1.3. Geodesics 13 Figure 1.7. Pseudosphere To determine a geodesic on the pseudosphere, we must thus find a curve, u = u(t) and v = v(t), that minimizes the arc-length integral s = a tb ta cot2 u ˙ 2 + sin2 u ˙ 2 dt (1.39) subject to the boundary conditions u(ta) = ua , v(ta) = va , u(tb) = ub , v(tb) = vb . (1.40) Alternatively, we may look for a curve, v = v(u), that minimizes the integral s = a ub ua cot2 u + sin2 u dv du 2 du (1.41) subject to the boundary conditions v(ua) = va, v(ub) = vb . (1.42) For other examples, see Exercise 1.6.6.
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