a reparametrization

If two people walk along the same trail but keep their own watches and walking speeds, they describe the identical path with different timetables. Swapping one timetable for another, while keeping the path itself untouched, is a reparametrization. It is the formal way of saying 'same curve, different schedule', and it is the tool that lets us tell which features of a curve are real geometry and which are mere artifacts of how we chose to trace it.

Precisely, given a parametrized curve r(t) defined for t in some interval, a reparametrization is a new curve r-tilde(s) = r(t(s)), where t = t(s) is a smooth function with a smooth inverse that converts the new parameter s into the old parameter t. The condition that makes this legitimate is that t'(s) is never zero — it must be either always positive (preserving direction of travel) or always negative (reversing it). The chain rule then shows the new velocity is r-tilde'(s) = r'(t) t'(s): the direction is unchanged (or exactly reversed), only the speed is rescaled. The image — the set of points visited — is identical. For instance, starting from r(t) = (cos t, sin t) and setting t = 2s gives r-tilde(s) = (cos 2s, sin 2s), the same circle traced twice as fast.

Reparametrization is the engine behind the central strategy of curve theory: because you can freely re-schedule a regular curve, you can choose the cleanest schedule of all, the arc-length parametrization, in which speed is constantly 1. Quantities that survive every reparametrization — like the image, the tangent line, and the curvature — are the true geometric invariants; quantities that change, like the speed |r'(t)| or the particular t-value at a point, are bookkeeping. Honest caveat: a reparametrization that flips direction (t'(s) < 0) reverses the unit tangent and, for plane curves, flips the sign of the signed curvature, so orientation is a genuine choice you must keep track of.