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Conical coordinates
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<h2 id="basic-definitions">Basic definitions</h2> <p>The conical coordinates ( r , μ , ν ) {\displaystyle (r,\mu ,\nu )} are defined by </p> x = r μ ν b c {\displaystyle x={\frac {r\mu \nu }{bc}}} y = r b ( μ 2 − b 2 ) ( ν 2 − b 2 ) ( b 2 − c 2 ) {\displaystyle y={\frac {r}{b}}{\sqrt {\frac {\left(\mu ^{2}-b^{2}\right)\left(\nu ^{2}-b^{2}\right)}{\left(b^{2}-c^{2}\right)}}}} z = r c ( μ 2 − c 2 ) ( ν 2 − c 2 ) ( c 2 − b 2 ) {\displaystyle z={\frac {r}{c}}{\sqrt {\frac {\left(\mu ^{2}-c^{2}\right)\left(\nu ^{2}-c^{2}\right)}{\left(c^{2}-b^{2}\right)}}}} <p>with the following limitations on the coordinates </p> ν 2 < c 2 < μ 2 < b 2 . {\displaystyle \nu ^{2}<c^{2}<\mu ^{2}<b^{2}.} <p>Surfaces of constant r are spheres of that radius centered on the origin </p> x 2 + y 2 + z 2 = r 2 , {\displaystyle x^{2}+y^{2}+z^{2}=r^{2},} <p>whereas surfaces of constant μ {\displaystyle \mu } and ν {\displaystyle \nu } are mutually perpendicular cones </p> x 2 μ 2 + y 2 μ 2 − b 2 + z 2 μ 2 − c 2 = 0 {\displaystyle {\frac {x^{2}}{\mu ^{2}}}+{\frac {y^{2}}{\mu ^{2}-b^{2}}}+{\frac {z^{2}}{\mu ^{2}-c^{2}}}=0} <p>and </p> x 2 ν 2 + y 2 ν 2 − b 2 + z 2 ν 2 − c 2 = 0. {\displaystyle {\frac {x^{2}}{\nu ^{2}}}+{\frac {y^{2}}{\nu ^{2}-b^{2}}}+{\frac {z^{2}}{\nu ^{2}-c^{2}}}=0.} <p>In this coordinate system, both <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> and the <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a> are separable. </p> <h2 id="scale-factors">Scale factors</h2> <p>The scale factor for the radius r is one (<i>hr</i> = 1), as in <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>. The scale factors for the two conical coordinates are </p> h μ = r μ 2 − ν 2 ( b 2 − μ 2 ) ( μ 2 − c 2 ) {\displaystyle h_{\mu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\mu ^{2}\right)\left(\mu ^{2}-c^{2}\right)}}}} <p>and </p> h ν = r μ 2 − ν 2 ( b 2 − ν 2 ) ( c 2 − ν 2 ) . {\displaystyle h_{\nu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\nu ^{2}\right)\left(c^{2}-\nu ^{2}\right)}}}.} <h2 id="bibliography">Bibliography</h2> <ul><li><a href="/facts/Philip_M._Morse/CoY0Lmop">Morse PM</a>, <a href="/facts/Herman_Feshbach/0TKoFTNw">Feshbach H</a> (1953). <i>Methods of Theoretical Physics, Part I</i>. New York: McGraw-Hill. p. 659. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-043316-X. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/52011515">52011515</a>. {{cite book}}: ISBN / Date incompatibility (help)</li> <li><a href="/facts/Henry_Margenau/18Bs6tP0">Margenau H</a>, Murphy GM (1956). <a href="https://archive.org/details/mathematicsofphy0002marg"><i>The Mathematics of Physics and Chemistry</i></a>. New York: D. van Nostrand. pp. <a href="https://archive.org/details/mathematicsofphy0002marg/page/183">183</a>–184. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/55010911">55010911</a>.</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. 179. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/59014456">59014456</a>. ASIN B0000CKZX7.</li> <li>Sauer R, Szabó I (1967). <i>Mathematische Hilfsmittel des Ingenieurs</i>. New York: Springer Verlag. pp. 991–100. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/67025285">67025285</a>.</li> <li>Arfken G (1970). <i>Mathematical Methods for Physicists</i> (2nd ed.). Orlando, FL: Academic Press. pp. 118–119. ASIN B000MBRNX4.</li> <li>Moon P, Spencer DE (1988). "Conical Coordinates (r, θ, λ)". <i>Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions</i> (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 37–40 (Table 1.09). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-18430-2.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/ConicalCoordinates.html">MathWorld description of conical coordinates</a></li></ul>