36 Chapter 2. <strong>Surfaces</strong>: Local TheoryavubuFigure 1.3S<strong>in</strong>ce x u × x v = s<strong>in</strong> u(s<strong>in</strong> u cos v, s<strong>in</strong> u s<strong>in</strong> v, cos u) =(s<strong>in</strong> u)x(u, v), the parametrization isregular away from u =0,π, which we’ve excluded anyhow because x fails to be one-to-oneat such po<strong>in</strong>ts. The u-curves are the so-called l<strong>in</strong>es of longitude <strong>and</strong> the v-curves are thel<strong>in</strong>es of latitude on the sphere.(e) Another <strong>in</strong>terest<strong>in</strong>g parametrization of the sphere is given by stereographic projection. (Cf.Exercise 1.1.1.) We parametrize the unit sphere less the north pole (0, 0, 1) by the xy-plane,(0,0,1)(x,y,z)(u,v,0)Figure 1.4assign<strong>in</strong>g to each (u, v) the po<strong>in</strong>t (≠ (0, 0, 1)) where the l<strong>in</strong>e through (0, 0, 1) <strong>and</strong> (u, v, 0)<strong>in</strong>tersects the unit sphere, as pictured <strong>in</strong> Figure 1.4. We leave it to the reader to derive thefollow<strong>in</strong>g formula <strong>in</strong> Exercise 1:(x(u, v) =2uu 2 + v 2 +1 ,2vu 2 + v 2 +1 , u2 + v 2 )− 1u 2 + v 2 . ▽+1For our last examples, we give two general classes of surfaces that will appear throughout ourwork.
§1. Parametrized <strong>Surfaces</strong> <strong>and</strong> the <strong>First</strong> Fundamental Form 37Example 2. Let I ⊂ R be an <strong>in</strong>terval, <strong>and</strong> let α(u) =(0,f(u),g(u)), u ∈ I, bearegularparametrized plane curve with f>0. Then the surface of revolution obta<strong>in</strong>ed by rotat<strong>in</strong>g α aboutthe z-axis is parametrized byx(u, v) = ( f(u) cos v, f(u) s<strong>in</strong> v, g(u) ) ,u ∈ I, 0 ≤ v