Epilogue

Summary

The ellipsoidal coordinate system replaces the spherical system whenever the geometrical standards of the space depend on direction. In other words, when the space exhibits some kind of anisotropy. This anisotropy is characterized by three orthogonal directions, specifying the principal directions, and the unit lengths along these directions, specifying the semi-axes of the reference ellipsoid. Hence, the reference ellipsoid encodes the complete structure of the anisotropic behavior of the space and defines the appropriate coordinate system. One of the variables of the ellipsoidal system, denoted by ρ, specifies a family of ellipsoids and therefore corresponds to the radial variable of the spherical system. The other two variables, denoted by μ and ν, specify a point on the ellipsoid and therefore they correspond to the spherical angular variables. Since the variables vary in successive intervals of the real line in the order (ρ, μ, ν), it is customary to refer to them in this particular order. We should keep in mind, however, that this order corresponds to a sinistral system. The order that leads to a dextral system is (ρ, μ, ν). The ellipsoidal system stems out of three couples of foci, two of which lie along the longest semi-axis and one lies along the intermediate semi-axis of the reference ellipsoid. These six foci define the focal ellipse, which has the two focal distances as its axes and the third one as its own focal distance.