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Apple (geometry)
In geometry, a lemon is a geometric shape, 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 (or arc). The surface of revolution of the complementary arc of the same circle, through the same axis, is called an apple. The apple and lemon together make up a self-crossing torus, the surface of revolution of the entire circle, with the apple as the outer shell of the torus and the lemon as its inner shell. The lemon forms the boundary of a convex set, while its surrounding apple is non-convex. The ball in North American Football (ball), football has a shape resembling a geometric lemon. However, although used with a related meaning in geometry, the term "football" is more commonly used to refer to a surface of revolution whose Gaussian curvature is positive and constant curvature, constant, formed from a more complicated curve than a circular arc. Alternatively, a foo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemon (geometry)
In geometry, a lemon is a geometric shape, 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 (or arc). The surface of revolution of the complementary arc of the same circle, through the same axis, is called an apple. The apple and lemon together make up a self-crossing torus, the surface of revolution of the entire circle, with the apple as the outer shell of the torus and the lemon as its inner shell. The lemon forms the boundary of a convex set, while its surrounding apple is non-convex. The ball in North American football has a shape resembling a geometric lemon. However, although used with a related meaning in geometry, the term "football" is more commonly used to refer to a surface of revolution whose Gaussian curvature is positive and constant, formed from a more complicated curve than a circular arc. Alternatively, a football may refer to a more abstract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CFL Football
The Canadian Football League (CFL; french: Ligue canadienne de football—LCF) is a professional sports league in Canada. The CFL is the highest level of competition in Canadian football. The league consists of nine teams, each located in a city in Canada. They are divided into two divisions: four teams in the East Division (CFL), East Division and five teams in the West Division (CFL), West Division. As of 2022, it features a 21-week season (sport), regular season in which each team plays 18 games with three bye (sports), bye weeks. This season traditionally runs from mid-June to early November. Following the regular season, six teams compete in the league's three-week #Playoffs, playoffs, which culminate in the Grey Cup championship game in late November. The Grey Cup is one of Canada's largest annual sports and television events. The CFL was officially named on January 19, 1958, upon the merger between the East Division (CFL), Interprovincial Rugby Football Union or "Big Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sears–Haack Body
The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a given body length and given volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl–Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947. The theory indicates that the wave drag scales as the square of the second derivative of the area distribution, D_\text \sim S''(x)2 (see full expression below), so for low wave drag it is necessary that S(x) be smooth. Thus, the Sears–Haack body is pointed at each end and grows smoothly to a maximum and then decreases smoothly toward the second point. Useful formulas The cross-sectional area of a Sears–Haack body is : S(x) = \frac x(1-x) = \pi R_\text^2 x(1-x), its volume is : V = \frac R_\text^2 L, its radius is : r(x) = R_\text x(1-x), the derivative (sl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature. Classification The Riemannian manifolds of constant curvature can be classified into the following three cases: * elliptic geometry – constant positive sectional curvature * Euclidean geometry – constant vanishing sectional curvature * hyperbolic geometry – constant negative sectional curvature. Properties * Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel \nabla \mathrm=0. * Every space of constant curvature is locally maximally symmetric, i.e. it has \frac n (n+1) number of local isometries, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the '' Theorema egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the '' Theorema egregium'' in 1827. Informal definition At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Football (ball)
A football is a ball inflated with air that is used to play one of the various sports known as football. In these games, with some exceptions, goals or points are scored only when the ball enters one of two designated goal-scoring areas; football games involve the two teams each trying to move the ball in opposite directions along the field of play. The first balls were made of natural materials, such as an inflated pig bladder, later put inside a leather cover, which has given rise to the American slang-term "pigskin". Modern balls are designed by teams of engineers to exacting specifications, with rubber or plastic bladders, and often with plastic covers. Various leagues and games use different balls, though they all have one of the following basic shapes: # a sphere: used in association football and Gaelic football # a prolate spheroid (elongated sphere) #* either with rounded ends: used in the rugby codes and Australian football #* or with more pointed ends: used in American ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. 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 spindle 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. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Torus-spindle
Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object that bears a defined relationship to a unit of measure used for calibration of measuring devices * Standard (timber unit), an obsolete measure of timber used in trade * Breed standard (also called bench standard), in animal fancy and animal husbandry * BioCompute Standard, a standard for next generation sequencing * ''De facto'' standard, product or system with market dominance * Gold standard, a monetary system based on gold; also used metaphorically for the best of several options, against which the others are measured * Internet Standard, a specification ratified as an open standard by the Internet Engineering Task Force * Learning standards, standards applied to education content * Standard displacement, a naval term describing the wei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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