DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Parametrized Surfaces and the First Fundamental Form 35vxx vx u ×x vv-curvex uuu-curveFigure 1.1(a) The graph of a function f : U → R, z = f(x, y), is parametrized by x(u, v) =(u, v, f(u, v)).Note that x u × x v =(−f u , −f v , 1) ≠ 0, sothis is always a regular parametrization.(b) The helicoid, asshown in Figure 1.2, is the surface formed by drawing horizontal rays fromFigure 1.2the axis of the helix α(t) =(cos t, sin t, bt) topoints on the helix:x(u, v) =(u cos v, u sin v, bv), u ≥ 0, v∈ R.Note that x u × x v =(b sin v, −b cos v, u) ≠ 0. The u-curves are rays and the v-curves arehelices.(c) The torus (surface of a doughnut) is formed by rotating a circle of radius b about a circleof radius a > b lying in an orthogonal plane, as pictured in Figure 1.3. The regularparametrization is given byx(u, v) =((a + b cos u) cos v, (a + b cos u) sin v, b sin u), 0 ≤ u, v < 2π.Then x u × x v = −b(a + b cos u) ( cos u cos v, cos u sin v, sin u ) , which is never 0.(d) The standard parametrization of the unit sphere Σ is given by spherical coordinates (φ, θ) ↔(u, v):x(u, v) =(sin u cos v, sin u sin v, cos u),0