Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in theperpendicular z {displaystyle z} -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci F 1 {displaystyle F_{1}} and F 2 {displaystyle F_{2}} are generally taken to be fixed at − a {displaystyle -a} and + a {displaystyle +a} , respectively, on the x {displaystyle x} -axis of the Cartesian coordinate system. Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in theperpendicular z {displaystyle z} -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci F 1 {displaystyle F_{1}} and F 2 {displaystyle F_{2}} are generally taken to be fixed at − a {displaystyle -a} and + a {displaystyle +a} , respectively, on the x {displaystyle x} -axis of the Cartesian coordinate system. The most common definition of elliptic cylindrical coordinates ( μ , ν , z ) {displaystyle (mu , u ,z)} is where μ {displaystyle mu } is a nonnegative real number and ν ∈ [ 0 , 2 π ) {displaystyle u in [0,2pi )} .