DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

6 Chapter 1. CurvesAlternatively, since tan(θ/2) = e t ,wehavesin θ =2sin(θ/2) cos(θ/2) =2et1+e 2t = 2e t = secht+ e−t cos θ = cos 2 (θ/2) − sin 2 (θ/2) = 1 − e2t1+e 2t = e−t − e te t = − tanh t,+ e−t and so we can parametrize the tractrix instead byβ(t) = ( t − tanh t, secht), t ≥ 0. ▽The fundamental concept underlying the geometry of curves is the arclength of a parametrizedcurve.Definition. If α: [a, b] → R 3 is a parametrized curve, then for any a ≤ t ≤ b, wedefine itsarclength from a to t to be s(t) =∫ ta‖α ′ (u)‖du.arclength of its trajectory—is the integral of its speed.That is, the distance a particle travels—theAn alternative approach is to start with the followingDefinition. Let α: [a, b] → R 3 be a (continuous) parametrized curve. Given a partition P ={a = t 0