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

ν 2 = 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]

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

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

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

The infinitesimal Euclidean distance between two points in these coordinates

{\begin{aligned}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{aligned}}

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

$s_{12}^{2}=(\xi _{1}-\xi _{2})^{2}+(\psi _{1}-\psi _{2})^{2}.$

References

↑ 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. arXiv:0908.3143. Bibcode:2009ApPhL..95c4106D. 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&ndash, 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&ndash, 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

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[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. arXiv:0908.3143. Bibcode:2009ApPhL..95c4106D. doi:10.1063/1.3180815. ISSN 0003-6951.