Paraboloidal coordinates

Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {displaystyle (mu , u ,lambda )} that generalize two-dimensional parabolic coordinates. They possess elliptic paraboloids as one-coordinate surfaces. As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates, both of which are also generalizations of two-dimensional parabolic coordinates. The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids. Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {displaystyle (mu , u ,lambda )} that generalize two-dimensional parabolic coordinates. They possess elliptic paraboloids as one-coordinate surfaces. As such, they should be distinguished from parabolic cylindrical coordinates and parabolic rotational coordinates, both of which are also generalizations of two-dimensional parabolic coordinates. The coordinate surfaces of the former are parabolic cylinders, and the coordinate surfaces of the latter are circular paraboloids. Differently from cylindrical and rotational parabolic coordinates, but similarly to the related ellipsoidal coordinates, the coordinate surfaces of the paraboloidal coordinate system are not produced by rotating or projecting any two-dimensional orthogonal coordinate system. The Cartesian coordinates ( x , y , z ) {displaystyle (x,y,z)} can be produced from the ellipsoidal coordinates ( μ , ν , λ ) {displaystyle (mu , u ,lambda )} by the equations