Modelling with Parametric Equations | 参数方程建模

📚 Modelling with Parametric Equations | 参数方程建模

Parametric equations are a powerful way to describe curves by writing x and y as separate functions of a third variable, usually t or θ. In Edexcel A Level Mathematics, parametric modelling appears in curve sketching, differentiation, integration, and applied problems such as projectile motion.

参数方程是一种强大的曲线描述方式，它将 x 和 y 分别写成第三个变量（通常是 t 或 θ）的函数。在 Edexcel A Level 数学中，参数建模出现在曲线作图、微分、积分以及抛体运动等应用问题中。

1. Parametric equations and why they matter | 参数方程及其重要性

A parametric model expresses both coordinates as functions of a single parameter. The standard form is:

参数模型用一个参数表示两个坐标。标准形式为：

x = f(t), y = g(t)

The parameter commonly represents time, but for geometric curves it may be an angle or a dimensionless variable. Parametric form also gives a natural direction of travel as the parameter increases.

参数通常表示时间，但对于几何曲线，它也可以是角度或无量纲变量。参数形式还给出了参数增大时的自然运动方向。

2. Converting between parametric and Cartesian forms | 参数式与笛卡尔式互化

To convert a parametric equation to Cartesian form, eliminate the parameter. For x = 2 cos θ and y = 3 sin θ, use the identity cos² θ + sin² θ = 1 to obtain:

将参数方程化为笛卡尔式的方法是消去参数。对于 x = 2 cos θ 和 y = 3 sin θ，利用恒等式 cos² θ + sin² θ = 1 可得到：

(x/2)² + (y/3)² = 1

If x = t² and y = 2t, then t = y/2 and x = (y/2)², so x = y²/4. Always check whether squaring or taking roots changes the domain or range.

若 x = t²，y = 2t，则 t = y/2，x = (y/2)²，所以 x = y²/4。一定要检查平方或开方是否改变了定义域或值域。

3. Sketching parametric curves | 参数曲线草图

Build a table of t, x, and y for key values, then plot the points. If possible, identify the general shape from the Cartesian equation first.

取关键的 t 值制作 x、y 数值表并描点。如果可能，先从笛卡尔方程判断大致形状。

Mark arrows to show the direction of increasing t. For x = t² and y = 2t, the curve enters from the right, moves left to the vertex at (0,0), then moves right again.

用箭头标出 t 增大的方向。对于 x = t²，y = 2t，曲线从右侧进入，向左移动到顶点 (0,0)，然后再向右移动。

4. First derivative: gradient of a parametric curve | 一阶导数：参数曲线的斜率

The chain rule gives the gradient as the quotient of two derivatives with respect to the parameter:

链式法则给出斜率，即两个关于参数的导数之商：

dy/dx = (dy/dt) ÷ (dx/dt)

This formula is valid only when dx/dt ≠ 0. For x = t² and y = 2t, we have dx/dt = 2t and dy/dt = 2, so dy/dx = 1/t.

该公式仅在 dx/dt ≠ 0 时成立。对于 x = t²，y = 2t，有 dx/dt = 2t，dy/dt = 2，因此 dy/dx = 1/t。

This derivative is the basis for finding tangents, normals, and stationary points on curves that are not easily expressed as y = F(x

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