1.5. Recommended reading 19 “Each railway is in a long tunnel, perfectly straight: so of course the middle of it is nearer the centre of the globe than the two ends: so every train runs half-way down-hill, and that gives it force enough to run the other half up-hill.” To which a protagonist replies: “Thank you. I understand that perfectly,” said Lady Muriel. “But the velocity, in the middle of the tunnel, must be something fearful! You can also find a homework problem, about a tunnel-train between Minneapolis and Chicago, in Brooke and Wilcox (1929). See also Kirmser (1966). Edwards’ (1965) article reignited keen interest in gravity-powered transportation and inspired the articles by Cooper (1966a,b), Venezian (1966), Mallett (1966), Laslett (1966), and Patel (1967) on the terres- trial brachistochrone. Aravind (1981) applied John Bernoulli’s optical method to the terrestrial brachistochrone and Prussing (1976), Chan- der (1977), McKinley (1979), and Denman (1985) pointed out that terrestrial brachistochrones are also tautochrones. Stalford and Gar- rett (1994) analyzed the terrestrial brachistochrone using differential geometry and optimal control theory. Struik (1933), Carath´ eodory (1937), and Kline (1972) summarize the early history of the study of geodesics. Geodesics are an important topic in differential geometry (Struik, 1961 Oprea, 2007), Riemann- ian geometry (Berger, 2003), and geometric modeling (Patrikalakis and Maekawa, 2002). See Bliss (1902) for examples of geodesics on a toroidal anchor ring and Sneyd and Peskin (1990) for examples of geodesic trajectories on general tubular surfaces. Isenberg (1992) and Oprea (2000) provide interesting and read- able introductions to the science and mathematics of soap films. Barbosa and Colares (1986), Nitsche (1989), Fomenko (1990), and Fomenko and Tuzhilin (1991) do an excellent job of presenting the history and theory of minimal surfaces.
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