Arc Length with Vector Functions

Arc length is simply the length of a curved path, the number you'd get by laying a string along it and then straightening the string out against a ruler. ("Arc" is just an old word for a piece of a curve.) It's how you'd find the length of a spiral handrail or how far a drone actually flew. The formula $\int_a^b\|\vec r\,'(t)\|\,dt$ looks dense, but it's just $\|\vec r\,'(t)\|$ (your speed at each moment) times $dt$ (a tiny slice of time), all added up by the integral. Distance equals speed times time, one moment at a time.

Think about why that works. Over a tiny time $dt$, the point moves along a piece of curve so short it's practically straight, in the direction of $\vec r\,'(t)$. Its length is speed times time, $\|\vec r\,'(t)\|\,dt$. Writing out the speed with the Pythagorean theorem gives the square root of the three squared derivatives. If you've seen arc length for parametric curves in Calculus II, this is the same formula with a third component under the root.

Arc length parametrization. Here's a neat trick. The arc length function $s(t)$ tells you how far along the curve you are at time $t$. Solve $s = s(t)$ for $t$ and substitute that into $\vec r$. The curve keeps exactly the same shape, but now it's traced at speed 1, so the parameter $s$ is literally distance along the curve: $s = 3$ means "3 units along." The reason to bother is that it strips out how fast you happen to be moving and leaves only the shape. That makes the curvature formulas in the next lesson cleanest in this parameter.

CNC machines and 3D printers move tool heads along curves at constant speed by parametrizing paths by arc length.