📚 Parametric Equations | 参数方程
Parametric equations allow us to express the coordinates of points on a curve using a third variable, often called the parameter. This powerful method makes it possible to describe complex motion and curves that are difficult to represent with a single Cartesian equation. In IB and CCEA A-level mathematics, parametric equations form a crucial bridge between algebra, calculus, and real-world kinematics.
参数方程允许我们使用第三个变量（通常称为参数）来表示曲线上点的坐标。这种方法可以描述复杂的运动和那些难以用单一笛卡尔方程表示的曲线。在 IB 和 CCEA A-level 数学中，参数方程是代数、微积分和现实运动学之间的重要桥梁。
1. Introduction to Parametric Equations | 参数方程简介
A parametric equation defines both x and y in terms of a third variable t: x = f(t), y = g(t). As t varies, the point (x, y) traces out a curve in the plane. This representation is particularly useful when a relationship cannot be expressed as a single function y = f(x), or when describing motion where t represents time.
参数方程使用第三个变量 t 来定义 x 和 y：x = f(t), y = g(t)。随着 t 的变化，点 (x, y) 在平面上描绘出一条曲线。当无法用单个函数 y = f(x) 表达关系时，或者当描述运动且 t 表示时间时，这种表示方式特别有用。
For example, the unit circle can be written as x = cos t, y = sin t, where t ∈ [0, 2π). Each value of t corresponds to an angle measured anticlockwise from the positive x-axis.
例如，单位圆可以写成 x = cos t, y = sin t，其中 t ∈ [0, 2π)。每个 t 值对应于从正 x 轴逆时针测量的角度。
2. Sketching Parametric Curves | 绘制参数曲线
To sketch a parametric curve, we can create a table of t, x, and y values. Choose a range of t, compute the corresponding coordinates, and plot them in order. Often, indicating the direction of increasing t with arrows helps to interpret the curve as a path.
要绘制参数曲线，我们可以制作一个包含 t、x 和 y 值的表格。选择一个 t 的范围，计算相应的坐标，并按顺序描点。通常，用箭头标出 t 增加的方向，有助于将曲线理解为一条路径。
For x = t² − 2, y = t + 1, with t from −3 to 3, we get a right‑opening parabola. The vertex is at (−2, 1) when t = 0. The direction arrows show the curve is traced from left to right as t increases.
对于 x = t² − 2，y = t + 1，t 从 −3 到 3，我们得到一条向右开口的抛物线。当 t = 0 时顶点位于 (−2, 1)。方向箭头显示随着 t 增加，曲线从左向右绘制。
3. Eliminating the Parameter | 消去参数
In many exam questions, you are asked to find the Cartesian equation of a curve given in parametric form. This involves eliminating the parameter t to obtain a relationship directly between x and y. Common techniques include substitution, using trigonometric identities, or solving for t from one equation.
在许多考题中，会要求找出以参数形式给出的曲线的笛卡尔方程。这需要通过消去参数 t 来得到 x 和 y 之间的直接关系。常用的技巧包括代入法、使用三角恒等式，或从一个方程解出 t。
For x = 2 cos t, y = 3 sin t, we use cos² t + sin² t = 1. Thus (x/2)² + (y/3)² = 1, which is an ellipse. For x = 1/t, y = t², we can write t = 1/x and substitute to get y = 1/x².
对于 x = 2 cos t，y = 3 sin t，使用 cos² t + sin² t = 1，得到 (x/2)² + (y/3)² = 1，这是一个椭圆。对于 x = 1/t，y = t²，可以写出 t = 1/x 并代入，得到 y = 1/x²。
4. First Derivative (dy/dx) | 一阶导数
The derivative dy/dx for parametric equations is found using the chain rule: dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0. This gradient function describes the slope of the tangent at any point on the curve.
参数方程的一阶导数 dy/dx 是通过链式法则求得的：dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。这个梯度函数描述了曲线上任意一点切线的斜率。
If x = t³ and y = t² + 1, then dx/dt = 3t², dy/dt = 2t, so dy/dx = 2t / 3t² = 2/(3t). At t = 2, the gradient is 2/(6) = 1/3.
如果 x = t³，y = t² + 1，则 dx/dt = 3t²，dy/dt = 2t，因此 dy/dx = 2t / 3t² = 2/(3t)。在 t = 2 处，梯度为 2/6 = 1/3。
5. Second Derivative (d²y/dx²) | 二阶导数
To find the second derivative d²y/dx², we differentiate dy/dx with respect to x, but since dy/dx is expressed in terms of t, we use: d²y/dx² = d(dy/dx)/dt ÷ dx/dt. This measures the concavity and rate of change of the gradient.
要求二阶导数 d²y/dx²，需对 dy/dx 关于 x 求导，但由于 dy/dx 是用 t 表示的，所以我们使用：d²y/dx² = d(dy/dx)/dt ÷ dx/dt。它衡量的是梯度的变化率和曲线的凹凸性。
From the previous example, dy/dx = 2/(3t) = (2/3) t⁻¹, so d(dy/dx)/dt = −(2/3) t⁻². Then d²y/dx² = [−(2/3) t⁻²] / (3t²) = −2/(9 t⁴). A negative value indicates the curve is concave down.
接上例，dy/dx = 2/(3t) = (2/3) t⁻¹，因此 d(dy/dx)/dt = −(2/3) t⁻²。于是 d²y/dx² = [−(2/3) t⁻²] / (3t²) = −2/(9 t⁴)。负值表明曲线在该点处是凹向下的。
6. Tangents and Normals | 切线与法线
Once the gradient m = dy/dx at a specific t is known, the equation of the tangent can be written in point–slope form: y − y₁ = m (x − x₁). The normal line has gradient −1/m (provided m ≠ 0) and passes through the same point.
一旦知道了特定 t 处的梯度 m = dy/dx，就可以用点斜式写出切线方程：y − y₁ = m (x − x₁)。法线的梯度为 −1/m（只要 m ≠ 0），且穿过同一点。
For x = t², y = 2t at t = 1, the point is (1, 2). dx/dt = 2t → 2, dy/dt = 2, so m = 1. The tangent is y − 2 = 1(x − 1) ⇒ y = x + 1. The normal is y − 2 = −1(x − 1) ⇒ y = −x + 3.
对于 x = t²，y = 2t，在 t = 1 处，点为 (1, 2)。dx/dt = 2t → 2，dy/dt = 2，因此 m = 1。切线为 y − 2 = 1(x − 1) ⇒ y = x + 1。法线为 y − 2 = −1(x − 1) ⇒ y = −x + 3。
7. Arc Length | 弧长
The length of a parametric curve from t = a to t = b is given by the integral s = ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt. Unlike Cartesian arc length, no rearrangement is needed—just differentiate both parametric expressions and integrate the square root of the sum of squares.
从 t = a 到 t = b 的参数曲线弧长由积分 s = ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt 给出。与笛卡尔弧长不同，这里不需要重新排列——只需对两个参数表达式求导，再对它们的平方和开方进行积分。
For x = 3 cos t, y = 3 sin t from t = 0 to π/2, we have dx/dt = −3 sin t, dy/dt = 3 cos t. The integrand is √[9 sin² t + 9 cos² t] = 3. Thus s = ∫₀^(π/2) 3 dt = (3π/2). This matches a quarter of the circumference of a circle with radius 3.
对于 x = 3 cos t，y = 3 sin t，t 从 0 到 π/2，有 dx/dt = −3 sin t，dy/dt = 3 cos t。被积函数为 √[9 sin² t + 9 cos² t] = 3。因此 s = ∫₀^(π/2) 3 dt = (3π/2)，这正好是半径为 3 的圆的四分之一周长。
8. Area Under a Curve | 曲线下的面积
To find the area bounded by a parametric curve and the x‑axis, we use the formula A = ∫ y dx = ∫ y (dx/dt) dt, with the limits of t corresponding to the required x‑values. If the curve loops back, careful handling of sign and limits is needed.
要求由参数曲线与 x 轴所围成的面积，我们使用公式 A = ∫ y dx = ∫ y (dx/dt) dt，其中 t 的积分限对应所需的 x 值。如果曲线出现回路，则需要小心处理符号和积分限。
For the cycloid x = a(t − sin t), y = a(1 − cos t) between t = 0 and 2π, the area under one arch is A = ∫₀²π a(1 − cos t) · a(1 − cos t) dt = a² ∫₀²π (1 − cos t)² dt = 3πa².
对于摆线 x = a(t − sin t)，y = a(1 − cos t)，t 在 0 到 2π 之间，一个拱下的面积为 A = ∫₀²π a(1 − cos t) · a(1 − cos t) dt = a² ∫₀²π (1 − cos t)² dt = 3πa²。
9. Applications in Kinematics | 运动学应用
In mechanics, parametric equations physically represent the trajectory of a particle, with t as time. The velocity components are vₓ = dx/dt and v_y = dy/dt. Speed is the magnitude √(vₓ² + v_y²). Acceleration components are aₓ = d²x/dt², a_y = d²y/dt².
在力学中，参数方程实际表示质点的运动轨迹，t 作为时间。速度分量为 vₓ = dx/dt 和 v_y = dy/dt。速率是速度的大小 √(vₓ² + v_y²)。加速度分量为 aₓ = d²x/dt²，a_y = d²y/dt²。
If a particle moves according to x = 2t, y = t² − 4t, differentiate to find vₓ = 2, v_y = 2t − 4. Speed at t = 3 is √(2² + 2²) = √8 = 2√2 m/s. The Cartesian path is found by eliminating t: t = x/2 gives y = (x/2)² − 4(x/2) = x²/4 − 2x, a parabola.
若质点按 x = 2t，y = t² − 4t 运动，求导得 vₓ = 2，v_y = 2t − 4。在 t = 3 时速率为 √(2² + 2²) = √8 = 2√2 m/s。消去 t 可得笛卡尔轨迹：t = x/2 代入 y = (x/2)² − 4(x/2) = x²/4 − 2x，这是一条抛物线。
10. Conversion Between Cartesian and Parametric Forms | 笛卡尔形式与参数形式的转换
Converting from Cartesian to parametric introduces a parameter. A common technique is to let x = t (or a function of t) and solve for y. For a circle x² + y² = r², we can set x = r cos θ, y = r sin θ. For a parabola y = x², a simple parametrisation is x = t, y = t². Always check that the domain of t covers the required portion of the curve.
从笛卡尔形式转换为参数形式需要引入一个参数。常用的技巧是令 x = t（或 t 的函数）并解出 y。对于圆 x² + y² = r²，可设 x = r cos θ，y = r sin θ。对于抛物线 y = x²，简单的参数化是 x = t，y = t²。务必检查 t 的定义域是否覆盖了曲线的所需部分。
For a line segment from (1,2) to (4,6), a parametric form is x = 1 + 3t, y = 2 + 4t, t ∈ [0,1]. This is extremely useful in vector equations and motion along a straight line.
对于从 (1,2) 到 (4,6) 的线段，参数形式为 x = 1 + 3t，y = 2 + 4t，t ∈ [0,1]。这在向量方程和沿直线运动中非常有用。
11. Common Parametric Forms (Circle, Ellipse, Cycloid) | 常见参数形式（圆、椭圆、摆线）
Recognising standard parametric forms saves time in exams. Circle centred at (h,k) radius r: x = h + r cos t, y = k + r sin t. Ellipse: x = h + a cos t, y = k + b sin t. Cycloid: x = a(t − sin t), y = a(1 − cos t). Parabola (sideways): x = at², y = 2at.
识别标准的参数形式可以在考试中节省时间。圆心为 (h,k)、半径为 r 的圆：x = h + r cos t，y = k + r sin t。椭圆：x = h + a cos t，y = k + b sin t。摆线：x = a(t − sin t)，y = a(1 − cos t)。侧开口抛物线：x = at²，y = 2at。
For the parabola x = 2t², y = 4t, the Cartesian equation is y² = 8x. The focus and directrix can be found more easily after elimination. Knowing these standard forms helps quickly identify the type of conic section.
对于抛物线 x = 2t²，y = 4t，其笛卡尔方程为 y² = 8x。消参后可以更容易地找到焦点和准线。熟知这些标准形式有助于快速识别圆锥曲线的类型。
12. Exam Tips and Summary | 考试技巧与总结
Always be mindful of the parameter range when sketching or integrating. Recall that dy/dx is undefined when dx/dt = 0, indicating a vertical tangent. When eliminating the parameter, check for domain restrictions that might exclude parts of the Cartesian curve.
在绘图或积分时务必注意参数的范围。记住当 dx/dt = 0 时 dy/dx 无定义，这表示存在垂直切线。消去参数时，需检查定义域的限制，这些限制可能会排除笛卡尔曲线的一部分。
Practice converting between forms and calculating derivatives/areas in both directions. A clear table of key formulas is invaluable:
勤加练习形式转换以及两个方向的导数与面积计算。整理一个清晰的公式表会非常有价值：
| Concept | Formula |
| First derivative | dy/dx = (dy/dt) / (dx/dt) |
| Second derivative | d²y/dx² = (d/dt[dy/dx]) / (dx/dt) |
| Arc length | s = ∫ √[(dx/dt)² + (dy/dt)²] dt |
| Area under curve | A = ∫ y (dx/dt) dt |
Mastering parametric equations opens the door to advanced topics like polar coordinates and differential geometry. Keep your work organised, label all steps clearly, and remember that every t tells a story about the shape and motion of the curve.
掌握参数方程将为极坐标和微分几何等高级主题打开大门。保持解题过程条理清晰，明确标注每一步，并记住每一个 t 都讲述了曲线形状和运动的故事。
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