Parametric Equations and Applications | 参数方程及其应用

📚 Parametric Equations and Applications | 参数方程及其应用

In coordinate geometry, we usually describe a curve by a Cartesian equation involving x and y. However, many curves are best described by expressing x and y in terms of a third variable, called a parameter. This approach, known as parametric equations, is powerful for modelling motion and for representing complicated curves.

在坐标几何中，我们通常用含 x 和 y 的直角坐标方程来描述曲线。然而，许多曲线更适合通过一个称为参数的第三变量来表达 x 和 y。这种表示方法称为参数方程，它在描述运动、表示复杂曲线等方面功能强大。

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1. What Are Parametric Equations? | 什么是参数方程？

A parametric equation defines the coordinates (x, y) as functions of a parameter, often written as x = f(t), y = g(t). As the parameter t varies, the point (x, y) traces out a curve. The domain of t must be stated so that the entire curve or the required portion of it is described correctly.

参数方程将坐标 (x, y) 表示为某个参数的函数，通常写作 x = f(t), y = g(t)。随着参数 t 变化，点 (x, y) 描出一条曲线。需要指明 t 的取值范围，才能正确描述整条曲线或其中某一段。

For example, the circle of radius r centred at the origin can be written as x = r cos θ, y = r sin θ, where θ is the parameter. When θ ranges from 0 to 2π, the point moves once around the circle in the anticlockwise direction.

例如，以原点为圆心、半径为 r 的圆可写成 x = r cos θ, y = r sin θ，其中 θ 为参数。当 θ 从 0 变到 2π 时，点沿逆时针方向绕圆一周。

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2. Converting Parametric to Cartesian Form | 参数方程与直角坐标方程的互化

To eliminate the parameter, solve one equation for t and substitute into the other, or use a trigonometric identity. For the circle x = r cos θ, y = r sin θ, squaring and adding gives x² + y² = r². For the parabola x = at², y = 2at, eliminating t yields y² = 4ax.

消去参数时，可以从一个方程解出 t 再代入另一个，或利用三角恒等式。对于圆 x = r cos θ, y = r sin θ，平方相加得 x² + y² = r²。对于抛物线 x = at², y = 2at，消去 t 得到 y² = 4ax。

Another example is x = t² + 1, y = t − 2. From the second equation t = y + 2, so x = (y + 2)² + 1, which is a Cartesian equation of a parabola.

另一个例子：x = t² + 1, y = t − 2。由第二个方程得 t = y + 2，所以 x = (y + 2)² + 1，这是抛物线的直角坐标方程。

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3. Common Parametric Curves | 常见曲线的参数方程

The following parametric forms appear frequently in A-level mathematics. It is useful to recognize them and remember their Cartesian equivalents.

以下参数形式在 A-level 数学中经常出现。认识它们并记住对应的直角坐标方程非常有用。

• Circle: x = r cos θ, y = r sin θ → x² + y² = r². | 圆：x = r cos θ, y = r sin θ → x² + y² = r²。
• Ellipse: x = a cos θ, y = b sin θ → x²/a² + y²/b² = 1. | 椭圆：x = a cos θ, y = b sin θ → x²/a² + y²/b² = 1。
• Parabola: x = at², y = 2at → y² = 4ax. | 抛物线：x = at², y = 2at → y² = 4ax。
• Hyperbola: x = a sec θ, y = b tan θ → x²/a² − y²/b² = 1. | 双曲线：x = a sec θ, y = b tan θ → x²/a² − y²/b² = 1。
• Cycloid: x = a(θ − sin θ), y = a(1 − cos θ). | 摆线：x = a(θ − sin θ), y = a(1 − cos θ)。

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4. Differentiation of Parametric Equations | 参数方程求导

If x = f(t) and y = g(t), then the chain rule gives the gradient function as

若 x = f(t) 且 y = g(t)，由链式法则可得导数公式为

dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0.

dy/dx = (dy/dt) ÷ (dx/dt)，其中 dx/dt ≠ 0。

For example, if x = 3t² and y = 2t³, then dx/dt = 6t, dy/dt = 6t². Thus dy/dx = (6t²)/(6t) = t. This result can also be checked by converting to y as a function of x directly.

例如，若 x = 3t², y = 2t³，则 dx/dt = 6t, dy/dt = 6t²。因此 dy/dx = (6t²)/(6t) = t。这个结果也可以通过直接转化为 y 关于 x 的函数来检验。

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5. Tangents and Normals | 切线与法线

The gradient of the tangent at a point with parameter t is given by dy/dx at that point. The tangent line equation is y − y₁ = m(x − x₁), where m = dy/dx and (x₁, y₁) is the point.

参数 t 对应点处的切线斜率为该点的 dy/dx。切线方程为 y − y₁ = m(x − x₁)，其中 m = dy/dx，(x₁, y₁) 为切点。

If m ≠ 0, the normal has gradient −1/m, so its equation is y − y₁ = −(1/m)(x − x₁). If m = 0, the normal is vertical.

若 m ≠ 0，法线斜率为 −1/m，因此法线方程为 y − y₁ = −(1/m)(x − x₁)。若 m = 0，则法线为竖直直线。

Example: For x = 2t, y = t² at t = 1, we have x = 2, y = 1. dx/dt = 2, dy/dt = 2t, so at t = 1, dy/dx = (2×1)/2 = 1. The tangent is y − 1 = 1(x − 2), i.e. y = x − 1.

例如：对于 x = 2t, y = t²，在 t = 1 处，x = 2, y = 1。dx/dt = 2, dy/dt = 2t，所以 t = 1 时，dy/dx = (2×1)/2 = 1。切线为 y − 1 = 1(x − 2)，即 y = x − 1。

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6. Second Derivatives | 二阶导数

The second derivative with respect to x is found by first differentiating dy/dx with respect to t, then dividing by dx/dt:

关于 x 的二阶导数，需先对 t 求 dy/dx 的导数，再除以 dx/dt：

d²y/dx² = [d/dt(dy/dx)] ÷ (dx/dt).

d²y/dx² = [d/dt(dy/dx)] ÷ (dx/dt)。

This formula is often used to determine concavity: if d²y/dx² > 0, the curve is concave up; if d²y/dx² < 0, it is concave down. Remember that d²y/dx² is not equal to (d²y/dt²)/(d²x/dt²).

这个公式常用于判断凹凸性：若 d²y/dx² > 0，曲线凹向上；若 d²y/dx² < 0，曲线凹向下。注意 d²y/dx² 并不等于 (d²y/dt²)/(d²x/dt²)。

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7. Integration and Area under a Curve | 积分与曲线下面积

The area under a parametric curve is given by

参数曲线下的面积为

A = ∫ y dx = ∫ y (dx/dt) dt,

A = ∫ y dx = ∫ y (dx/dt) dt，

where the limits of integration are values of t corresponding to the required x-values. Since dx/dt may be negative, take the absolute value of the integral if necessary.

其中积分限是与所需 x 值对应的 t 值。由于 dx/dt 可能为负，必要时对积分结果取绝对值。

For example, using the cycloid x = a(θ − sin θ), y = a(1 − cos θ), the area under one arch can be computed by integrating a(1 − cos θ)·a(1 − cos θ) dθ from 0 to 2π, which gives 3πa².

例如，对于摆线 x = a(θ − sin θ), y = a(1 − cos θ)，一个拱形下的面积可由 y(dx/dθ) 对 θ 从 0 到 2π 积分得到，结果为 3πa²。

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8. Applications in Kinematics | 运动学中的应用

In kinematics, the position of a particle is often given parametrically as x(t), y(t). Then the velocity components are dx/dt and dy/dt, and the speed is

在运动学中，质点的位置通常用参数式 x(t), y(t) 表示。速度分量为 dx/dt 和 dy/dt，速率为

speed = √((dx/dt)² + (dy/dt)²).

速率 = √((dx/dt)² + (dy/dt)²)。

The acceleration components are d²x/dt² and d²y/dt², and the magnitude of acceleration is √((d²x/dt²)² + (d²y/dt²)²). These ideas are central to motion in a plane.

加速度分量为 d²x/dt² 和 d²y/dt²，加速度大小为 √((d²x/dt²)² + (d²y/dt²)²)。这些概念是平面运动的核心内容。

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9. Applications in Projectile Motion | 抛体运动中的应用

Projectile motion is a classic example of parametric modelling. If a particle is launched with speed u at an angle θ above the horizontal, then ignoring air resistance, its position at time t is

抛体运动是参数建模的经典例子。若质点以速率 u、与水平方向夹角 θ 抛出，忽略空气阻力时，其 t 时刻的位置为

x = u cos θ · t, y = u sin θ · t − ½ g t²,

x = u cos θ · t，y = u sin θ · t − ½ g t²，

where g is the acceleration due to gravity. Eliminating t gives the trajectory, which is a parabola: y = x tan θ − (g x²)/(2u² cos²θ).

其中 g 是重力加速度。消去 t 可得轨迹方程，它是一个抛物线：y = x tan θ − (g x²)/(2u² cos²θ)。

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10. Advantages and Tips | 参数方程的优势与解题技巧

Parametric equations often simplify the calculation of gradients and areas, especially for curves like cycloids. When solving problems, always note the range of t, check that dx/dt ≠ 0 before differentiating, and use identities such as cos²θ + sin²θ = 1 to eliminate parameters.

参数方程常常简化斜率和面积的计算，尤其对于摆线等曲线。解题时，要注意 t 的取值范围，在求导前检查 dx/dt ≠ 0，并利用 cos²θ + sin²θ = 1 等恒等式消去参数。

Common mistakes include forgetting to divide by dx/dt when finding dy/dx, misusing the second derivative formula, and ignoring negative area contributions. Practising the conversion between parametric and Cartesian forms will help you recognise hidden curves and verify answers.

常见错误包括：求 dy/dx 时忘记除以 dx/dt；误用二阶导数公式；忽略面积中的负贡献。多做参数方程与直角坐标方程的互化练习，有助于识别隐藏的曲线并验证答案。

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