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 , μ , ν ) {\displaystyle (r,\mu ,\nu )} are defined by

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)}}}}

with the following limitations on the coordinates

ν 2 < c 2 < μ 2 < b 2 . {\displaystyle \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 , {\displaystyle x^{2}+y^{2}+z^{2}=r^{2},}

whereas surfaces of constant μ {\displaystyle \mu } and ν {\displaystyle \nu } are mutually perpendicular cones

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}

and

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.}

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 (hr = 1), as in spherical coordinates. The scale factors for the two conical coordinates are

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)}}}}

and

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)}}}.}

References

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
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Orthogonal coordinate systems
Two dimensional
  • Cartesian
  • Polar (Log-polar)
  • Parabolic
  • Bipolar
  • Elliptic
Three dimensional