8 1. Introduction x y A B Figure 1.4. Hypocycloid with inner radius R − S AB π The simplest case arises when the surface is a level set for one of the coordinates in a system of orthogonal curvilinear coordinates. The arc length can then be written using the scale factors of the coordinate system. Consider, for example, two points, A and B, on a sphere of radius R centered at the origin. We wish to join A and B by the shortest, continuously differentiable curve lying on the sphere. We start by specifying position, r(x, y, z) = x i + y j + z k , (1.13) using the Cartesian coordinates x, y, and z and Cartesian basis vec- tors i, j, and k. For points on the surface of a sphere, we now switch to the spherical coordinates r, θ, and φ (see Figure 1.6). Since x = r sin θ cos φ , y = r sin θ sin φ , z = r cos θ , (1.14)
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