Conical coordinates

Coordinate surfaces of the conical coordinates. The constants b and c were chosen as 1 and 2, respectively. The red sphere represents r = 2, the blue elliptic cone aligned with the vertical z-axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) x-axis corresponds to ν² = 2/3. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.26, -0.78, 1.34). The elliptic cones intersect the sphere in spherical conics.

Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius r) and by two families of perpendicular elliptic cones, aligned along the z- and x-axes, respectively. The intersection between one of the cones and the sphere forms a spherical conic.

Basic definitions

The conical coordinates $(r,\mu ,\nu )$ are defined by

$x=\frac{r\mu \nu }{bc}$

$y=\frac{r}{b}\sqrt{\frac{({\mu }^2-b^2)({\nu }^2-b^2)}{(b^2-c^2)}}$

$z=\frac{r}{c}\sqrt{\frac{({\mu }^2-c^2)({\nu }^2-c^2)}{(c^2-b^2)}}$

with the following limitations on the coordinates

${\nu }^2<c^2<{\mu }^2<b^2.$

Surfaces of constant r are spheres of that radius centered on the origin

$x^2+y^2+z^2=r^2,$

whereas surfaces of constant $\mu$ and $\nu$ are mutually perpendicular cones

$\frac{x^2}{{\mu }^2}+\frac{y^2}{{\mu }^2-b^2}+\frac{z^2}{{\mu }^2-c^2}=0$

and

$\frac{x^2}{{\nu }^2}+\frac{y^2}{{\nu }^2-b^2}+\frac{z^2}{{\nu }^2-c^2}=0.$

In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.

Scale factors

The scale factor for the radius r is one (h_r = 1), as in spherical coordinates. The scale factors for the two conical coordinates are

$h_{\mu }=r\sqrt{\frac{{\mu }^2-{\nu }^2}{(b^2-{\mu }^2)({\mu }^2-c^2)}}$

and

$h_{\nu }=r\sqrt{\frac{{\mu }^2-{\nu }^2}{(b^2-{\nu }^2)(c^2-{\nu }^2)}}.$

Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 659. ISBN 0-07-043316-X. 
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 183–184. https://archive.org/details/mathematicsofphy0002marg. 
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 179. ASIN B0000CKZX7. 
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 991–100. 
• Arfken G (1970). Mathematical Methods for Physicists (2nd ed.). Orlando, FL: Academic Press. pp. 118–119. ASIN B000MBRNX4. 
• Moon P, Spencer DE (1988). "Conical Coordinates (r, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 37–40 (Table 1.09). ISBN 978-0-387-18430-2. 

External links

• MathWorld description of conical coordinates