Name: polar curve distance with calculus
Uploaded: 2026-06-08T09:38:49.530Z
Duration: 1 min 26 s
Channel: JoVE

假设一架无人机被派往茂密的森林中搜寻一名失踪的徒步者。为了确保对地面每一平方米都进行检查，无人机沿一条不断扩展的螺旋路径飞行。在此，徒步者的最后已知位置作为极坐标曲线的极点。无人机的位置由其与该点的距离以及从正 x 轴逆时针测量的旋转角度来确定。为了确定无人机是否有足够的电量完成任务，救援团队必须使用弧长函数计算需要飞行的总距离。推导过程首先利用极坐标与直角坐标之间的关系，求出水平和垂直位置的变化率。将这些速率平方后相加，可使该公式完全用极坐标表示。此处使用了勾股定理恒等式来简化表达式。该恒等式将复杂的三角函数项合并，得到曲线总长度的简化公式。通过求解从初始角度到最终角度区间内的积分，该团队可以准确计算出无人机完成飞行所需的时间。

利用极坐标积分求解弧长

Integration Applied to Polar Coordinates to Find Arc Lengths

在极坐标系中，平面曲线通过一个固定点（称为极点）的径向距离 r 和从参考方向测量的角度 θ 来描述。该坐标系特别适用于涉及旋转的路径，例如搜索无人机所遵循的扩展螺旋轨迹。若将徒步者最后已知的位置视为极点，则无人机在任意时刻的位置可用极坐标方程 r = f(θ) 表示，其中从极点出发的距离随无人机逆时针旋转而变化。为了计算总行进距离，可以使用以下方法将极坐标曲线与直角坐标系关联起来\begin{equation*}x = r\cos\theta\end{equation*}\begin{equation*}y = r\sin\theta\end{equation*}由于 r 取决于 θ，两者均 x 和 y 也随……而变化 θ求导得\begin{equation*}\jfrac{dx}{d\theta} = \jfrac{dr}{d\theta}\cos\theta - r\sin\theta\end{equation*}\begin{equation*}\jfrac{dy}{d\theta} = \jfrac{dr}{d\theta}\sin\theta + r\cos\theta\end{equation*}这些表达式描述了无人机水平和垂直位置随角度变化的速率。矩形弧长公式为\begin{equation*}ds = \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\, d\theta\end{equation*}代入导数并化简可得\begin{equation*}S = \int_{{\theta}_{1}}^{{\theta}_{2}} \sqrt{r^2 + \left(\jfrac{dr}{d\theta}\right)^2} \, d\theta\end{equation*}简化之所以发生，是因为混合的三角函数项相互抵消，且该恒等式\begin{equation*}\sin^2\theta + \cos^2\theta = 1\end{equation*}合并剩余项。对于无人机的螺旋航线，通过从初始角度到最终角度计算该积分，可得到总的飞行距离，从而使得救援团队能够评估电池容量是否足以完成任务。

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integration-applied-to-polar-coordinates-to-find-arc-lengths

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Parametric Equations and Polar Coordinates

参数方程与极坐标

parametric-equations-and-polar-coordinates

KEY TERMS AND CONCEPTS

关键术语与概念

对极坐标进行积分以求得弧长

polar curve distance with calculus

Polar coordinates can be used to measure the distance along a curve that turns around a fixed point. In this system, a point is described by the radial distance r from the pole and the angle θ from a reference direction. That makes it useful for motion such as an expanding spiral, where distance and direction change together.

A search drone gives a clear example of a polar path. If the hiker’s last known position is the pole, the drone’s location at any moment can be written as the polar equation r = f(θ). As the drone moves counterclockwise, its distance from the pole changes with the angle.

To find the total distance traveled, the polar curve is linked to rectangular coordinates. The relationships x = r cos θ and y = r sin θ show how the position in the plane depends on θ. Because r changes with θ, both x and y also change as the angle changes.

Differentiating gives the rates of change of the horizontal and vertical positions. The result is dx/dθ = (dr/dθ) cos θ - r sin θ and dy/dθ = (dr/dθ) sin θ + r cos θ. These expressions describe how the drone moves in each direction as it rotates.

The rectangular arc length formula is then used. It begins with ds = \sqrt{(dx/dθ)^2 + (dy/dθ)^2}\, dθ. After substituting the derivatives and simplifying, the polar arc length formula becomes S = \int_{θ_1}^{θ_2} \sqrt{r^2 + (dr/dθ)^2} \, dθ.

The simplification works because the mixed trigonometric terms cancel. The identity sin^2θ + cos^2θ = 1 also combines the remaining terms. Evaluating the integral from the initial angle to the final angle gives the total flight distance, which can help a rescue team estimate whether the drone has enough battery for the mission.

Polar Curve Distance with Calculus

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key terms and concepts

Learn how polar coordinates and calculus work together to find the distance along a spiral path using the arc length integral and derivative of r.

Polar Curve Distance with Calculus | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

Polar Curve Distance with Calculus

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