10 1. Introduction x y z r φ θ Figure 1.6. Spherical coordinates The element of arc length in spherical coordinates is given by ds = √ dr · dr = hr 2 dr2 + h2 θ dθ2 + h2 φ dφ2 (1.19) = dr2 + r2 dθ2 + r2 sin2 θ dφ2 . For a sphere of radius r = R, this element reduces to ds = R dθ2 + sin2 θ dφ2 . (1.20) If we assume that φ = φ(θ), finding the curve that minimizes the arc length between the points A = (θa,φa) and B = (θb,φb) simplifies to finding the function φ(θ) that minimizes the integral s = B A ds = R θB θA 1 + sin2 θ (dφ/dθ)2 dθ (1.21) subject to the boundary conditions φ(θa) = φa , φ(θb) = φb . (1.22)
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