c1fa74…: “Out from a wonderful expression”

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Consider the curve shown below, which is traced out by $\vec {p}(t)$ .

If the curve does use arclength as a parameter, what must be true?

$\left |\vec {p}'(t)\right |$ should be $1$ for all $t$ . $\left |\vec {p}'(t)\right |$ must be $1$ for some $t>0$ , but not necessarily for all $t$ . $\int _0^t \left |\vec {p}'(\tau )\right | \d \tau = t$ for all $t$ . $\int _0^t \left |\vec {p}'(\tau )\right | \d \tau = t$ for some $t>0$ , but not necessarily for all $t$ . $|\vec {p}(t)| = t$ for all $t$ . $|\vec {p}(t)| = t$ for some $t>0$ , but not necessarily for all $t$ .

Is the curve parameterized by arclength?

Yes. No. It does, but only for some values of $t$ .

If $\vec {p}(t)$ is a parameterization of the curve that uses arclength as a parameter, then $\vec {p}(1)$ should give the $x,y$ -coordinates after we have travelled $1$ unit along the curve. From the picture, we have travelled much farther than $1$ unit, since the only way we can start at the origin, travel $1$ unit, and arrive at $(1,0)$ is to travel along the line from $(0,0)$ to $(1,0)$ .