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 taco-shaped curves.

Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius r) and by two families of perpendicular cones, aligned along the z- and x-axes, respectively.

Basic definitions

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

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

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

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

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

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

and

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

Alternate Definition

An alternative set of conical coordinates have been derived^[1]

$\begin{align} \xi &= r\cos(\phi\sin\theta)\\ \psi &= r\sin(\phi\sin\theta) \\ \zeta &= \theta, \end{align}$

where $\{r, \theta, \phi\}$ are spherical polar coordinates. The corresponding inverse relations are

$\begin{align} r & = \sqrt{\xi^2 + \psi^2}\\ \phi & = \frac{1}{\sin\zeta}\arctan(\frac{\psi}{\xi}) \\ \theta & = \zeta. \end{align}$

The infinitesimal Euclidean distance between two points in these coordinates $\begin{align} ds^2 & = d\xi^2 + d\psi^2+(\xi^2 + \psi^2)(1+\arctan(\frac{\psi}{\xi})^2\cot\zeta^2) d\zeta^2 \\ &+ 2\psi\arctan(\frac{\psi}{\xi})\cot\zeta d\xi d\zeta - 2\xi\arctan(\frac{\psi}{\xi})\cot\zeta d\psi d\zeta. \end{align}$

If the path between any two points is constrained to surface of the cone given by $\zeta = \frac{\pi}{4}$ then the geodesic distance between any two points

$\{\xi_1,\psi_1,\zeta_1=\frac{\pi}{4}\}$ and $\{\xi_2,\psi_2,\zeta_2=\frac{\pi}{4}\}$ is

$s^2_{12} = (\xi_1-\xi_2)^2+(\psi_1-\psi_2)^2.$

References

1. Drake, Samuel Picton; Anderson, Brian D. O.; Yu, Changbin (2009-07-20). "Causal association of electromagnetic signals using the Cayley–Menger determinant". Applied Physics Letters 95 (3): 034106. doi:10.1063/1.3180815. ISSN 0003-6951.

Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 659. ISBN 0-07-043316-X. LCCN 52011515.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 183–184. LCCN 55010911.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 179. LCCN 59014456. ASIN B0000CKZX7.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 991–100. LCCN 67025285.
• 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