DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

8 Chapter 1. CurvesAn important observation from a theoretical standpoint is that any regular parametrized curvecan be reparametrized by arclength. For if α is regular, the arclength function s(t) =∫ ta‖α ′ (u)‖duis an increasing function (since s ′ (t) =‖α ′ (t)‖ > 0 for all t), and therefore has an inverse functiont = t(s). Then we can consider the parametrizationNote that the chain rule tells us thatβ(s) =α(t(s)).β ′ (s) =α ′ (t(s))t ′ (s) =α ′ (t(s))/s ′ (t(s)) = α ′ (t(s))/‖α ′ (t(s))‖is everywhere a unit vector; in other words, β moves with speed 1.EXERCISES 1.1*1. Parametrize the unit circle (less the point (−1, 0)) by the length t indicated in Figure 1.11.(x,y)(−1,0)tFigure 1.11♯ 2.Consider the helix α(t) =(a cos t, a sin t, bt). Calculate α ′ (t), ‖α ′ (t)‖, and reparametrize α byarclength.3. Let α(t) = ( √3 1cos t + √ 1 12sin t, √3 1cos t, √3 cos t − √ 12sin t ) .reparametrize α by arclength.Calculate α ′ (t), ‖α ′ (t)‖, and*4. Parametrize the graph y = f(x), a ≤ x ≤ b, and show that its arclength is given by thetraditional formula∫ b √length = 1+ ( f ′ (x) ) 2 dx.a5. a. Show that the arclength of the catenary α(t) =(t, cosh t) for 0 ≤ t ≤ b is sinh b.b. Reparametrize the catenary by arclength. (Hint: Find the inverse of sinh by using thequadratic formula.)*6. Consider the curve α(t) =(e t ,e −t , √ 2t). Calculate α ′ (t), ‖α ′ (t)‖, and reparametrize α byarclength, starting at t =0.