7 Contours of constant u , with u ∈ [0 , ∞ ], are nested el- lipses with common foci located at ( x ; y ) = ( ± f ; 0). Con- tours of constant v , with v ∈ [0 , 2 π ], are hyperbolae. Together these contours form an orthogonal coordinate system. See Fig. 4. Data is to be interpolated onto the ellipse whose cross section is that of the elliptical cylinder of Fig. 3. See Fig. 5. EllipticCylindricalCoord_1000.gif (GIF Image, 475x412 pixels) FIG. 4: Elliptical coordinates showing contours of constant u and constant v [7]. The foci are at ( ± a, 0), and in our case a = f . FIG. 5: A square or rectangular grid in the x , y plane for a fixed z value on the 3-dimensional grid. Values at data points near the ellipse are to be interpolated onto the ellipse. For our work we will need the unit vector ˆ e$_{u}$ , the unit vector (outwardly) normal to the surface of the elliptical cylinder. Write r = x ˆ e$_{x}$ + y ˆ e$_{y}$ + z ˆ e$_{z}$ = f cosh( u ) cos( v ) ˆ e$_{x}$ + f sinh( u ) sin( v ) ˆ e$_{y}$ + z ˆ e$_{z}$ . (25) Then, by definition, we have the result ˆ e$_{u}$ = ( ∂ r /∂u ) / || ( ∂ r /∂u ) || = sinh( u ) cos( v ) ˆ e$_{x}$ + cosh( u ) sin( v ) ˆ e$_{y}$ [cosh $^{2}$( u ) − cos $^{2}$( v )] 1 / 2 . (26) [( ∂$_{u}$ ) 2 + ( ∂$_{v}$ ) $^{2}$][ P ( u ) Q ( v )] = (36) ( k $^{2}$f$^{2}$/ 4)[2 cosh(2 u ) − 2 cos(2 v )] P ( u ) Q ( v ) . so that the requirement (34) can be rewritten in the form cosh $^{2}$( u ) − cos $^{2}$( v ) = (1 / 2)[cosh(2 u ) − cos(2 v )] (35) We also observe that there is the trigonometric identity [( ∂$_{u}$ ) 2 + ( ∂$_{v}$ ) $^{2}$][ P ( u ) Q ( v )] = k $^{2}$f$^{2}$[cosh$^{2}$( u ) − cos $^{2}$( v )] P ( u ) Q ( v ) . (34) where the functions P and Q are yet to be determined. Employing the Ansatz (33) in Laplace’s equation and use of (32b) yields the requirement ψ ∼ P ( u ) Q ( v ) exp( ikz ) (33) Let us seek to construct harmonic functions of the form 3. Mathieu Equations ∇ $^{2}$ψ = (1 /f $^{2}$)[cosh$^{2}$( u ) − cos $^{2}$( v )] − $^{1}$[( ∂$_{u}$ ) 2 + ( ∂$_{v}$ ) $^{2}$] ψ + ( ∂$_{z}$ ) $^{2}$ψ. (32b) B$_{u}$ = ˆ e$_{u}$ · B = ( ∇ ψ )$_{u}$ = (1 /f )[cosh $^{2}$( u ) − cos $^{2}$( v )] − 1 / $^{2}$( ∂ψ/∂u ) , (32a) From this relation we infer the results ds 2 ⊥ = dx 2 + dy 2 = f $^{2}$sinh( u + iv ) sinh( u − iv )( du 2 + dv $^{2}$) = f $^{2}$[cosh$^{2}$( u ) − cos $^{2}$( v )]( du 2 + dv $^{2}$) . (31) Now form the product of (29) and (30) to get the trans- verse line-element relation dx − idy = f sinh( ¯ w )( du − idv ) . (30) and the complex conjugate result dx + idy = f sinh( w )( du + idv ) (29) Form differentials of both sides of (28). Doing so gives the result ζ = f cosh( w ) . (28) In these variables, the relations (24) can be written in the more compact form w = u + iv. (27b) and ζ = x + iy, (27a) It is also convenient to employ the complex variables