|
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. 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 topological genus, ''g'', of 1 or greater. The Euler characteristic χ of a ''g'' holed toroid is 2(1−''g''). 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. Equations A toroid is specified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Polyhedron
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topology (Mathematics), topological Genus (mathematics), genus () of 1 or greater. Notable examples include the Császár polyhedron, Császár and Szilassi polyhedron, Szilassi polyhedra. Variations in definition Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link (geometry), link of the vertex. For toroidal polyhedra, this manifold is an orientability, orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus. In this area, it is important to distinguish embedding, embedded toroidal polyhedra, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Inductors And Transformers
Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal (ring or donut) shape. They are passivity (engineering), passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated core, laminated iron, iron powder, or Ferrite (magnet), ferrite, around which wire is wound. Although closed-core inductors and transformers often use cores with a rectangular shape, the use of toroidal-shaped cores sometimes provides superior electrical performance. The advantage of the toroidal shape is that, due to its symmetry, the amount of magnetic flux that escapes outside the core (leakage flux) can be made low, potentially making it more efficient and making it emit less electromagnetic interference (EMI). Toroidal inductors and transformers are used in a wide range of electronic circuits: power supply, power supplies, inverters, and amplifiers, which in turn are used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroid By Zureks
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. 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 topological genus, ''g'', of 1 or greater. The Euler characteristic χ of a ''g'' holed toroid is 2(1−''g''). 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. Equations A toroid is specified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Propeller
A toroidal propeller is a type of propeller that is ring-shaped with each blade forming a closed loop. The propellers are significantly quieter at audible frequency ranges, between 20 Hz and 20 kHz, while generating comparable thrust to traditional propellers. In practice, toroidal propellers reduce noise pollution in both aviation and maritime transport. History In the centuries after Archimedes invented the Archimedes' screw, developments of propeller design led to the torus marine propeller, then described as a propeller featuring "double blades". It was invented in the early 1890s by Charles Myers from Manchester affiliated with Fawcett, Preston and Company. The design was successfully trialed on several English steam tugboats and passenger ferries at the time. In the 1930s, Friedrich Honerkamp patented a toroidal fan, and Rene Louis Marlet patented a toroidal aircraft propeller. The marine propeller was patented again in the late 1960s by Australian enginee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, 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. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a Lemon (geometry), spindle torus (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a ''solid torus'' include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels. Topological properties The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S^1 \times S^1, the ordinary torus. Since the disk D^2 is contractible, the solid torus has the homotopy type o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Revolution
A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersecting the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the ''solid of revolution''. Examples of surfaces of revolution generated by a straight line are cylinder (geometry), cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). Properties The sections of the surface of revolution made by planes through the axis are called ''meridional sections''. Any meridional section can be consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus's Centroid Theorem
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640. The first theorem The first theorem states that the surface area ''A'' of a surface of revolution generated by rotating a plane curve ''C'' about an axis external to ''C'' and on the same plane is equal to the product of the arc length ''s'' of ''C'' and the distance ''d'' traveled by the geometric centroid of ''C'': A = sd. For example, the surface area of the torus with minor radius ''r'' and major radius ''R'' is A = (2\pi r)(2\pi R) = 4\pi^2 R r. Proof A curve given by the positive function f(x) is bounded by two points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi (Greek alphabet, Greek lower-case letter chi (letter), chi). The Euler characteristic was originally defined for polyhedron, polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology (mathematics), homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
