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Parametric Equations | 参数方程

2026-08-02 18:32 · 本文部分内容由AI辅助创作

📚 Parametric Equations | 参数方程

nly denoted as t (the parameter). This technique is especially useful for modelling paths, circles, and motions that cannot be expressed as a single function y = f(x). In the Edexcel A-Level Mathematics syllabus, a solid grasp of parametric equations is required for curve sketching, differentiation, integration, and solving geometric problems involving tangents and areas.

参数方程通过将 x 和 y 坐标分别表示为第三个变量（通常记为参数 t）的函数，提供了一种灵活描述曲线的方法。对于无法写成单一函数 y = f(x) 的路径、圆和运动，这种技巧尤其有用。在 Edexcel A-Level 数学课程中，牢固掌握参数方程对于绘制曲线、求导、积分以及解决涉及切线和面积的几何问题是必需的。

1. Introduction to Parametric Equations | 参数方程简介

s the parameter. As t varies over a given domain, the point (x, y) traces a curve in the plane. Unlike Cartesian equations, parametric forms can easily represent multi-valued or looping curves.

参数方程定义了两个函数：x = f(t) 和 y = g(t)，其中 t 为参数。当 t 在给定范围内变化时，点 (x, y) 在平面上描绘出一条曲线。与直角坐标方程不同，参数形式可以轻松表示多值或循环曲线。

unit circle anticlockwise. The choice of parameter often reflects time in motion problems.

例如，x = t², y = 2t 描绘出一条侧向的抛物线，而 x = cos t, y = sin t 在 0 ≤ t < 2π 时按逆时针方向描绘出单位圆。在运动问题中，参数的选择通常反映时间。

2. Sketching Parametric Curves | 绘制参数曲线

To sketch a parametric curve, create a table of values for t, x, and y. Select sensible t‑values, calculate the corresponding coordinates, and plot them. Then join the points smoothly, respecting the direction as t increases.

要绘制参数曲线，可制作一个包含 t、x 和 y 的数值表格。选择合适的 t 值，计算对应的坐标并描点，然后按 t 增加的方向光滑连接各点。

For the curve x = t² – 1, y = t + 2 with –2 ≤ t ≤ 2, you would tabulate t = –2, –1, 0, 1, 2 to obtain points (–3,0), (0,1), (–1,2), (0,3), (3,4). The resulting shape is a right‑opening parabola. Arrows on the sketch must indicate the increasing‑t direction.

对于曲线 x = t² – 1, y = t + 2，其中 –2 ≤ t ≤ 2，可列出 t = –2, –1, 0, 1, 2，得到点 (–3,0), (0,1), (–1,2), (0,3), (3,4)。所得的形状是一条向右开口的抛物线。必须在草图上用箭头标明 t 增加的方向。

3. Eliminating the Parameter | 消去参数

Eliminating the parameter means finding a Cartesian equation (relating x and y directly) from the parametric form. This often reveals the type of curve and its restrictions. The method involves solving for t from one equation and substituting into the other, or using trigonometric identities.

消去参数意味着从参数形式中找出直接联系 x 和 y 的直角坐标方程。这通常会揭示曲线的类型及其限制。方法包括从一个方程解出 t 再代入另一个，或利用三角恒等式。

Example: Given x = 2t + 1, y = t² – 4, isolate t = (x – 1)/2, then substitute into y: y = [(x – 1)/2]² – 4 = (x – 1)²/4 – esian equation is a parabola. However, the domain of x may be restricted by the original t‑range.

示例：给定 x = 2t + 1, y = t² – 4，解出 t = (x – 1)/2，代入 y：y = [(x – 1)/2]² – 4 = (x – 1)²/4 – 4。这个直角坐标方程是一条抛物线。但 x 的定义域可能受到原始 t 范围的限制。

1, an ellipse. Always note the range of θ to identify which part of the ellipse is traced.

对于 x = a cos θ, y = b sin θ 这类三角参数方程，利用 cos²θ + sin²θ = 1 得到 (x/a)² + (y/b)² = 1，即椭圆。务必留意 θ 的范围以确定描绘了椭圆的哪一部分。

4. Cartesian Equation from Parametric Equations | 从参数方程到直角坐标方程

main for x (or y). For instance, if t ≥ 0 in x = t², y = t³, then x ≥ 0, and the Cartesian equation y² = x³ (for x ≥ 0) describes the right branch of the semi‑cubical parabola.

消参后，直角坐标方程可能只代表整条曲线的一部分。你必须将 t 的定义域转化为 x（或 y）的对应定义域。例如，若在 x = t², y = t³ 中 t ≥ 0，则 x ≥ 0，此时直角坐标方程 y² = x³（x ≥ 0）描述半立方抛物线的右支。

Checking the domain avoids drawing extra branches that do not belong to the original parametric curve. Always test a couple of t‑values to confirm that the Cartesian equation’s full graph does not exceed the parametric path.

检查定义域可以避免画出不属于原始参数曲线的多余分支。始终测试几个 t 值，以确认直角坐标方程的完整图形没有超出参数路径。

5. Differentiation: dy/dx in Parametric Form | 求导：参数形式的 dy/dx

y/dt) / (dx/dt), provided dx/dt ≠ 0. This formula gives the derivative as a function of t.

要在不消去参数的情况下求参数曲线的梯度，可利用链式法则：dy/dx = (dy/dt) ÷ (dx/dt) = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。该公式给出一个关于 t 的导数函数。

ve dy/dt = 2, dx/dt = 2t, so dy/dx = 2 / (2t) = 1/t. At t = 2, the gradient is 1/2. At t = –1, the gradient is –1, revealing how the slope changes along the curve.

对于 x = t², y = 2t，有 dy/dt = 2，dx/dt = 2t，因此 dy/dx = 2 / (2t) = 1/t。在 t = 2 处，梯度为 1/2。在 t = –1 处，梯度为 –1，这揭示了斜率沿曲线如何变化。

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6. Tangents and Normals | 切线与法线

Once dy/dx is known at a specific t‑value, the tangent line at the point (x(t), y(t)) can be written using point–slope form: y – y₁ = m (x – x₁), where m = dy/dx evaluated at that t. The normal line has gradient –1/m (if m ≠ 0) and a similar equation.

一旦知道了特定 t 值处的 dy/dx，就可以用点斜式写出点 (x(t), y(t)) 处的切线方程：y – y₁ = m (x – x₁)，其中 m 是在该 t 值处求得的 dy/dx。法线具有梯度 –1/m（若 m ≠ 0）和类似的方程。

ple: For x = t², y = 2t at t = 1, the point is (1, 2), and dy/dx = 1/1 = 1. The tangent: y – 2 = 1(x – 1) → y = x + 1. The normal: gradient –1, equation y – 2 = –1(x – 1) → y = –x + 3.

示例：对于 x = t², y = 2t 在 t = 1 处，点为 (1, 2)，且 dy/dx = 1/1 = 1。切线：y – 2 = 1(x – 1) → y = x + 1。法线：梯度 –1，方程为 y – 2 = –1(x – 1) → y = –x + 3。

answer in the form requested, often ax + by + c = 0 or y = mx + c. Check that the given t‑value indeed yields the claimed point.

最终答案务必用题目要求的形式表示，通常是 ax + by + c = 0 或 y = mx + c。检查给定的 t 值是否确实给出所声称的点。

7. Second Derivative d²y/dx² | 二阶导数

x) = [d/dt(dy/dx)] / (dx/dt). Differentiate the first derivative with respect to t, then divide by dx/dt.

要确定参数曲线上驻点的凹凸性或性质，需要用到二阶导数。它由 d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)] / (dx/dt) 给出。先对 t 求一阶导数的导数，然后除以 dx/dt。

Using x = t², y = 2t: dy/dx = 1/t. Then d/dt(1/t) = –1/t², and dx/dt = 2t, so d²y/dx² = (–1/t²) / (2t) = –1/(2t³). At t = 1, d²y/dx² = –1/2, indicating the curve is concave down there.

以 x = t², y = 2t 为例：dy/dx = 1/t。则 d/dt(1/t) = –1/t²，而 dx/dt = 2t，因此 d²y/dx² = (–1/t²) / (2t) = –1/(2t³)。在 t = 1 处，d²y/dx² = –1/2，表明该处曲线向下凹。

8. Integrating Parametric Equations: Area Under a Curve | 参数方程的积分：曲线下方面积

The area enclosed by a parametric curve and the x‑ an be found by converting the integral ∫ y dx into an integral with respect to t: Area = ∫x=ab y dx = ∫t=t₁t₂ y (dx/dt) dt, where t₁ and t₂ are the parameter values corresponding to x = a and x = b.

参数曲线与 x 轴围成的面积可以通过将积分 ∫ y dx 转换为关于 t 的积分求得：面积 = ∫x=ab y dx = ∫t=t₁t₂ y (dx/dt) dt，其中 t₁ 和 t₂ 是对应于 x = a 和 x = b 的参数值。

example: Find the area under the curve x = t², y = t³ for 0 ≤ t ≤ 2. The curve meets the x‑ nterval, Area = ∫₀⁴ y dx = ∫₀² t³ * (2t) dt = ∫₀² 2t⁴ dt = [2/5 t⁵]₀² = 64/5 square units.

示例：求曲线 x = t², y = t³ 在 0 ≤ t ≤ 2 下方与 x 轴围成的面积。曲线在 t = 0 (x = 0) 和 t = 2 (x = 4) 处与 x 轴相交。由于该区间上 y ≥ 0，面积 = ∫₀⁴ y dx = ∫₀² t³ * (2t) dt = ∫₀² 2t⁴ dt = [2/5 t⁵]₀² = 64/5 平方单位。

9. Area Between Parametric Curve and Axes | 参数曲线与坐标轴围成面积

For areas involving the y‑ dy/dt) dt, with limits in y. Similarly, for regions between two curves expressed parametrically, subtract the lower y‑values and integrate the difference with respect to x after converting to t.

对于涉及 y 轴的面积，交换角色：面积 = ∫ x dy = ∫ x (dy/dt) dt，并采用 y 的上下限。类似地，对于由两条参数曲线界定的区域，用 y 值相减，并在转换为 t 之后对 x 积分其差值。

If the curve crosses the x‑axis, split the interval and use absolute values to ensure all area contributions are positive. The parametric approach remains robust as long as you carefully translate the x‑limits into t‑limits.

如果曲线穿过 x 轴，要拆分区间并取绝对值，以确保所有面积贡献均为正。只要谨慎地将 x 积分限转换为 t 积分限，参数方法就依然稳健。

10. Limits of Integration and Direction | 积分限与方向

When setting up a parametric area integral, note the direction of traversal. If x increases as t increases, t₁ < t₂ yields the standard left‑to‑right area. If x decreases (dx/dt < 0), you may need to reverse the t‑limits to ensure a positive dx in the original sense, or simply swap the limits.

建立参数面积积分时，注意遍历方向。如果 x 随 t 增加而增加，则 t₁ < t₂ 给出标准的从左到右的面积。如果 x 递减（dx/dt < 0），可能需要反转 t 积分限以确保原本意义上的 dx 为正，或者直接交换积分限。

For the curve x = 2 – y = t², from t = 0 to t = 2, x decreases from 2 to 0. The area between the curve and the x‑axis can be found by Area = ∫x=20 y dx = ∫t=02 t² * (–1) dt = – gives the positive area. Alternatively, use limits t=2 to t=0 in the integrand with dx/dt.

对于曲线 x = 2 – t, y = t²，t 从 0 到 2，x 从 2 减小到 0。曲线与 x 轴之间的面积可由面积 = ∫x=20 y dx = ∫t=02 t² * (–1) dt = –∫₀² t² dt 求得。取绝对值即得正面积。或者，利用 dx/dt 将被积函数的积分限设为 t=2 到 t=0。

11. Practical Examples and Exam Tips | 实例与考试技巧

Typical Edexcel questions may ask you to find the Cartesian equation, the gradient at a point, the equation of a tangent, or the area bounded by the curve and coordinate axes. Always start by identifying the allowed range of t and draw a quick sketch.

典型的 Edexcel 考题可能要求你求直角坐标方程、某点处的梯度、切线方程，或曲线与坐标轴围成的面积。始终从确定 t 的允许范围开始，并快速画出草图。

f a tangent or the region for integration. In differentiation, be careful to present dy/dx in terms of t or x as required—some questions ask for the gradient as a function of x after substituting back.

消参时要注意定义域：t ≥ 0 可能限制 x ≥ 0，这会影响切线的有效性或积分区域。在求导中，注意根据要求将 dy/dx 表示为 t 或 x 的函数——有些题目可能要求代回后用 x 表示梯度。