Toroidal coordinates

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 F 1 {displaystyle F_{1}} and F 2 {displaystyle F_{2}} in bipolar coordinates become a ring of radius a {displaystyle a} in the x y {displaystyle xy} plane of the toroidal coordinate system; the z {displaystyle z} -axis is the axis of rotation. The focal ring is also known as the reference circle. 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 F 1 {displaystyle F_{1}} and F 2 {displaystyle F_{2}} in bipolar coordinates become a ring of radius a {displaystyle a} in the x y {displaystyle xy} plane of the toroidal coordinate system; the z {displaystyle z} -axis is the axis of rotation. The focal ring is also known as the reference circle. The most common definition of toroidal coordinates ( σ , τ , ϕ ) {displaystyle (sigma , au ,phi )} is where the σ {displaystyle sigma } coordinate of a point P {displaystyle P} equals the angle F 1 P F 2 {displaystyle F_{1}PF_{2}} and the τ {displaystyle au } coordinate equals the natural logarithm of the ratio of the distances d 1 {displaystyle d_{1}} and d 2 {displaystyle d_{2}} to opposite sides of the focal ring The coordinate ranges are − π < σ ≤ π {displaystyle -pi <sigma leq pi } and τ ≥ 0 {displaystyle au geq 0} and 0 ≤ ϕ < 2 π . {displaystyle 0leq phi <2pi .} Surfaces of constant σ {displaystyle sigma } correspond to spheres of different radii that all pass through the focal ring but are not concentric. The surfaces of constant τ {displaystyle au } are non-intersecting tori of different radii that surround the focal ring. The centers of the constant- σ {displaystyle sigma } spheres lie along the z {displaystyle z} -axis, whereas the constant- τ {displaystyle au } tori are centered in the x y {displaystyle xy} plane. 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