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.

They describe a three-dimensional coordinate system in which the poloidal direction follows a small circular ring around the surface, while the toroidal direction follows a large circular ring around the torus, encircling the central void.

6 days ago · The toroidal coordinates are defined by x = (asinhvcosphi)/(coshv-cosu) (1) y = (asinhvsinphi)/(coshv-cosu) (2) z = (asinu)/(coshv-cosu), (3) where sinhz is the hyperbolic sine and coshz is the hyperbolic cosine. The coordinates satisfy u in [0,2pi), v in [0,infty), and...

Coordinate systems used in toroidal systems: (X, Y, Z) [1] , where [2] is called the toroidal angle (and not the cylindrical angle, at least not in the context of magnetic confinement). Cylindrical symmetry (symmetry under rotation over ) is referred to as axisymmetry. , where.

Jun 6, 2020 · The system of toroidal coordinates is orthogonal. The Lamé coefficients are: $$ L _ \sigma = L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } , $$ $$ L _ \phi = \frac{a ^ {2} \sinh ^ {2} \tau }{( \cosh \tau - \cos \sigma ) ^ {2} } . $$ The Laplace operator is:

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. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation.

In this document, we discuss coordinates that are useful for toroidal magnetic con nement systems with nested ux surfaces. Throughout, we let denote a poloidal angle (the short way around the torus) and let denote a toroidal angle (the long way around the torus). We assume both and are periodic with period 2 .

"Proper" toroidal coordinates are not frequently utilized for describing the physics of toroid ally confined plasmas. However, they have been employed occasionally (Shafranov 1960 and Fielding and Hitchon 1980). Furthermore, these coordinates have their own intrinsic value, since many problems in mathe­

May 26, 1999 · Toroidal Coordinates A system of Curvilinear Coordinates for which several different notations are commonly used. In this work ( , , ) is used, whereas Arfken (1970) uses ( , , ).

Toroidal coordinates are defined by three parameters: the radius r, and two angles: θ (around a circle in the x, y-plane) and φ (around the z-axis). Adjust the sliders to see how the surface depends on each parameter.