- Conical coordinates
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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.
Contents
Basic definitions
The conical coordinates (r,μ,ν) are defined by
with the following limitations on the coordinates
- ν2 < c2 < μ2 < b2
Surfaces of constant r are spheres of that radius centered on the origin
- x2 + y2 + z2 = r2
whereas surfaces of constant μ and ν are mutually perpendicular cones
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
References
Bibliography
- Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 659. ISBN [[Special:BookSources/0-07-043316-X, LCCN 52-11515|0-07-043316-X, LCCN 52-11515]].
- Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 183–184. LCCN 55-10911.
- Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 179. LCCN 59-14456, ASIN B0000CKZX7.
- Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 991–100. LCCN 67-25285.
- Arfken G (1970). Mathematical Methods for Physicists (2nd ed. 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. ed.). New York: Springer-Verlag. pp. 37–40 (Table 1.09). ISBN 978-0387184302.
External links
Orthogonal coordinate systems Two dimensional orthogonal coordinate systems Three dimensional orthogonal coordinate systems Cartesian coordinate system • Cylindrical coordinate system • Spherical coordinate system • Parabolic cylindrical coordinates • Paraboloidal coordinates • Oblate spheroidal coordinates • Prolate spheroidal coordinates • Ellipsoidal coordinates • Elliptic cylindrical coordinates • Toroidal coordinates • Bispherical coordinates • Bipolar cylindrical coordinates • Conical coordinates • Flat-ring cyclide coordinates • Flat-disk cyclide coordinates • Bi-cyclide coordinates • Cap-cyclide coordinates • Concave bi-sinusoidal single-centered coordinates • Concave bi-sinusoidal double-centered coordinates • Convex inverted-sinusoidal spherically-aligned coordinates • Quasi-random-intersection cartesian coordinatesCategories:- Coordinate systems
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