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Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis separating its two foci. The foci are located at a distance 1 from the vertical z-axis. The red sphere is the σ=30° isosurface, the blue torus is the τ=0.5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the x-z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.996, -1.725, 1.911).
Illustration of toroidal coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis separating its two foci. two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. The foci are located at a distance 1 from the vertical z-axis. The red sphere is the σ=30° isosurface, the blue torus is the τ=0. 5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the x-z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0. 996, -1. 725, 1. 911).

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. Thus, the two foci F1 and F2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. The focal ring is also known as the reference circle.

Contents

Definition

The most common definition of toroidal coordinates (σ,τ,φ) is

x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \cos \phi
y = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \sin \phi
z = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}

where the σ coordinate of a point P equals the angle F1PF2 and the τ coordinate equals the natural logarithm of the ratio of the distances d1 and d2 to opposite sides of the focal ring

\tau = \ln \frac{d_{1}}{d_{2}}.

The coordinate ranges are -\pi<\sigma\le\pi and \tau\ge 0 and 0\le\phi < 2\pi.

Coordinate surfaces

Rotating this two-dimensional bipolar coordinate system about the vertical axis produces the three-dimensional toroidal coordinate system above. A circle on the vertical axis becomes the red sphere, whereas a circle on the horizontal axis becomes the blue torus.
Rotating this two-dimensional bipolar coordinate system about the vertical axis produces the three-dimensional toroidal coordinate system above. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. A circle on the vertical axis becomes the red sphere, whereas a circle on the horizontal axis becomes the blue torus. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

Surfaces of constant σ correspond to spheres of different radii

\left( x^{2} + y^{2} \right) +\left( z - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}

that all pass through the focal ring but are not concentric. The surfaces of constant τ are non-intersecting tori of different radii

z^{2} +\left( \sqrt{x^{2} + y^{2}} - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau}

that surround the focal ring. The centers of the constant-σ spheres lie along the z-axis, whereas the constant-τ tori are centered in the xy plane.

Inverse transformation

The (σ, τ, φ) coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle φ is given by the formula

\tan \phi = \frac{y}{x}

The cylindrical radius ρ of the point P is given by

ρ2 = x2 + y2

and its distances to the foci in the plane defined by φ is given by

d_{1}^{2} = (\rho + a)^{2} + z^{2}
d_{2}^{2} = (\rho - a)^{2} + z^{2}
Geometric interpretation of the coordinates σ and τ of a point P. Observed in the plane of constant azimuthal angle φ, toroidal coordinates are equivalent to bipolar coordinates. The angle σ is formed by the two foci in this plane and P, whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.
Geometric interpretation of the coordinates σ and τ of a point P. Observed in the plane of constant azimuthal angle φ, toroidal coordinates are equivalent to bipolar coordinates. two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system. The angle σ is formed by the two foci in this plane and P, whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.

The coordinate τ equals the natural logarithm of the focal distances

\tau = \ln \frac{d_{1}}{d_{2}}

whereas the coordinate σ equals the angle between the rays to the foci, which may be determined from the law of cosines

\cos \sigma = -\frac{4a^{2} - d_{1}^{2} - d_{2}^{2}}{2 d_{1} d_{2}}

where the sign of σ is determined by whether the coordinate surface sphere is above or below the x-y plane. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general

Scale factors

The scale factors for the toroidal coordinates σ and τ are equal

h_{\sigma} = h_{\tau} = \frac{a}{\cosh \tau - \cos\sigma}

whereas the azimuthal scale factor equals

h_{\phi} = \frac{a \sinh \tau}{\cosh \tau - \cos\sigma}

Thus, the infinitesimal volume element equals

dV= \frac{a^{3}\sinh \tau}{\left( \cosh \tau - \cos\sigma \right)^{3}} d\sigma d\tau d\phi

and the Laplacian is given by

\nabla^{2} \Phi =\frac{\left( \cosh \tau - \cos\sigma \right)^{3}}{a^{2}\sinh \tau} \left[ \sinh \tau \frac{\partial}{\partial \sigma}\left( \frac{1}{\cosh \tau - \cos\sigma}\frac{\partial \Phi}{\partial \sigma}\right) + \frac{\partial}{\partial \tau}\left( \frac{\sinh \tau}{\cosh \tau - \cos\sigma}\frac{\partial \Phi}{\partial \tau}\right) + \frac{1}{\sinh \tau \left( \cosh \tau - \cos\sigma \right)}\frac{\partial^{2} \Phi}{\partial \phi^{2}}\right]

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ,φ) by substituting the scale factors into the general formulae found in orthogonal coordinates. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2.

Toroidal Harmonics

Standard separation

The 3-variable Laplace equation

\nabla^2\Psi=0

admits solution via separation of variables in toroidal coordinates. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an Making the substitution

V=U\sqrt{\cosh\tau-\cos\sigma}

A separable equation is then obtained. A particular solution obtained by separation of variables is:

V= \sqrt{\cosh\tau-\cos\sigma}\,\,S_\nu(\sigma)T_{\mu\nu}(\tau)\Phi_\mu(\phi)\,

where each function is a linear combination of:

S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
T_{\mu\nu}(\tau)=P_{\nu-1/2}^\mu(\cosh\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\nu-1/2}^\mu(\cosh\tau)
\Phi_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}

Where P and Q are associated Legendre functions of the first and second kind. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an Note This article describes a very general class of functions These Legendre functions are often referred to as toroidal harmonics.

Toroidal harmonics have many interesting properties. If you make a variable substitution \,\!1<z=\cosh\eta\, then, for instance, with vanishing order (the convention is to not write the order when it vanishes) and \,\!n=0

Q_{-\frac12}(z)=\sqrt{\frac{2}{1+z}}K\left(\sqrt{\frac{2}{1+z}}\right)

and

P_{-\frac12}(z)=\frac{2}{\pi}\sqrt{\frac{2}{1+z}}K \left( \sqrt{\frac{z-1}{z+1}} \right)

where \,\!K and \,\!E are the complete elliptic integrals of the first and second kind respectively. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. The rest of the toroidal harmonics can be obtained, for instance, in terms of the complete elliptic integrals, by using recurrence relations for associated Legendre functions.

The classic applications of toroidal coordinates are in solving partial differential equations, e. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i g. , Laplace's equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation, for which toroidal coordinates does not allow a separation of variables. In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an The Helmholtz equation, named for Hermann von Helmholtz, is the Elliptic partial differential equation (\nabla^2 + k^2 A = 0 In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an Typical examples would be the electric potential and electric field of a conducting torus, or in the degenerate case, a conducting ring. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can

An alternative separation

Alternatively, a different substitution may be made (Andrews 2006)

V=\frac{U}{\sqrt{\rho}}

where

\rho=\sqrt{x^2+y^2}=\frac{\cosh\tau-\cos\sigma}{a\sinh\tau}

Again, a separable equation is obtained. A particular solution obtained by separation of variables is then:

V= \frac{a}{\rho}\,\,S_\nu(\sigma)T_{\mu\nu}(\tau)\Phi_\mu(\phi)\,

where each function is a linear combination of:

S_\nu(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
T_{\mu\nu}(\tau)=P_{\mu-1/2}^\nu(\coth\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{\mu-1/2}^\nu(\coth\tau)
\Phi_\mu(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}

Note that although the toroidal harmonics are used again for the T  function, the argument is cothτ rather than coshτ and the μ and ν indices are exchanged. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an This method is useful for situations in which the boundary conditions are independent of the spherical angle θ, such as the charged ring, an infinite half plane, or two parallel planes. For identities relating the toroidal harmonics with argument hyperbolic cosine with those of argument hyperbolic cotangent, see the Whipple formulae. In the theory of Special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression

See also


References

Bibliography

External links

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