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Toroid

A toroid is a doughnut-shaped geometric figure generated by revolving a closed about an in its plane that does not intersect the , enclosing a solid volume often resembling a ring or when the curve is circular. In physics and engineering, the term commonly refers to a , consisting of multiple turns of wire wound tightly around a ring-shaped to form a closed loop, which produces a concentrated confined within the . Geometrically, the toroid generalizes the —the specific case where a circle is rotated about an axis in its plane at a fixed from —yielding a with applications in , where it serves as a fundamental non-spherical shape. The and surface area of a toroidal solid can be calculated using integrals based on the generating curve, with the standard having V = 2\pi^2 R r^2 and surface area A = 4\pi^2 R r, where R is the from of the tube to the center of the and r is the tube radius. In , a functions as a bent , offering cylindrical that results in a \mathbf{B} inside the approximately and equal to B = \frac{\mu_0 N I}{2\pi R} (where \mu_0 is the permeability of free space, N is the number of turns, I is the current, and R is the mean radius), while the field outside is ideally zero due to the closed path enclosing no net current for external Amperian loops. This containment of the field minimizes leakage and interference, making toroids superior to straight solenoids for applications requiring efficient energy storage and transfer. Toroidal coils are widely employed in as inductors and transformers due to their compact design, low , and high efficiency. Key uses include power supplies, audio amplifiers, and devices like MRI machines, where they enable precise and voltage stepping without external field disruptions. In advanced physics, toroidal geometries model confinement in reactors, such as tokamaks, leveraging their ability to sustain stable, looping magnetic fields.

Definition and Basic Concepts

Mathematical Definition

In mathematics, a toroid is defined as a obtained by rotating a closed about an lying in its plane that does not intersect the , thereby generating a surface featuring a central . This construction distinguishes the toroid from simpler surfaces like spheres or cylinders, as the non-intersecting ensures the resulting encloses an empty interior . The generating is typically a or a , with the yielding the most common form known as the . The key parameters characterizing a standard toroid are the major radius R, defined as the distance from the axis of rotation to the center of the generating , and the minor r, the of the generating curve itself. For a non-self-intersecting ring-shaped toroid, the condition R > r must hold, preventing the surface from crossing itself. Visually, this produces a doughnut-like shape with a genus of , meaning it possesses exactly one hole, a that underscores its fundamental role in surface classification. The term "toroid" was coined in the late , with its first recorded use in , to generalize surfaces beyond the specific circular case of the . For the basic toroid surface generated by a , the implicit equation in three-dimensional Cartesian coordinates is given by \left( \sqrt{x^2 + y^2} - R \right)^2 + z^2 = r^2, where the axis of rotation aligns with the z-axis. This equation encapsulates the surface's geometry, with points satisfying it lying on the toroid.

Relation to the Torus

The represents a specific instance of a , generated by revolving a of (the ) around an in its that lies outside the itself, resulting in a with uniform tube thickness. This configuration distinguishes the from more general , where the generating curve may deviate from a , leading to variable cross-sections. Topologically, the torus and the standard toroid share identical properties, including an of 0 and a of 1, which characterize them as orientable surfaces with a single hole. These invariants ensure that the standard can be embedded in three-dimensional without self-intersection, provided the major radius exceeds the minor radius to avoid overlap. In mathematical usage, the term "torus" is typically reserved for this ideal circular surface, particularly in and , where it serves as the fundamental example of a genus-1 surface. In contrast, "toroid" encompasses a broader class, including non-circular surfaces of revolution, solid volumes with ring-like topology (such as the ), and even polyhedral approximations that are topologically equivalent to a . This distinction arises from the : "torus" derives from the Latin torus, meaning "swelling," "bulge," or "cushion," originally referring to architectural moldings and adopted in mathematics around the ; "toroid," coined later in 1886 from New Latin roots, emphasizes the ring-like (-oid) form more generally. A example of the appears in toroidal coordinates (\eta, \tau, \phi), where surfaces of constant \tau define toroidal surfaces, with the standard given by \tau = constant, parameterized as: \begin{align*} x &= \frac{a \sinh \tau \cos \phi}{\cosh \tau - \cos \eta}, \\ y &= \frac{a \sinh \tau \sin \phi}{\cosh \tau - \cos \eta}, \\ z &= \frac{a \sin \eta}{\cosh \tau - \cos \eta}, \end{align*} where a > 0 is a scale factor, \eta \in [0, 2\pi), \tau \in [0, \infty), and \phi \in [0, 2\pi).

Geometric Properties

Parametric Equations

The parametric equations for the surface of a standard toroid, generated by revolving a circle of radius r (the minor radius) centered at (R, 0, 0) in the xz-plane around the z-axis, where R is the major radius, are given by \begin{align*} x &= (R + r \cos \theta) \cos \phi, \\ y &= (R + r \cos \theta) \sin \phi, \\ z &= r \sin \theta, \end{align*} with angular parameters \theta, \phi \in [0, 2\pi). These equations parameterize points on the toroidal surface, enabling computational generation and visualization in three-dimensional space. The derivation begins with the parametric form of the generating circle in the xz-plane: x = R + r \cos \theta, z = r \sin \theta, y = 0. Revolving this circle around the z-axis introduces the azimuthal angle \phi, transforming the x- and y-coordinates via rotation: x' = x \cos \phi, y' = x \sin \phi, while z remains unchanged, yielding the full parametric equations. An equivalent implicit equation for the circular toroid, eliminating the parameters, is \left( \sqrt{x^2 + y^2} - R \right)^2 + z^2 = r^2. This form describes the set of points at a fixed distance r from a circle of radius R in the xy-plane. For non-circular generating curves, the parametric equations adapt by replacing the circular terms with the appropriate parametrization of the cross-section. For an elliptical cross-section with semi-major axis a along x and semi-minor axis b along z, the equations become x = (R + a \cos \theta) \cos \phi, y = (R + a \cos \theta) \sin \phi, z = b \sin \theta. Similarly, for polygonal paths, the generating curve can be parameterized piecewise using linear segments between vertices of the , then revolved around the axis to form the toroid surface. The shape of the toroid varies with the ratio of R to r. When R > r, the surface forms a non-self-intersecting toroid. As R approaches r, the inner pinches to a point, creating a horn toroid at R = r. For R < r, self-intersections occur along the inner surface, resulting in a spindle toroid. For example, with R = 2 and r = 1, the surface remains a smooth without overlap; but with R = 1 and r = 2, the parameters produce a self-intersecting spindle shape.

Surface Area and Volume Calculations

The surface area and volume of a toroid are computed using its parametric equations, which provide the foundation for integration over the surface or solid. For the standard ring toroid, generated by revolving a circle of radius r (tube radius) around an axis at distance R (major radius) from its center, with R > r, explicit closed-form formulas exist. These derive from either or direct surface and volume integrals. Pappus's centroid theorem, attributed to the Greek mathematician Pappus of (c. 290–350 AD), simplifies the calculations for surfaces and solids of revolution. For the surface area S, the theorem states that S equals the arc length of the generating curve times the distance traveled by its . The generating circle has arc length $2\pi r, and its centroid travels a circular path of radius R with length $2\pi R, yielding S = (2\pi r)(2\pi R) = 4\pi^2 R r. Similarly, for the volume V of the solid toroid, the theorem uses the area of the generating disk \pi r^2 and the same centroid path length, giving V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2. Direct derivations via confirm these results and offer insight into the . The equations for the surface are \mathbf{r}(\theta, \phi) = ((R + r \cos \theta) \cos \phi, (R + r \cos \theta) \sin \phi, r \sin \theta), where $0 \leq \theta, \phi < 2\pi. The surface area element is dS = \| \mathbf{r}_\theta \times \mathbf{r}_\phi \| \, d\theta \, d\phi = r (R + r \cos \theta) \, d\theta \, d\phi. Integrating over the parameter domain gives S = \int_0^{2\pi} \int_0^{2\pi} r (R + r \cos \theta) \, d\theta \, d\phi = 4\pi^2 R r. For the volume, methods like the washer or shell integration, or the divergence theorem applied to the solid, yield V = 2\pi^2 R r^2; Pappus provides an equivalent non-integral approach emphasizing the centroid's role. These formulas assume a circular cross-section and R > r to ensure a non-self-intersecting ring without a hole . For non-circular toroids, where the generating is an arbitrary closed , closed forms are unavailable, and the surface area requires numerical of the general surface S = \iint_D \| \mathbf{r}_u \times \mathbf{r}_v \| \, du \, dv over the parameter domain D, often using quadrature methods or . Volume computations similarly rely on numerical or techniques for the enclosed region. When R < r, as in spindle toroids, the surface self-intersects, invalidating the standard formulas for the enclosed volume; modifications, such as subtracting overlapping regions or using inclusion-exclusion, are needed to compute the actual solid volume accurately.

Types and Variations

Ring Torus

The ring torus, also known as the standard or common torus, is generated by revolving a of radius r about an coplanar with the circle but displaced by a distance R from its center, where R > r to ensure the surface does not self-intersect. This condition prevents the generating circle from crossing the of , producing a smooth, doughnut-like ring without overlaps or singularities. The resulting surface is a characterized by its around the central . Geometrically, the ring torus features a tube whose centerline maintains a constant distance R from the axis of , forming a circular path of major R. The cross-section to this centerline is uniformly circular with r, ensuring consistent thickness throughout the structure. This uniformity contributes to the ring torus's topological of 1, distinguishing it as a compact orientable surface with a single hole. Visually, the absence of intersections or cusps renders it stable and aesthetically simple, ideal for physical prototypes and simulations where smooth is essential. A key special property of the ring torus is its central role in standard coordinates, a curvilinear system where surfaces of constant coordinate values describe shapes aligned with the ring geometry. These coordinates facilitate analysis in fields requiring , such as and around objects. Regarding mappings, certain abstract representations of the admit isometric immersions into higher-dimensional spheres under specific metric conditions, though the embedded ring torus in exhibits varying . For quantitative aspects, substituting torus parameters into surface area formulas—typically derived or —yields precise closed-form results: area , of . These expressions highlight the of the shape, with area growing linearly in both radii quadratically in the minor radius.

Spindle and Horn Tori

Spindle tori and horn tori represent degenerate cases of the standard generated by revolving a of r around an at distance R from the 's , where R \leq r. In these configurations, the generating intersects or touches the of , leading to self-intersections or singularities that distinguish them from the non-degenerate ring . The spindle arises when R < r, causing the revolved surface to self-intersect and form a characteristic "apple" shape. The self-intersection occurs at two singular points located on the of rotation, symmetric about the equatorial plane, where the surface crosses itself without a well-defined tangent plane. These points correspond to the locations where the generating intersects the before . The horn torus forms when R = r, with the generating circle tangent to the axis at a single point. This results in a horn-like cusp singularity at the origin, where the inner equator degenerates to a point and the surface develops a conical pinch without a tangent plane. The cusp manifests as a sharp spike directed toward the axis. The parametric equations for both spindle and horn tori are identical to those of the ring torus: \begin{align*} x &= (R + r \cos \theta) \cos \phi, \\ y &= (R + r \cos \theta) \sin \phi, \\ z &= r \sin \theta, \end{align*} where \theta, \phi \in [0, 2\pi). For R \leq r, the radial term R + r \cos \theta becomes non-positive for certain \theta, specifically zero when \cos \theta = -R/r. This yields self-intersection points at (x, y, z) = (0, 0, \pm r \sqrt{1 - (R/r)^2}) for the spindle torus. At the bifurcation point R = r, these two points coincide at the origin, transitioning from double singularities to a single cusp. This parametric adjustment highlights a geometric bifurcation as R decreases through r, closing the central hole and introducing singularities. Geometrically, the axis intersection alters the embedding while preserving the topological genus of 1, equivalent to a smooth torus, though the resulting surfaces are immersed with singularities that prevent them from being smooth manifolds. In the spindle torus, the self-intersections divide the surface into inner ("lemon") and outer ("apple") regions connected at the singular points, confining geodesics to disjoint components. The horn torus features a single connected surface with the cusp acting as an infinite potential barrier for geodesics, eliminating unbound paths. These properties make spindle and horn tori valuable in analyzing singularity formation and geodesic behavior on surfaces of revolution.

Square and Other Non-Circular Toroids

A square toroid is generated by revolving a square closed curve around an axis lying in the plane of the square but external to it, typically parallel to two opposite sides, yielding a surface composed of four distinct portions: two cylindrical surfaces from the sides parallel to the axis and two annular disk-like surfaces from the perpendicular sides. This configuration produces a frame-like structure with sharp edges at the corners of the original square, distinguishing it from the smooth circular torus. The parametric representation for a square toroid requires a piecewise definition corresponding to the four linear segments of the square generator. For a square of side length s centered at distance R from the revolution axis (with R > s/2), the parameter \theta (revolution angle, $0 \leq \theta < 2\pi) is used alongside a segment parameter \phi (0 to 1 for each side). For the bottom side (z = -s/2, x from R - s/2 to R + s/2), the equations are x = (R + (s \phi - s/2)) \cos \theta, y = (R + (s \phi - s/2)) \sin \theta, z = -s/2; similar linear adjustments apply to the other sides by shifting in z or x. This piecewise linear form in the generator parameter introduces discontinuities in the first derivative at corners, complicating differentiability compared to circular cases. Properties of the square toroid exhibit increased complexity in curvature distribution, with zero Gaussian curvature along the cylindrical portions and concentrated strains at the revolved corners, often requiring approximations like filleting in practical models. Volume calculations leverage , yielding V = 2\pi R s^2 for a solid square toroid, where R is the centroid distance (mean radius) and s^2 the cross-sectional area, simplifying integration over the non-smooth boundary. Surface area approximations sum contributions from each revolved segment: $2\pi (R - s/2) s + 2\pi (R + s/2) s + 2 \times 2\pi R s for the two cylinders and two annuli, respectively, though exact values demand careful handling of edge effects. Other non-circular toroids include elliptical variants, formed by revolving an ellipse around an external axis coplanar with it. An elliptical toroid has parametric equations x = (c + a \cos v) \cos u, y = (c + a \cos v) \sin u, z = b \sin v, where u, v \in [0, 2\pi), c > a is the distance from the axis to the ellipse center, a and b are the semi-major and semi-minor axes of the elliptical cross-section (with eccentricity e = \sqrt{1 - (b/a)^2} for a > b). Increasing eccentricity stretches the cross-section, altering the surface's aspect ratio and enhancing asymmetry in curvature, which affects metrics like mean curvature more variably than in circular toroids. Volume follows Pappus's theorem as V = 2\pi c \cdot \pi a b, emphasizing the role of the elliptical area \pi a b. Historically, square and other non-circular toroids served as simplified models in early for toroidal coils and structural frames, facilitating manual calculations of or stresses before tools enabled precise circular designs; for instance, square cross-sections approximated uniform in pre-digital prototypes.

Physical Applications

In

In , a toroid serves as for a , formed by winding insulated wire uniformly around a ring-shaped to generate a confined primarily within the interior. This configuration exploits the closed-loop of the toroid to produce a nearly uniform inside while minimizing the field outside, approaching zero due to the symmetric cancellation of lines. The concept of the traces its origins to the early 19th century, when developed the first practical device in 1831, known as the "induction ring," which consisted of an wound with two separate coils to demonstrate mutual . This invention laid the groundwork for efficient inductive devices by showing how a changing current in one coil could induce voltage in another without direct electrical connection. For a tightly wound carrying I with N total turns, the strength B at a radial r from the center (where r is between the inner and outer radii of the toroid) is given by Ampère's law as B = \frac{\mu_0 N I}{2\pi r}, where \mu_0 is the permeability of free space; this field varies inversely with r but remains azimuthal and contained within the core for an ideal case. Toroidal solenoids find primary applications in transformers and inductors, where the primary winding carries the input and a secondary winding captures the induced voltage, enabling efficient power transfer across voltage levels. Compared to linear solenoids, toroidal designs offer significant advantages, including reduced magnetic leakage —often less than 1% outside the core—due to the closed magnetic path, which minimizes and improves energy efficiency in compact devices. To enhance performance, toroidal cores are typically constructed from ferromagnetic materials such as laminated iron, ferrite, or powdered iron, which increase the \mu_r (often 100–10,000 times that of air) and thereby amplify the and without saturation at typical operating currents. The self- L of such a toroid, assuming a thin with mean R and cross-sectional area A, is approximated by L = \frac{\mu_0 N^2 A}{2\pi R}, where the formula scales with N^2 to reflect the mutual reinforcement of flux from multiple turns; for ferromagnetic cores, \mu_0 is replaced by \mu = \mu_r \mu_0 to account for the material's enhancement.

In Plasma Physics and Fusion

In plasma physics and fusion research, toroids play a central role in magnetic confinement devices designed to sustain high-temperature plasmas for controlled nuclear fusion. The toroidal geometry, resembling a doughnut shape, allows for the creation of nested magnetic flux surfaces that prevent plasma particles from escaping due to the Lorentz force acting on charged ions and electrons. This configuration is essential for achieving the extreme conditions required for fusion reactions, where plasma temperatures exceed 100 million degrees Celsius and densities must be maintained long enough for deuterium-tritium nuclei to fuse. Devices like tokamaks and stellarators exploit this shape to generate helical magnetic field lines that wrap around the plasma, providing stability against both radial and axial losses. Tokamaks confine plasma using a combination of toroidal and poloidal magnetic fields: the toroidal field B_{\tor} is produced by external coils encircling the torus, while the poloidal field B_{\pol} arises from an induced plasma current flowing toroidally. This interplay creates helical field lines, characterized by the safety factor q = \frac{r B_{\tor}}{R B_{\pol}}, where r is the minor radius and R is the major radius of the toroid; q represents the number of toroidal turns a field line makes per poloidal turn and must typically exceed 2 at the plasma edge to avoid disruptive instabilities. Stellarators, in contrast, achieve similar helical confinement without relying on plasma currents, using twisted external coils to produce both field components directly, which enhances steady-state operation but increases design complexity. The toroidal curvature introduces geometric effects into the magnetohydrodynamic (MHD) equations, such as the Shafranov shift, which displaces the plasma equilibrium away from the geometric center and influences pressure gradients. Key challenges in toroidal confinement include MHD instabilities driven by the non-uniformity of the toroid. instabilities occur when the safety factor [q](/page/Q) falls below critical values, causing helical deformations of the plasma column that can lead to sudden reconfiguration or disruption. Ballooning instabilities, exacerbated by the toroid's varying (stronger on the inner side), arise from adverse pressure gradients in regions of weak magnetic , potentially ejecting blobs outward. These modes are analyzed through the MHD , where toroidal effects modify the stability criteria compared to cylindrical approximations. Historical milestones trace back to the in the 1950s, where and proposed the concept, leading to the first operational device, T-1, in 1958 at the , which demonstrated plasma confinement times of milliseconds. The ITER project exemplifies modern toroidal fusion efforts, featuring a large toroidal vacuum chamber with a major radius of 6.2 meters to house a volume of 830 cubic meters, aiming to produce 500 MW of from 50 MW input, with first plasma now scheduled for 2034–2035 as of 2025. This international endeavor integrates advanced superconducting magnets to sustain fields up to 13 , addressing scalability issues in prior devices; recent milestones include the completion of the central solenoid in 2025. Related compact toroids, such as spheromaks and field-reversed configurations (FRCs), lack a central column; FRCs are prolate configurations that reverse the field at the edge while maintaining poloidal dominance inside, enabling higher values ( pressure over magnetic pressure) for efficient confinement in smaller volumes. Spheromaks are formed by magnetized guns and exhibit force-free equilibria with both toroidal and poloidal fields, though they face tilt instabilities absent in axisymmetric tokamaks.