Parametric Equations: Concepts and Basic Properties | 参数方程的概念与基本性质

📚 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。

For example, the equations x = t² and y = 2t define a parabola. When t = 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。

Some curves are very difficult to express as a single Cartesian equation y = f(x). The cycloid, traced by a point on the rim of a rolling wheel, is one famous example. In 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。

It is essential to show the direction of motion by adding arrows. Moreover, the domain of t determines which part of the curve actually appears. For instance, restricting 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。

For example, suppose x = 2t + 1 and y = t² − 3. From the first equation, t = (x − 1)/2. Substituting into the 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

When the parametric equations involve sine and cosine, the direct substitution method is often clumsy. 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。

For a circle with radius r, write x = r cos t and y = r sin t. Squaring and adding 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。

For an ellipse, use 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。

Consider the unit circle x = cos t, y = sin t. If 0 ≤ t < 2π, the whole circle is traced once. If 0 ≤ t ≤ π, 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

If x = f(t) and y = g(t), then the gradient 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

This formula is valid only when dx/dt is not zero. If dx/dt = 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

Once the gradient m = dy/dx is known at 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)

The normal line is perpendicular to the tangent. When m ≠ 0, the gradient of the normal is −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

The area under a parametric curve is found by integration with respect to x, rewritten using the parameter 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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Parametric Equations: Concepts and Basic Properties | 参数方程的概念与基本性质