Geodesics

On a sphere all geodesics are arcs of great circles. On a surface of constant negative curvature (like (c)) geodesics diverge exponentially, as noted in early work on chaos theory (see page 971). The path of a geodesic can in general be found by requiring that the analog of acceleration vanishes for it. In the case of a surface defined by z==f[x, y] this is equivalent to solving

x''[t] == -((Derivative[1, 0][f][x[t], y[t]] (Derivative[1][y][t]^{2} Derivative[0, 2][f][x[t], y[t]] + 2 Derivative[1][x][t] Derivative[1][y][t] Derivative[1, 1][f][x[t], y[t]] + Derivative[1][x][t]^{2} Derivative[2, 0][f][x[t], y[t]]))/ (1 + Derivative[0, 1][f][x[t], y[t]]^{2} + Derivative[1, 0][f][x[t], y[t]]^{2}))

together with the corresponding equation for y'', as already noted by Leonhard Euler in 1728 in connection with his development of the calculus of variations.