What is Length of an arc? — Mathematics

Definition 24.9 University Mathematics — Year 1 · Chapter 24 — Plane Curves

The length of a $C^1$ arc $f \colon \intcc{a}{b} \to \R^2$ is the integral of the speed:

L = \int_a^b \norm{f'(t)}\,\dd t = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\;\dd t .

(Motivation: on a small interval, $M(t + h) \approx M(t) + h f'(t)$ , so the arc is close to a polygon whose segment lengths sum to a Riemann sum of $\norm{f'}$ , Theorem 15.20.) For a polar curve $r = r(\theta)$ , the velocity $r'\vec u + r\vec v$ has orthogonal components, so

L = \int_{\theta_1}^{\theta_2} \sqrt{r'(\theta)^2 + r(\theta)^2}\;\dd\theta .

The length does not depend on the (monotone, $C^1$ ) parametrization chosen: substituting $t = \varphi(s)$ in the integral (Theorem 15.15) multiplies $f'$ by $\varphi'$ and $\dd t$ by $\varphi'^{-1}$ .

One arch of the cycloid (radius R = 1), the rolling circle at t = 2, and the two guide lines of the weekend problem: the chord MC to the contact point is normal to the curve, the chord MT to the top of the circle is tangent. — One arch of the cycloid (radius $R = 1$ ), the rolling circle at $t = 2$ , and the two guide lines of the weekend problem: the chord $MC$ to the contact point is *normal* to the curve, the chord $MT$ to the top of the circle is *tangent*.

Examples

Example 24.10 (Sanity check: the circle)

For $f(t) = (R\cos t, R\sin t)$ on $\intcc{0}{2\pi}$ : $\norm{f'} = R$ , so $L = 2\pi R$ — the definition returns the circumference. Reparametrization test: the map $g(t) = (R\cos 2t, R\sin 2t)$ on $\intcc{0}{\pi}$ draws the same circle at doubled speed $\norm{g'} = 2R$ , and

\int_0^{\pi} 2R\,\dd t = 2\pi R

again: half the time, twice the speed, same length — the invariance promised in the definition, watched once on numbers. (Running $g$ on all of $\intcc{0}{2\pi}$ would give $4\pi R$ : a curve traversed twice is twice as long as a journey; length measures the parametrized path, and honest bookkeeping of the interval is part of the computation.) For the astroid $(\cos^3t, \sin^3t)$ : $\norm{f'} = 3\abs{\sin t\cos t} = \tfrac32\abs{\sin 2t}$ , and by symmetry $L = 4\int_0^{\pi/2} \tfrac32\sin 2t\,\dd t = 6$ : a curve drawn inside the unit circle, of length $6 < 2\pi$ . The weekend problem measures the most famous arch of all.

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Examples

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