ 0，切线是竖直的。若两者均为零，则需进一步分析（通常是尖点或自交点）。
5. Equations of Tangents and Normals | 切线和法线方程
To find the tangent at a point given by a specific t-value, calculate the gradient m = dy/dx at that t, then use the point-slope form: y − y₁ = m(x − x₁), where (x₁, y₁) is the point obtained from the parametric equations.
求某给定 t 值对应点的切线时，先计算该 t 处的梯度 m = dy/dx，然后利用点斜式：y − y₁ = m(x − x₁)，其中 (x₁, y₁) 为由参数方程得到的点坐标。
For the normal line, the gradient is −1/m (provided m ≠ 0), and the equation uses the same point with y − y₁ = (−1/m)(x − x₁).
对于法线，梯度为 −1/m（假设 m ≠ 0），方程使用同一点，表达式为 y − y₁ = (−1/m)(x − x₁)。
Example: For x = 4 cos t, y = 3 sin t at t = π/4, the point is (2√2, (3√2)/2). dx/dt = −4 sin t, dy/dt = 3 cos t → m = (3 cos t)/(−4 sin t) = −(3/4) cot t. At t = π/4, m = −3/4. Tangent: y − (3√2)/2 = (−3/4)(x − 2√2).
示例：对 x = 4 cos t, y = 3 sin t 在 t = π/4 处，点为 (2√2, (3√2)/2)。dx/dt = −4 sin t, dy/dt = 3 cos t → m = (3 cos t)/(−4 sin t) = −(3/4) cot t。在 t = π/4 处，m = −3/4。切线方程：y − (3√2)/2 = (−3/4)(x − 2√2)。
6. Second Derivative: d²y/dx² | 二阶导数：d²y/dx²
The second derivative measures the concavity of a parametric curve and is found by differentiating dy/dx with respect to x:
d²y/dx² = d(dy/dx)/dx = [d(dy/dx)/dt] / (dx/dt).
二阶导数衡量参数曲线的凹凸性，通过对 x 再求一次 dy/dx 得到：
d²y/dx² = d(dy/dx)/dx = [d(dy/dx)/dt] / (dx/dt)。
Procedure: first compute dy/dx as a function of t; differentiate that expression with respect to t; then divide by dx/dt. Do not simply divide d²y/dt² by d²x/dt² — that is a common mistake.
步骤：先计算出关于 t 的函数 dy/dx；对该表达式关于 t 求导；再除以 dx/dt。切勿将 d²y/dt² 直接除以 d²x/dt² —— 这是一个常见错误。
Example: x = t − 1, y = t² + t. Then dx/dt = 1, dy/dt = 2t + 1 → dy/dx = 2t + 1. d(dy/dx)/dt = 2, so d²y/dx² = 2 / 1 = 2, confirming the curve is always concave up.
示例：x = t − 1, y = t² + t。dx/dt = 1, dy/dt = 2t + 1 → dy/dx = 2t + 1。d(dy/dx)/dt = 2，故 d²y/dx² = 2 / 1 = 2，说明曲线始终向上凹。
7. Area Under a Parametric Curve | 参数曲线下的面积
To find the area bounded by a parametric curve and the x-axis, use the substitution formula:
Area = ∫ y dx = ∫t₁t₂ y(t) x'(t) dt.
求参数曲线与 x 轴所围面积，可使用代换公式：
面积 = ∫ y dx = ∫t₁t₂ y(t) x'(t) dt。
The limits t₁ and t₂ correspond to the x-bounds of the region. You must ensure that x(t) moves monotonically (or split the interval) to avoid sign errors. The result is signed area, so take absolute values if total area is required.
积分限 t₁ 和 t₂ 对应区域的 x 边界。必须确保 x(t) 单调变化（否则需拆分区间），以避免符号错误。该结果表示带符号的面积，若需求总面积应取绝对值。
Example: Find the area of the ellipse x = a cos t, y = b sin t from t = 0 to t = π (upper half). dx/dt = −a sin t. Then Area = ∫0π b sin t · (−a sin t) dt = −ab ∫0π sin²t dt = −ab × (π/2) = −abπ/2. The negative sign indicates orientation; total area = abπ (the well-known formula).
示例：求椭圆 x = a cos t, y = b sin t 从 t = 0 到 t = π 的上半部分面积。dx/dt = −a sin t。则面积 = ∫0π b sin t · (−a sin t) dt = −ab ∫0π sin²t dt = −ab × (π/2) = −abπ/2。负号表示方向；总面积为 abπ（熟知的公式）。
8. Volumes of Revolution | 旋转体体积
When a parametric curve is rotated about the x-axis, the volume is given by:
V = π ∫ y² dx = π ∫t₁t₂ [y(t)]² x'(t) dt.
将参数曲线绕 x 轴旋转时，体积公式为：
V = π ∫ y² dx = π ∫t₁t₂ [y(t)]² x'(t) dt。
If rotating about the y-axis, use V = π ∫ x² dy = π ∫t₁t₂ [x(t)]² y'(t) dt, with appropriate limits. These integrals follow directly from the standard volume of revolution formulae by substituting dx = x'(t) dt.
若绕 y 轴旋转，可使用 V = π ∫ x² dy = π ∫t₁t₂ [x(t)]² y'(t) dt，并选取合适的上下限。这些积分直接来自标准旋转体体积公式，只需代入 dx = x'(t) dt 即可。
Be careful with signs: if x(t) is decreasing, the dx term will be negative, but squaring y eliminates sign issues with the integrand itself. The limits t₁ and t₂ must correspond to the region being rotated.
注意符号：如果 x(t) 下降，dx 项为负，但由于对 y 平方，被积函数本身的符号不受影响。积分限 t₁ 和 t₂ 必须与旋转区域对应。
9. Arc Length | 弧长
The length of a parametric curve from t = a to t = b is:
s = ∫ab √[(dx/dt)² + (dy/dt)²] dt.
参数
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