DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Examples, Arclength Parametrization 97. Find the arclength of the tractrix, given in Example 2, starting at (0, 1) and proceeding to anarbitrary point.♯ 8.Let P, Q ∈ R 3 and let α: [a, b] → R 3 be any parametrized curve with α(a) =P , α(b) =Q.Let v = Q − P . Prove that length(α) ≥‖v‖, sothat the line segment from P to Q gives theshortest possible path. (Hint: Consider∫ baα ′ (t) · vdt and use the Cauchy-Schwarz inequalityu · v ≤‖u‖‖v‖. Ofcourse, with the alternative definition on p. 6, it’s even easier.)9. Consider a uniform cable with density δ hanging in equilibrium. As shown in Figure 1.12, thetension forces T(x +∆x), −T(x), and the weight of the piece of cable lying over [x, x +∆x]all balance. If the bottom of the cable is at x =0,T 0 is the magnitude of the tension there,T(x+∆x)θ +∆θ−T(x)θ −gδ∆sxx+∆xFigure 1.12and the cable is the graph y = f(x), show that f ′′ (x) = gδ √1+fT ′ (x) 2 . (Remember that0tan θ = f ′ (x).) Letting ∫ C = T 0 /gδ, show that f(x) =C cosh(x/C) +c for some constant c.du(Hint: To integrate √ , make the substitution u = sinh v.)1+u 210. As shown in Figure 1.13, Freddy Flintstone wishes to drive his car with square wheels along astrange road. How should you design the road so that his ride is perfectly smooth, i.e., so thatthe center of his wheel travels in a horizontal line? (Hints: Start with a square with verticesOCQPFigure 1.13at (±1, ±1), with center C at the origin. If α(s) =(x(s),y(s)) is an arclength parametrization