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Bispherical coordinates
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<h2 id="definition">Definition</h2> <p>The most common definition of bispherical coordinates ( τ , σ , ϕ ) {\displaystyle (\tau ,\sigma ,\phi )} is </p> x = a sin σ cosh τ − cos σ cos ϕ , y = a sin σ cosh τ − cos σ sin ϕ , z = a sinh τ cosh τ − cos σ , {\displaystyle {\begin{aligned}x&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi ,\\y&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi ,\\z&=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }},\end{aligned}}} <p>where the σ {\displaystyle \sigma } coordinate of a point P {\displaystyle P} equals the angle F 1 P F 2 {\displaystyle F_{1}PF_{2}} and the τ {\displaystyle \tau } coordinate equals the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> of the ratio of the distances d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} to the foci </p> τ = ln d 1 d 2 {\displaystyle \tau =\ln {\frac {d_{1}}{d_{2}}}} <p>The coordinates ranges are −∞ < τ {\displaystyle \tau } < ∞, 0 ≤ σ {\displaystyle \sigma } ≤ π {\displaystyle \pi } and 0 ≤ ϕ {\displaystyle \phi } ≤ 2 π {\displaystyle \pi } . </p> <h3>Coordinate surfaces</h3> <p>Surfaces of constant σ {\displaystyle \sigma } correspond to intersecting tori of different radii </p> z 2 + ( x 2 + y 2 − a cot σ ) 2 = a 2 sin 2 σ {\displaystyle z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\cot \sigma \right)^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}} <p>that all pass through the foci but are not concentric. The surfaces of constant τ {\displaystyle \tau } are non-intersecting spheres of different radii </p> ( x 2 + y 2 ) + ( z − a coth τ ) 2 = a 2 sinh 2 τ {\displaystyle \left(x^{2}+y^{2}\right)+\left(z-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}} <p>that surround the foci. The centers of the constant- τ {\displaystyle \tau } spheres lie along the z {\displaystyle z} -axis, whereas the constant- σ {\displaystyle \sigma } tori are centered in the x y {\displaystyle xy} plane. </p> <h3>Inverse formulae</h3> <p>The formulae for the inverse transformation are: </p> σ = arccos ( R 2 − a 2 Q ) , τ = arsinh ( 2 a z Q ) , ϕ = arctan ( y x ) , {\displaystyle {\begin{aligned}\sigma &=\arccos \left({\dfrac {R^{2}-a^{2}}{Q}}\right),\\\tau &=\operatorname {arsinh} \left({\dfrac {2az}{Q}}\right),\\\phi &=\arctan \left({\dfrac {y}{x}}\right),\end{aligned}}} <p>where R = x 2 + y 2 + z 2 {\textstyle R={\sqrt {x^{2}+y^{2}+z^{2}}}} and Q = ( R 2 + a 2 ) 2 − ( 2 a z ) 2 . {\textstyle Q={\sqrt {\left(R^{2}+a^{2}\right)^{2}-\left(2az\right)^{2}}}.} </p> <h3>Scale factors</h3> <p>The scale factors for the bispherical coordinates σ {\displaystyle \sigma } and τ {\displaystyle \tau } are equal </p> h σ = h τ = a cosh τ − cos σ {\displaystyle h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}} <p>whereas the azimuthal scale factor equals </p> h ϕ = a sin σ cosh τ − cos σ {\displaystyle h_{\phi }={\frac {a\sin \sigma }{\cosh \tau -\cos \sigma }}} <p>Thus, the infinitesimal volume element equals </p> d V = a 3 sin σ ( cosh τ − cos σ ) 3 d σ d τ d ϕ {\displaystyle dV={\frac {a^{3}\sin \sigma }{\left(\cosh \tau -\cos \sigma \right)^{3}}}\,d\sigma \,d\tau \,d\phi } <p>and the Laplacian is given by </p> ∇ 2 Φ = ( cosh τ − cos σ ) 3 a 2 sin σ [ ∂ ∂ σ ( sin σ cosh τ − cos σ ∂ Φ ∂ σ ) + sin σ ∂ ∂ τ ( 1 cosh τ − cos σ ∂ Φ ∂ τ ) + 1 sin σ ( cosh τ − cos σ ) ∂ 2 Φ ∂ ϕ 2 ] {\displaystyle {\begin{aligned}\nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sin \sigma }}&\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)\right.\\[8pt]&{}\quad +\left.\sin \sigma {\frac {\partial }{\partial \tau }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sin \sigma \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]\end{aligned}}} <p>Other differential operators such as ∇ ⋅ F {\displaystyle \nabla \cdot \mathbf {F} } and ∇ × F {\displaystyle \nabla \times \mathbf {F} } can be expressed in the coordinates ( σ , τ ) {\displaystyle (\sigma ,\tau )} by substituting the scale factors into the general formulae found in <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal coordinates</a>. </p> <h2 id="applications">Applications</h2> <p>The classic applications of bispherical coordinates are in solving <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a>, e.g., <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a>, for which bispherical coordinates allow a <a href="/facts/Separation_of_variables/oJdkw5xq">separation of variables</a>. However, the <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a> is not separable in bispherical coordinates. A typical example would be the <a href="/facts/Electric_field/tt4MooBs">electric field</a> surrounding two conducting spheres of different radii. </p> <h2 id="bibliography">Bibliography</h2> <ul><li>Morse PM, Feshbach H (1953). <i>Methods of Theoretical Physics, Parts I and II</i>. New York: McGraw-Hill. pp. 665–666, 1298–1301.</li> <li>Korn GA, <a href="/facts/Theresa_M._Korn/qKomWYbN">Korn TM</a> (1961). <i>Mathematical Handbook for Scientists and Engineers</i>. New York: McGraw-Hill. p. 182. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/59014456">59014456</a>.</li> <li>Zwillinger D (1992). <i>Handbook of Integration</i>. Boston, MA: Jones and Bartlett. p. 113. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-86720-293-9.</li> <li>Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)". <i>Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions</i> (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-02732-7.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/BisphericalCoordinates.html">MathWorld description of bispherical coordinates</a></li></ul>