A toroid using a square.A torus is a type of toroid.
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface.[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topologicalgenus, g, of 1 or greater. The Euler characteristicχ of a g holed toroid is 2(1-g).[2]
The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and are not toroids.
Toroidal structures occur in both natural and synthetic materials.[3]
Equations
A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):
Square toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
Circular toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
The orbital period is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
The Method of Mechanical Theorems, also referred to as The Method, is one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures (centroid) and the law of the lever, which were demonstrated by Archimedes in On the Equilibrium of Planes.
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line, which may not intersect the generatrix. The surface created by this revolution and which bounds the solid is the surface of revolution.
A surface of revolution is a surface in Euclidean space created by rotating a curve one full revolution around an axis of rotation . The volume bounded by the surface created by this revolution is the solid of revolution.
Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs to obtain hollow solids of revolutions. This is in contrast to shell integration, which integrates along an axis perpendicular to the axis of revolution.
A Gabriel's horn is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Judgment Day. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
In mathematics, Pappus's centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A right circular cylinder is a cylinder whose generatrices are perpendicular to the bases. Thus, in a right circular cylinder, the generatrix and the height have the same measurements. It is also less often called a cylinder of revolution, because it can be obtained by rotating a rectangle of sides and around one of its sides. Fixing as the side on which the revolution takes place, we obtain that the side , perpendicular to , will be the measure of the radius of the cylinder.
In a toroidal fusion power reactor, the magnetic fields confining the plasma are formed in a helical shape, winding around the interior of the reactor. The safety factor, labeled q or q(r), is the ratio of the times a particular magnetic field line travels around a toroidal confinement area's "long way" (toroidally) to the "short way" (poloidally).
In geometry, a mylar balloon is a surface of revolution. While a sphere is the surface that encloses a maximal volume for a given surface area, the mylar balloon instead maximizes volume for a given generatrix arc length. It resembles a slightly flattened sphere.
In geometry, a lemon is a geometric shape that is constructed as the surface of revolution of a circular arc of angle less than half of a full circle rotated about an axis passing through the endpoints of the lens. The surface of revolution of the complementary arc of the same circle, through the same axis, is called an apple.
A toroidal planet is a hypothetical type of telluric exoplanet with a toroidal or doughnut shape. While no firm theoretical understanding as to how toroidal planets could form naturally is necessarily known, the shape itself is potentially quasistable, and is analogous to the physical parameters of a speculatively constructible megastructure in self-suspension, such as a Dyson Ring, ringworld, Stanford torus or Bishop Ring.
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