Calculus / Chinese

Arc length of a plane curve

Charles队长 · Bilibili · 0:24

Bilibili · Charles队长Original video

IN THIS VIDEO

01Parametric curve
02Chord approximation
03Integrate speed

FROM THE LEARNING NOTES

\mathbf r(t)=(\varphi(t),\psi(t)),\quad t\in[a,b]

Parametric curve

SELECTED MOMENT

The integral of speed

SUGGESTED PREPARATION

Parametric equations
Pythagorean theorem
Derivatives and definite integrals

READ & KEEP

The explanation, unpacked.

Reviewed learning material · Video analysis · English

Read the full overview

The animation approximates a parametric curve by an inscribed polygonal path. Each chord length is the Euclidean length of the coordinate increment. For a C1 curve, refining the partition gives the integral of speed. Length counts the traversal, so repeatedly visiting an arc counts it repeatedly.

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Chapters

0:00A parametric curve 0:08Polygonal approximation 0:14The integral of speed

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

Write the curve as $x=φ(t)$ , $y=ψ(t)$ for t in [a,b]. Select points in parameter order and join them to form an inscribed polygonal path.

Each chord has length √(Δxᵢ²+Δyᵢ²). Adding these gives an approximation. At a finite partition, a chord is generally not the same length as its corresponding curved arc.

For a continuously differentiable curve, as the maximum parameter spacing tends to zero the polygonal length approaches the integral of √(φ′²+ψ′²), the speed. This computes traveled length without negative contributions. Repeated traversal of an arc counts that arc multiple times.

Knowledge cards

Parametric curve

One parameter controls both coordinates. The length formula applies to C1 or appropriate piecewise smooth parameterizations.

\mathbf r(t)=(\varphi(t),\psi(t)),\quad t\in[a,b]

Chord approximation

Use Euclidean distance for each chord. Exact arc length comes from the refined-partition limit, not equality at a finite subdivision.

\Delta s_i=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}

Integrate speed

Accumulate the magnitude of the parameter velocity. The norm keeps every contribution nonnegative.

s=\int_a^b\|\mathbf r\prime(t)\|\,dt=\int_a^b\sqrt{\varphi\prime(t)^2+\psi\prime(t)^2}\,dt

Traversal multiplicity

The formula counts the parameter path traveled. Retracing a segment counts it again; it does not deduplicate the geometric image.

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Definite integrals ApplicationAt 0:18
Why this connection?
The animation integrates speed to obtain the length of a continuously differentiable or suitably piecewise smooth parametric curve. Chord lengths approximate the traversal before refinement. The nonnegative integral counts retraced portions again rather than deduplicating the geometric image.

Arc length of a plane curve

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Parametric curve

Chord approximation

Integrate speed

Traversal multiplicity

Explore the knowledge in this video

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