📚 Parametric Equations: Concepts and Basic Properties | 参数方程的概念与基本性质
A parametric equation describes a curve by expressing the coordinates x and y as separate functions of a common variable, usually denoted by t. This third variable is called the parameter, and as t varies over a defined interval, the point (x, y) traces out the curve.
参数方程通过将坐标 x 与 y 分别表示为共同变量（通常记为 t）的函数来描述一条曲线。这个第三变量称为参数；当 t 在某个给定区间内变化时，点 (x, y) 就在平面上描绘出相应的曲线。
1. The General Form of Parametric Equations | 参数方程的一般形式
For a curve given parametrically, we write two equations involving the parameter t:
对于参数曲线，我们用两个含参数 t 的方程来表示：
x = f(t), y = g(t)
Each value of t produces exactly one ordered pair (x, y), which corresponds to one point on the curve. As t changes continuously, the point moves along the curve.
每一个 t 值恰好产生一个有序数对 (x, y)，对应曲线上的一个点。当 t 连续变化时，该点就沿着曲线运动。
0, the point is (0, 0); when t = 1, the point is (1, 2); when t = 2, the point is (4, 4).
例如，方程 x = t² 与 y = 2t 定义了一条抛物线。当 t = 0 时，点为 (0, 0)；当 t = 1 时，点为 (1, 2)；当 t = 2 时，点为 (4, 4)。
2. Why Use Parameters? | 为什么要引入参数
Parametric equations are especially useful for describing motion. If t represents time, then the position of a particle is naturally split into its horizontal component x(t) and vertical component y(t). This makes the model simple and direct.
参数方程特别适合描述运动。若 t 表示时间，则质点的位置自然分解为水平分量 x(t) 与竖直分量 y(t)，使得模型既简单又直观。
parametric form, the cycloid has the elegant equations x = r(t − sin t), y = r(1 − cos t).
某些曲线很难写成单一的直角坐标方程 y = f(x)。例如滚轮边缘上一点所描绘的摆线，就是著名例子。摆线的参数方程非常简洁：x = r(t − sin t)，y = r(1 − cos t)。
A parametric description also gives the curve an orientation: as t increases, the curve is traced in a particular direction, indicated by arrows when sketching.
参数描述还赋予曲线方向性：随着 t 增大，曲线按特定方向被描绘出来，绘图时需要用箭头标明这一方向。
3. Sketching Parametric Curves | 参数曲线的绘制
To sketch a parametric curve, choose a selection of t-values, compute the corresponding x and y values, and plot the resulting points in order. Joining the points in order of increasing t gives the shape of the curve.
绘制参数曲线时，先选取一系列 t 值，计算对应的 x 与 y 值，并按顺序描点。按照 t 增大的顺序连接这些点，即可得到曲线的形状。
tricting t to 0 ≤ t ≤ 1 produces only a segment of the full curve, not the whole parabola.
必须用箭头标出运动方向。此外，t 的定义域决定了曲线上实际出现的部分。例如，将 t 限制在 0 ≤ t ≤ 1 只能得到整条曲线的一段，而不是整条抛物线。
Always calculate at least five points, including the endpoints of the t-interval, before joining them smoothly.
连接各点之前，至少应计算五个点的坐标，并且要包括 t 区间端点的对应点。
Check whether x and y are increasing or decreasing between the chosen values, as this helps explain the shape and direction.
检查所选值之间 x 与 y 是递增还是递减，这有助于理解曲线的形状与方向。
4. Eliminating the Parameter | 消去参数
To convert a parametric equation into a Cartesian equation, eliminate t between the two equations. The simplest method is to solve one equation for t and substitute the result into the other equation.
要将参数方程化为直角坐标方程，需要在两个方程之间消去 t。最常用的方法是先从一个方程中解出 t，再将其代入另一个方程。
second equation gives y = ((x − 1)/2)² − 3, which simplifies to a quadratic function in x.
例如，设 x = 2t + 1，y = t² − 3。由第一个方程得 t = (x − 1)/2。代入第二个方程得 y = ((x − 1)/2)² − 3，化简后得到关于 x 的二次函数。
t = (x − 1)/2 ⇒ y = (x − 1)²/4 − 3
Always state any restriction on x or y after elimination. A parametric curve may represent only part of the Cartesian curve, depending on the interval of t.
消元后一定要说明 x 或 y 的限制条件。取决于 t 的区间，参数曲线可能只代表直角坐标曲线的一部分。
5. Using Trigonometric Identities to Eliminate t | 利用三角恒等式消去参数
msy. Instead, use the Pythagorean identity sin²t + cos²t = 1 to combine the two equations.
当参数方程包含正弦与余弦时，直接代入法往往较繁琐。这时应利用毕达哥拉斯恒等式 sin²t + cos²t = 1 来合并两个方程。
ng gives x² + y² = r² cos²t + r² sin²t = r². Hence the Cartesian equation is x² + y² = r².
对于半径为 r 的圆，令 x = r cos t，y = r sin t。两式平方后相加得 x² + y² = r²(cos²t + sin²t) = r²。因此直角坐标方程为 x² + y² = r²。
x = a cos t and y = b sin t. Dividing by a and b before squaring gives x²/a² + y²/b² = 1.
对于椭圆，使用 x = a cos t，y = b sin t。先将两式分别除以 a 与 b 再平方，可得 x²/a² + y²/b² = 1。
Circle: x² + y² = r²; Ellipse: x²/a² + y²/b² = 1
6. Domain, Range and the Interval of t | 定义域、值域与 t 的区间
The interval of the parameter t is part of the definition of a parametric curve. Changing the interval changes the curve, even if the same two equations are used.
参数 t 的区间是参数曲线定义的一部分。即使两个方程完全相同，改变 t 的区间也会改变对应的曲线。
only the upper semicircle appears. If 0 ≤ t ≤ π/2, only one quarter of the circle remains.
考虑单位圆 x = cos t，y = sin t。若 0 ≤ t < 2π，则整圆被描绘一次；若 0 ≤ t ≤ π，则只出现上半圆；若 0 ≤ t ≤ π/2，则只剩下四分之一圆。
After eliminating the parameter, the Cartesian equation may suggest a larger set of points than the original curve. The t-interval automatically restricts which x-values and y-values are allowed.
消去参数后，直角坐标方程可能暗示一个比原曲线更大的点集。t 的区间会自动限制 x 和 y 的取值范围，这一点在解题时不可忽略。
7. Differentiation of Parametric Functions | 参数方程的求导
nt dy/dx cannot be found directly because y is not expressed as a function of x. Instead, use the chain rule connecting the two derivatives.
若 x = f(t)，y = g(t)，由于 y 并未直接表示为 x 的函数，dy/dx 不能直接求出。此时需要利用链式法则将两个导数联系起来。
dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0
= 0 at a particular t-value, the tangent is vertical and dy/dx is undefined at that point.
该公式仅在 dx/dt 不为零时成立。若某点处 dx/dt = 0，则切线为竖直方向，该点处 dy/dx 无定义。
The second derivative is obtained by differentiating dy/dx with respect to t and then dividing by dx/dt again:
求二阶导数时，应先将 dy/dx 对 t 求导，再除以 dx/dt：
d²y/dx² = [d(dy/dx)/dt] ÷ (dx/dt)
A common mistake is to differentiate dy/dx directly with respect to x. This is incorrect, because dy/dx is still written in terms of t.
常见错误是直接把 dy/dx 对 x 求导。这样做是不对的，因为 dy/dx 仍然是以 t 为变量的表达式。
8. Tangents and Normals to Parametric Curves | 参数曲线的切线与法线
a parameter value t₀, the equation of the tangent at the point (x₀, y₀) uses the point-slope form:
一旦求得在参数值 t₀ 处的斜率 m = dy/dx，就可以利用点斜式写出点 (x₀, y₀) 处的切线方程：
y − y₀ = m(x − x₀)
1/m, so the normal has equation y − y₀ = −(1/m)(x − x₀). If m = 0, the normal is vertical.
法线与切线互相垂直。当 m ≠ 0 时，法线的斜率为 −1/m，因此法线方程为 y − y₀ = −(1/m)(x − x₀)。若 m = 0，则法线为竖直方向。
Remember that the point (x₀, y₀) is obtained by substituting t₀ into both x(t) and y(t). Never use the formula for the tangent at a point that has not been verified to lie on the curve.
注意，点 (x₀, y₀) 需要将 t₀ 分别代入 x(t) 与 y(t) 求得。切勿对未经检验是否位于曲线上的点使用切线公式。
9. Integration, Area and Arc Length | 积分、面积与弧长
ameter t. Since the substitution x = x(t) gives dx = (dx/dt) dt, the area formula becomes:
参数曲线下方的面积通过对 x 积分求得，并改写为关于参数 t 的积分。由代换 x = x(t) 可得 dx = (dx/dt) dt，因此面积公式为：
A = ∫ y (dx/dt) dt
The limits of integration must be the t-values corresponding to the first and last points of the curve, not x-values. Choosing the wrong direction reverses the sign of the area.
积分的上下限必须是对应曲线起点与终点的 t 值，而不是 x 值。选错方向会使面积取相反符号。
Similarly, the arc length of a parametric curve from t = a to t = b is given by:
类似地，参数曲线从 t = a 到 t = b 的弧长为：
L = ∫ √[(dx/dt)² + (dy/dt)²] dt
This formula arises from applying Pythagoras' theorem to infinitesimal horizontal and vertical changes, then integrating along the curve.
该公式来源于对无穷小平方向与竖直方向变化应用毕达哥拉斯定理，再沿曲线积分。
10. Common Parametric Curves | 常见参数曲线
The table below summarises standard parametric forms that appear frequently in A-Level examinations.
下表总结了 A-Level 考试中经常出现的标准参数形式。
| Curve | 曲线 | Parametric Equations | 参数方程 | Cartesian Form / Notes | 直角坐标形式 / 备注 |
|---|---|---|
| Circle | 圆 | x = r cos t, y = r sin t, 0 ≤ t < 2π | x² + y² = r² |
| Ellipse | 椭圆 | x = a cos t, y = b sin t | x²/a² + y²/b² = 1 |
| Parabola | 抛物线 | x = at², y = 2at | y² = 4ax |
| Cycloid | 摆线 | x = r(t − sin t), y = r(1 − cos t) | Not easily written in Cartesian form | 不易写成直角坐标形式 |
| Line segment | 线段 | x = x₀ + at, y = y₀ + bt | Gradient b/a, vector direction (a, b) | 斜率 b/a，方向向量 (a, b) |
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
The following points will help you avoid the most frequent pitfalls in parametric equation questions.
以下要点可以帮助你在参数方程题目中避免最常见的失分陷阱。
Always state the interval of t when writing a parametric equation. Without it, the curve is ambiguous.
写出参数方程时一定要注明 t 的区间。没有区间，曲线就不明确。
Never divide by dx/dt without checking that it is not zero. A vertical tangent requires separate treatment.
不可贸然除以 dx/dt 而不检查它是否为零。竖直切线需要单独处理。
Do not confuse dy/dt with dy/dx. The former is the rate of change of y with respect to t, the latter with respect to x.
不要把 dy/dt 与 dy/dx 混淆。前者是 y 对 t 的变化率，后者是 y 对 x 的变化率。
When finding the second derivative, remember to divide by dx/dt after differentiating with respect to t.
求二阶导数时，切记在对 t 求导之后还要再除以 dx/dt。
For area and arc length problems, convert the integration limits to t-values before substituting into the formula.
在面积与弧长问题中，应先求出积分上下限对应的 t 值，再代入公式计算。
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