12 1. Introduction with u(ta) = ua , v(ta) = va , u(tb) = ub , v(tb) = vb . (1.30) In this formulation, we have two dependent variables, u(t) and v(t), and one independent variable, t. If v can be written as a function of u, v = v(u), we can instead rewrite our integral as s = ub ua E + 2 F dv du + G dv du 2 du (1.31) with v(ua) = va, v(ub) = vb . (1.32) This is now a problem with one dependent variable and one indepen- dent variable. To make all this concrete, let us take, as an example, the pseu- dosphere (see Figure 1.7), half of the surface of revolution generated by rotating a tractrix about its asymptote. If the asymptote is the z-axis, we can write the equation for a pseudosphere, parametrically, as r(u, v) = a sin u cos v i + a sin u sin v j (1.33) + a cos u + ln tan u 2 k . Since ru = ∂r ∂u (1.34) = (a cos u cos v, a cos u sin v, −a sin u + a csc u) and rv = ∂r ∂v = (−a sin u sin v, a sin u cos v, 0) , (1.35) the first-order fundamental quantities reduce to E = ru · ru = a2 cot2 u , (1.36) F = ru · rv = 0 , (1.37) G = rv · rv = a2 sin2 u . (1.38)

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