the arc-length parametrization
Suppose you measure progress along a curve not by a clock, but by a tape measure: how much actual distance you have walked from a chosen starting mark. That distance is the arc length, usually written s. The arc-length parametrization is the schedule in which the parameter IS this measured distance. It is the fairest possible way to traverse a curve, because covering one unit of parameter always means covering exactly one unit of length — the bug crawls at constant unit speed and never speeds up or slows down.
First, the arc length of a regular curve r(t) from t = a to t = b is the integral s = integral from a to b of |r'(t)| dt, which adds up the speed over time to get total distance, just as distance = speed times time for a steady walk. Now define s(t) to be this length measured from a fixed start. Because the curve is regular, |r'(t)| > 0, so s(t) is strictly increasing and can be inverted to give t as a function of s; substituting yields a new parametrization r(s) in which, by the chain rule, the speed is exactly 1 for every s: |r'(s)| = 1. That is the meaning of unit speed. As a method: to arc-length parametrize, (1) compute |r'(t)|, (2) integrate to get s(t), (3) solve for t(s), (4) substitute back. For r(t) = (cos t, sin t), |r'(t)| = 1 already, so t is itself arc length and the circle is already unit-speed.
Why bother? Because once speed is constant 1, the formulas of curve theory shed all their clutter. The unit tangent is simply T = r'(s); the curvature is just kappa = |r''(s)|, the size of the acceleration; and the Frenet-Serret formulas take their cleanest form. Differentiating with respect to arc length strips away the arbitrary timetable and exposes the pure geometry. The one honest drawback is practical: the integral for s(t) often has no elementary closed form (even for an ellipse it leads to elliptic integrals, which is where the name comes from), so arc-length parametrization is a powerful theoretical normalization that you cannot always write down explicitly.
A circle of radius R, r(t) = (R cos t, R sin t), has |r'(t)| = R, so its arc length from 0 is s = R t. Inverting, t = s/R, and the unit-speed form is r(s) = (R cos(s/R), R sin(s/R)). Check: r'(s) = (-sin(s/R), cos(s/R)) has length exactly 1, as required.
Arc length s = integral of speed; reparametrizing by s makes the speed constantly 1.
Every regular curve CAN be parametrized by arc length in principle, but you often cannot write s(t) in elementary functions — the arc-length integral of an ellipse is genuinely non-elementary. Unit speed is a theoretical normalization, not always an explicit formula.