📚 Investigation 3: Parametric Equations | 参数方程探究
Parametric equations provide a powerful way to describe curves by expressing the coordinates x and y in terms of an independent parameter, often t. Unlike standard Cartesian equations, parametric forms can represent complex motions, such as the flight of a projectile or the orbit of a planet, where time is naturally the driving variable. In this investigation, we will explore how to work with parametric equations: eliminating the parameter, graphing, differentiating, finding areas, and computing arc lengths. These skills are essential for the IB Mathematics curriculum and for solving real-world problems in physics and engineering.
参数方程通过用一个独立参数（通常为t）表示坐标x和y，提供了一种描述曲线的强大方式。与标准笛卡尔方程不同，参数形式可以表示复杂的运动，例如抛射体的飞行或行星的轨道，其中时间自然成为驱动变量。在本次探究中，我们将深入探讨如何处理参数方程：消去参数、绘制图形、求导、计算面积以及求弧长。这些技能对于IB数学课程至关重要，也能帮助解决物理和工程中的实际问题。
1. Introduction to Parametric Equations | 参数方程简介
cos θ, y = sin θ with 0 ≤ θ < 2π. This representation offers greater flexibility than y = √1 − x² or its negative counterpart, and it naturally incorporates direction and orientation.
参数曲线由一对函数 x = f(t) 和 y = g(t) 定义，其中 t 是在某一区间内变化的参数。参数的选择可以是时间、角度或任何方便的变量。例如，圆可以表示为 x = cos θ, y = sin θ，其中 0 ≤ θ < 2π。这种表示比 y = √1 − x² 及其负值形式更灵活，而且自然地包含了方向和取向。
In IB Mathematics, parametric equations appear in both Analysis & Approaches and Applications & Interpretation courses. They allow us to model phenomena where x and y change interdependently but not as a single-valued function. Key advantages include the ability to trace the entire curve, handle vertical tangents, and describe motion with a velocity vector.
在IB数学课程中，参数方程同时出现在分析与方法和应用与解释两类课程中。它们能帮助我们为那些 x 与 y 相互依赖变化但不是单值函数的现象建模。主要优点包括能够描绘整条曲线、处理垂直切线，以及用速度矢量描述运动。
int (x, y). As t increases, the point moves along a straight line. This dynamic view is central to understanding parametric curves.
考虑参数形式 x = t + 1, y = 2t − 3。对应每个 t 值，我们会得到一个点 (x, y)。当 t 增加时，该点沿一条直线运动。这种动态视角对理解参数曲线至关重要。
2. Eliminating the Parameter | 消去参数
r. With trigonometric parameters, use identities such as cos²θ + sin²θ = 1 to eliminate θ.
消去参数是从参数方程推导出笛卡尔方程 y = F(x) 的过程。这通常能揭示曲线的几何形状。方法是从其中一个方程解出 t，然后代入另一个方程。对于三角函数参数，可利用恒等式如 cos²θ + sin²θ = 1 来消去 θ。
2)² + (y/3)² = 1, which is an ellipse. Similarly, from x = t², y = t³ (t ≥ 0), solving t = √x gives y = (√x)³ = x3/2, a power curve.
例如，给定 x = 2 cos θ, y = 3 sin θ，可写出 cos θ = x/2 和 sin θ = y/3。平方相加得 (x/2)² + (y/3)² = 1，这是一个椭圆。类似地，由 x = t², y = t³ (t ≥ 0)，解出 t = √x 得 y = (√x)³ = x3/2，这是一条幂函数曲线。
he point moves from left to right as t increases. Therefore, parametric forms preserve dynamical information that Cartesian forms may lose.
然而，消去参数有时会掩盖曲线的方向或限制定义域。从 x = t, y = t² 得到的笛卡尔方程 y = x² 无法显示当 t 增加时点是从左向右移动的。因此，参数形式保留了笛卡尔形式可能丢失的动态信息。
3. Graphing Parametric Curves | 绘制参数曲线
t) cannot be expressed simply as y = f(x) yet are easily plotted from the parametric form.
要绘制参数曲线，可先列出 t，x，y 的取值表，描点并用箭头标出 t 增加的方向。t 的定义域决定了要绘制的曲线部分。像摆线 (x = t − sin t, y = 1 − cos t) 这样的曲线难以简单表示为 y = f(x)，但可从参数形式轻松绘图。
Most graphing calculators and software accept parametric input. When analysing a curve, always note important points: where the curve crosses the axes, where it has horizontal or vertical tangents, and its starting and ending positions. These features are found by examining x(t) and y(t) and their derivatives.
大多数图形计算器和软件都接受参数输入。分析曲线时，务必关注关键点：曲线与坐标轴的交点、水平或垂直切线位置，以及起点和终点。这些特征可通过考察 x(t)、y(t) 及其导数得到。
1, 6). The shape is a rightward-opening parabola. Arrows indicate that as t increases, the point moves from the leftmost to the rightmost part of the curve.
例如，绘制 x = t − 1, y = t² + 2，其中 −2 ≤ t ≤ 2。列表得到点如 t = −2: (−3, 6)；t = 0: (−1, 2)；t = 2: (1, 6)。形状是一条向右开口的抛物线。箭头表明随着 t 增加，点从曲线最左侧移动到最右侧。
4. Differentiation of Parametric Functions | 参数函数的求导
The derivative dy/dx for a parametric curve is given by the chain rule:
dy/dx = (dy/dt) / (dx/dt)
provided dx/dt ≠ 0. This formula allows us to find the slope of the tangent at any point without eliminating the parameter. It is one of the most useful tools in parametric analysis.
参数曲线的导数 dy/dx 由链式法则给出：
dy/dx = (dy/dt) / (dx/dt)
前提是 dx/dt ≠ 0。该公式使我们无需消去参数即可求出任意点处切线的斜率，是参数分析中最有用的工具之一。
= 0 while dy/dt ≠ 0, which does not happen here except at t = 0 (corresponding to a cusp).
对于 x = 3t², y = 2t³，计算得 dx/dt = 6t，dy/dt = 6t²。因此 dy/dx = (6t²)/(6t) = t，即斜率等于参数值本身。水平切线出现在 dy/dt = 0 (t = 0) 时，斜率为 0。垂直切线出现在 dx/dt = 0 而 dy/dt ≠ 0 时，本例仅在 t = 0 时发生（对应一个尖点）。
√((dx/dt)² + (dy/dt)²), and the direction of motion is given by the vector (dx/dt, dy/dt).
这种求导技术在运动学中至关重要。若 t 表示时间，则 dx/dt 和 dy/dt 分别是速度的水平分量和垂直分量。粒子的速率是 √((dx/dt)² + (dy/dt)²)，运动方向由矢量 (dx/dt, dy/dt) 给出。
5. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative for a parametric curve is found by differentiating dy/dx with respect to x again using the chain rule:
d²y/dx² = d(dy/dx)/dt ÷ dx/dt
This derivative describes the concavity of the curve. A positive d²y/dx² indicates the curve is concave up; a negative value indicates concave down.
参数曲线的二阶导数通过再次利用链式法则对 dy/dx 关于 x 求导得到：
d²y/dx² = d(dy/dx)/dt ÷ dx/dt
该导数描述了曲线的凹凸性。d²y/dx² 为正表示曲线凹向上，为负表示凹向下。
ve up; for t < 0, concave down. This change in concavity at t = 0 corresponds to the cusp.
沿用前面例子 x = 3t², y = 2t³，有 dy/dx = t。则 d(dy/dx)/dt = 1，dx/dt = 6t。因此 d²y/dx² = 1/(6t)，当 t ≠ 0 时成立。对于 t > 0，曲线凹向上；对于 t < 0，凹向下。在 t = 0 处凹凸性的改变对应那个尖点。
Inflection points on a parametric curve can be found where d²y/dx² changes sign, provided the curve is smooth. Always check the domain of the parameter to avoid misinterpretation. Second derivative analysis is also helpful when optimizing functions in parametrics, such as maximizing the height of a projectile.
参数曲线上的拐点可出现在 d²y/dx² 变号的位置，且曲线需光滑。始终检查参数的定义域以避免误判。在参数函数优化中，二阶导数分析也很有用，例如求抛射体的最大高度。
6. Equations of Tangents and Normals | 切线与法线方程
) corresponding to t = t₀, the equation of the tangent line can be written using the point– d at t₀. The normal line is perpendicular to the tangent, so its slope is −1/m (if m ≠ 0).
一旦知道与 t = t₀ 对应的点 (x₀, y₀) 处的 dy/dx，便可用点斜式写出切线方程：y − y₀ = m(x − x₀)，其中 m 为在 t₀ 处的 dy/dx。法线垂直于切线，因此其斜率为 −1/m（若 m ≠ 0）。
ngent equation is y − 0 = 1(x − 2) → y = x − 2. The normal slope is −1, giving y = −x + 2.
例如，求曲线 x = t² + 1, y = t³ − t 在 t = 1 处的切线。此时 (x₀, y₀) = (2, 0)。dx/dt = 2t，dy/dt = 3t² − 1。t = 1 时，dx/dt = 2，dy/dt = 2，故 m = 2/2 = 1。切线方程为 y − 0 = 1(x − 2) → y = x − 2。法线斜率为 −1，方程为 y = −x + 2。
These equations are essential for modelling trajectories and for designing curves with specific properties in engineering. The ability to find tangents directly from parametric forms without eliminating the parameter simplifies work with complex curves like epitrochoids.
这些方程对于轨迹建模以及在工程中设计具有特定性质的曲线至关重要。无需消去参数即可直接从参数形式求切线，简化了类似于外旋轮线等复杂曲线的处理。
7. Area Under a Parametric Curve | 参数曲线下的面积
The area enclosed by a parametric curve and the x‑axis, or between two curves, can be calculated by integrating y with respect to x while expressing dx in terms of dt:
A = ∫ y dx = ∫ y(t) · (dx/dt) dt
The limits of integration are the t‑values corresponding to the x‑limits. Care must be taken with the orientation: if t runs in the reverse direction, dx/dt changes sign, and the integral yields a signed area.
参数曲线与 x 轴围成的面积，或两条曲线间的面积，可通过将 y 对 x 积分，同时将 dx 用 dt 表示来计算：
A = ∫ y dx = ∫ y(t) · (dx/dt) dt
积分限是与 x 限对应的 t 值。需注意方向：若 t 反向变化，dx/dt 会变号，积分结果为有向面积。
²π (1 − cos t)² dt. Evaluating this gives 3π, a remarkable result that the area under a cycloid arch is exactly three times the area of the generating circle.
假设要求摆线 x = t − sin t, y = 1 − cos t 在一个拱形 (0 ≤ t ≤ 2π) 下的面积。则 dx/dt = 1 − cos t。面积为 A = ∫₀²π (1 − cos t)(1 − cos t) dt = ∫₀²π (1 − cos t)² dt。积分结果为 3π，这个非凡的结论表明摆线一拱下的面积恰好是生成圆面积的三倍。
feasible method. This technique also applies to finding the area between a curve and the y‑ x dy = x(t) · (dy/dt) dt. Always sketch the region to confirm the limits and orientation.
当曲线不是关于 x 的简单函数时，参数积分往往是唯一可行的方法。此技术也可用于求曲线与 y 轴之间的面积，使用 x dy = x(t) · (dy/dt) dt。务必绘制区域草图以确认积分限和方向。
8. Arc Length of a Parametric Curve | 参数曲线的弧长
The length s of a parametric curve from t = a to t = b is given by the integral:
s = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
This formula arises from summing the lengths of infinitesimal hypotenuse segments. It works for any smooth curve, even those that loop or intersect themselves. No elimination of the parameter is needed.
参数曲线从 t = a 到 t = b 的长度 s 由下面积分给出：
s = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
该公式源于对微小的斜边线段长度求和。它适用于任何光滑曲线，即使曲线有环路或自相交。无需消去参数。
nd summing gives (dx/dt)² + (dy/dt)² = e²ᵗ( (cos t−sin t)² + (sin t+cos t)² ) = 2e²ᵗ. Thus √… = eᵗ√2, and the arc length is ∫₀π eᵗ√2 dt = √2(e^π − 1).
举例说明，求曲线 x = eᵗ cos t, y = eᵗ sin t 在 0 ≤ t ≤ π 上的长度。则 dx/dt = eᵗ(cos t − sin t), dy/dt = eᵗ(sin t + cos t)。平方求和得 (dx/dt)² + (dy/dt)² = e²ᵗ( (cos t−sin t)² + (sin t+cos t)² ) = 2e²ᵗ。因此 √… = eᵗ√2，弧长为 ∫₀π eᵗ√2 dt = √2(e^π − 1)。
Parametric arc length is widely used in physics for computing the distance travelled by a particle along a curved path. The integrand represents the speed if t is time. Thus, the arc length formula directly links geometry with kinematics.
参数弧长在物理学中广泛用于计算粒子沿曲线路径行进的距离。若 t 为时间，被积函数即表示速率。因此弧长公式将几何与运动学直接联系起来。
9. Parametric Equations in Kinematics | 运动学中的参数方程
x/dt², d²y/dt²). Parametric equations thus provide a natural language for describing motion. The path of the particle is the parametric curve itself.
在力学中，运动粒子的位置表示为 r(t) = (x(t), y(t))。速度是 v(t) = (dx/dt, dy/dt)，加速度是 a(t) = (d²x/dt², d²y/dt²)。因此参数方程为描述运动提供了自然的语言。粒子的轨迹就是参数曲线本身。
the speed is √[(v₀ cos α)² + (v₀ sin α − g t)²]. The maximum height occurs when dy/dt = 0.
当参数为时间 t 时，我们可以独立分析水平与垂直运动。对于抛体运动，常用 x = (v₀ cos α) t, y = (v₀ sin α) t − ½ g t²。求导可得速度分量，速率为 √[(v₀ cos α)² + (v₀ sin α − g t)²]。最大高度发生在 dy/dt = 0 时。
ound by eliminating t, reveals the parabolic shape: y = x tan α − (g x²) / (2 v₀² cos² α).
这种分量分离简化了复杂问题，例如击中斜坡上的目标或计算射程。通过消去 t 得到的轨迹笛卡尔方程揭示了其抛物线形状：y = x tan α − (g x²) / (2 v₀² cos² α)。
10. Investigation: Path of a Projectile | 探究：抛体运动路径
s at an angle of 30° to the horizontal, taking g = 9.8 m/s². The parametric equations are:
x = 20 cos 30° t, y = 20 sin 30° t − 4.9 t²
Using cos 30° = √3/2 ≈ 0.8660, sin 30° = 0.5, we have x = 17.32 t, y = 10 t − 4.9 t².
让我们探究以初速 20 m/s、与水平成 30° 角从原点发射的抛体路径，取 g = 9.8 m/s²。参数方程为：
x = 20 cos 30° t, y = 20 sin 30° t − 4.9 t²
代入 cos 30° = √3/2 ≈ 0.8660，sin 30° = 0.5，得 x = 17.32 t，y = 10 t − 4.9 t²。
2.041) ≈ 17.32 × 2.041 ≈ 35.3 m. The maximum height occurs at t = 1.02 s giving y ≈ 5.1 m.
为绘制路径，我们可取不同 t 值进行描点。粒子着地时 y = 0（排除 t = 0）：10 t − 4.9 t² = 0 → t = 10/4.9 ≈ 2.041 s。射程为 x(2.041) ≈ 17.32 × 2.041 ≈ 35.3 m。最大高度在 t = 1.02 s 时，y ≈ 5.1 m。
0163 x². This parabola accurately models the trajectory under constant gravity neglecting air resistance. Parametric form preserves the time dependence, which is crucial for timing events.
消去 t：t = x/17.32 → y = 10(x/17.32) − 4.9 (x/17.32)² → y = 0.577x − 0.0163 x²。该抛物线精确地模拟了恒重力下的轨迹，忽略空气阻力。参数形式保留了时间依赖性，这对于事件计时至关重要。
11. Solving Real-World Problems | 解决实际问题
Parametric equations model a vast array of real-world scenarios. Beyond projectiles, they describe the orbits of planets (with t as an angular parameter), the shape of a suspension bridge cable (catenary in parametric form), and the motion of a point on a rolling wheel (cycloid). In robotics, parametric curves define the path of robotic arms to achieve smooth motion.
参数方程可模拟大量实际场景。除了抛射运动，它们还描述行星轨道（t 为角参数）、悬索桥缆绳形状（参数形式的悬链线），以及滚动车轮上一点的运动（摆线）。在机器人学中，参数曲线定义了机械臂的路径以实现平滑运动。
One common application is in computer graphics: Bézier curves are parametric cubic polynomials used to design smooth shapes and fonts. The handle of a Bézier curve is essentially a velocity tangent derived from the parametric form. Understanding parametrics deepens your appreciation of these digital tools.
一个常见应用是在计算机图形学中：贝塞尔曲线是参数三次多项式，用于设计平滑的形状和字体。贝塞尔曲线的控制柄本质上是源于参数形式的速度切线。理解参数方程能加深你对这些数字工具的认知。
When analysing a problem, decide whether parametric representation simplifies the work. If the relationship between x and y is not a single function, or if motion is involved, parametrics are often the best choice. They also handle closed curves, like the ellipse or Lissajous figures, with ease.
分析问题时，需判断参数表示是否能简化工作。若 x 与 y 的关系不是单值函数，或涉及运动，参数方程通常是最佳选择。它们还能
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