---- > [!definition] Definition. ([[parameterization by arc length]]) A **parameterization by arc length** of a [[regular curve|regular]] [[parameterized curve]] $(\alpha, \alpha(I))$ is one satisfying (fixing origin point $t_{0}$) $s(t)=t-t_{0}$, i.e., one for which the time elapsed between two points is also the arc length between them. > Here $s$ denotes the [[arc length of a path|arc length]] function. ^fd36b9 > [!equivalence] > A parameterization $\alpha$ is by arc length if and only if it has unit speed ($\|\alpha'(t)\|_{2}=1$). ^a338b2 > [!note] Remark. > Every [[regular curve|regular]], [[smooth]] [[parameterized curve]] admits a [[parameterization by arc length|parameterization by arc length]] > [!intuition] > The idea being that if you travel at 1 unit per second, in $T$ seconds you will have traveled $T$ units. 'Time' and 'length' agree. > [!justification] Proof of Equivalence. > $\to.$ Suppose $\alpha$ is a parameterization by arc length, $t-t_{0} = s(t)$ for fixed constant $t_{0}$. Differentiating both sides yields $1=s'(t)$. By the [[fundamental theorem of calculus]], $s'(t)= \frac{d}{dt} \int _{t_{0}}^{t} \|\alpha'(\tau)\|_{2} \, d\tau= \|\alpha'(t)\|_{2}$. So the parameterization is unit-speed. > $\leftarrow.$ Suppose $\alpha$ is unit-speed, $\|\alpha'(t)\|_{2}=1.$ Then $s(t)=\int_{t_{0}}^{t} \underbrace{\|\alpha'(t)\|_{2}}_{=1}\, dt = t- t_{0}, $ thus the parameterization is in fact by arc length. > [!proposition] Procedure. (Reparameterizing by Arc Length) > [[smooth regular parameterized curves admit (re)parameterizations by arc length]] > ^7f92cf ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```