Elliptic coordinate system

In geometry, the elliptic(al) coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are 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. In geometry, the elliptic(al) coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are 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 coordinates ( μ , ν ) {displaystyle (mu , u )} is where μ {displaystyle mu } is a nonnegative real number and ν ∈ [ 0 , 2 π ] . {displaystyle u in .} On the complex plane, an equivalent relationship is