1.
Maclaurin spheroid
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A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes, in other words, an ellipsoid with two equal semi-diameters. If the ellipse is rotated about its axis, the result is a prolate spheroid. If the ellipse is rotated about its axis, the result is an oblate spheroid. If the generating ellipse is a circle, the result is a sphere, because of the combined effects of gravity and rotation, the Earths shape is not quite a sphere but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere, the current World Geodetic System model uses a spheroid whose radius is 6,378.137 km at the equator and 6,356.752 km at the poles. The semi-major axis a is the radius of the spheroid. There are two cases, c < a, oblate spheroid c > a, prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area S o b l a t e =2 π a 2 where e 2 =1 − c 2 a 2. The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. A prolate spheroid with c > a has surface area S p r o l a t e =2 π a 2 where e 2 =1 − a 2 c 2. The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a and these formulas are identical in the sense that the formula for Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity, both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid is 4π/3a2c ≈4. 19a2c, if A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is π/6A2C ≈0. 523A2C. Both of these curvatures are always positive, so every point on a spheroid is elliptic. These are just two of different parameters used to define an ellipse and its solid body counterparts. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal, deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects. An extreme example of a planet in science fiction is Mesklin, in Hal Clements novel Mission of Gravity. The prolate spheroid is the shape of the ball in several sports, several moons of the Solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids
2.
Quadric
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In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is an hypersurface in a space, and is defined as the zero set of an irreducible polynomial of degree two in D +1 variables. When the defining polynomial is not absolutely irreducible, the set is generally not considered as a quadric. The values Q, P and R are often taken to be real numbers or complex numbers. A quadric is an algebraic variety, or, if it is reducible. Quadrics may also be defined in spaces, see Quadric. Quadrics in the Euclidean plane are those of dimension D =1, in this case, one talks of conic sections, or conics. In three-dimensional Euclidean space, quadrics have dimension D =2 and they are classified and named by their orbits under affine transformations. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, each of these 17 normal forms correspond to a single orbit under affine transformations. In three cases there are no points, ε1 = ε2 =1, ε1 =0, ε2 =1. In one case, the cone, there is a single point. If ε4 =1, one has a line, for ε4 =0, one has a double plane. For ε4 =1, one has two intersecting planes and it remains nine true quadrics, a cone and three cylinders and five non-degenerated quadrics, which are detailed in the following table. In a three-dimensional Euclidean space there are 17 such normal forms, of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all, the quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the coordinates on RD+1 are one introduces new coordinates on RD+2 related to the original coordinates by x i = X i / X0. In the new variables, every quadric is defined by an equation of the form Q = ∑ i j a i j X i X j =0 where the coefficients aij are symmetric in i and j. Regarding Q =0 as an equation in projective space exhibits the quadric as an algebraic variety
3.
Surface (mathematics)
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In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line, there are several more precise definitions, depending on the context and the mathematical tools that are used for the study. Often, a surface is defined by equations that are satisfied by the coordinates of its points and this is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface, if the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 −1 =0. A surface may also be defined as the image, in space of dimension at least 3. In this case, one says that one has a parametric surface, for example, the unit sphere may be parametrized by the Euler angles, also called longitude u and latitude v by x = cos cos y = sin cos z = sin . Parametric equations of surfaces are often irregular at some points, for example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles. For the remaining two points, one has cos v =0, and the longitude u may take any values, also, there are surfaces for which there cannot exits a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations and this allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, in classical geometry, a surface is generally defined as a locus of a point or a line. A ruled surface is the locus of a moving line satisfying some constraints, in modern terminology, a surface is a surface. In this article, several kinds of surfaces are considered and compared, a non-ambiguous terminology is thus necessary for distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two and we call differential surfaces the surfaces that are differentiable manifolds. Every differential surface is a surface, but the converse is false. For simplicity, unless stated, surface will mean a surface in the Euclidean space of dimension 3 or in R3. A surface, that is not supposed to be included in another space, is called an abstract surface, the graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface, a plane is together an algebraic surface and a differentiable surface
4.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
5.
Ellipsoid
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An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes
6.
American football
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The offense must advance at least ten yards in four downs, or plays, or else they turn over the football to the opposing team, if they succeed, they are given a new set of four downs. Points are primarily scored by advancing the ball into the teams end zone for a touchdown or kicking the ball through the opponents goalposts for a field goal. The team with the most points at the end of a game wins, American football evolved in the United States, originating from the sports of association football and rugby football. The first game of American football was played on November 6,1869, during the latter half of the 1870s, colleges playing association football switched to the Rugby Union code, which allowed carrying the ball. American football as a whole is the most popular sport in the United States, Professional football and college football are the most popular forms of the game, with the other major levels being high school and youth football. As of 2012, nearly 1.1 million high school athletes and 70,000 college athletes play the sport in the United States annually, almost all of them men, in the United States, American football is referred to as football. The term football was established in the rulebook for the 1876 college football season. The terms gridiron or American football are favored in English-speaking countries where other codes of football are popular, such as the United Kingdom, Ireland, New Zealand, American football evolved from the sports of association football and rugby football. What is considered to be the first American football game was played on November 6,1869 between Rutgers and Princeton, two college teams, the game was played between two teams of 25 players each and used a round ball that could not be picked up or carried. It could, however, be kicked or batted with the feet, hands, head or sides, Rutgers won the game 6 goals to 4. Collegiate play continued for years in which matches were played using the rules of the host school. Representatives of Yale, Columbia, Princeton and Rutgers met on October 19,1873 to create a set of rules for all schools to adhere to. Teams were set at 20 players each, and fields of 400 by 250 feet were specified, Harvard abstained from the conference, as they favored a rugby-style game that allowed running with the ball. An 1875 Harvard-Yale game played under rugby-style rules was observed by two impressed Princeton athletes and these players introduced the sport to Princeton, a feat the Professional Football Researchers Association compared to selling refrigerators to Eskimos. Princeton, Harvard, Yale and Columbia then agreed to play using a form of rugby union rules with a modified scoring system. These schools formed the Intercollegiate Football Association, although Yale did not join until 1879, the introduction of the snap resulted in unexpected consequences. Prior to the snap, the strategy had been to punt if a scrum resulted in bad field position, however, a group of Princeton players realized that, as the snap was uncontested, they now could hold the ball indefinitely to prevent their opponent from scoring. In 1881, both teams in a game between Yale-Princeton used this strategy to maintain their undefeated records, each team held the ball, gaining no ground, for an entire half, resulting in a 0-0 tie
7.
Rugby football
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Rugby is a type of football developed at Rugby School in Rugby, Warwickshire, one of many versions of football played at English public schools in the 19th century. The two main types of rugby are rugby league and rugby union, although rugby league initially used rugby union rules, they are now wholly separate sports. Following the 1895 split in rugby football, the two rugby league and rugby union differed in administration only. Soon the rules of rugby league were modified, resulting in two different forms of rugby. After 100 years, in 1995 rugby union joined rugby league, the Greeks and Romans are known to have played many ball games, some of which involved the use of the feet. These games appear to have resembled rugby football, the Roman politician Cicero describes the case of a man who was killed whilst having a shave when a ball was kicked into a barbers shop. Roman ball games already knew the air-filled ball, the follis, episkyros is recognised as an early form of football by FIFA. In 1871, English clubs met to form the Rugby Football Union, in 1892, after charges of professionalism were made against some clubs for paying players for missing work, the Northern Rugby Football Union, usually called the Northern Union, was formed. The existing rugby union authorities responded by issuing sanctions against the clubs, players, after the schism, the separate clubs were named rugby league and rugby union. Rugby union is both a professional and amateur game, and is dominated by the first tier unions, Argentina, Australia, England, France, Ireland, Italy, New Zealand, Scotland, South Africa and Wales. Rugby Union is administered by World Rugby, whose headquarters are located in Dublin and it is the national sport in New Zealand, Wales, Fiji, Samoa, Tonga and Madagascar, and is the most popular form of rugby globally. The Olympic Games have admitted the seven-a-side version of the game, known as Rugby sevens, there was a possibility sevens would be a demonstration sport at the 2012 London Olympics but many sports including sevens were dropped. In Canada and the United States, rugby union evolved into gridiron football, during the late 1800s, the two forms of the game were very similar, but numerous rule changes have differentiated the gridiron-based game from its rugby counterpart. Rugby league is also both a professional and amateur game, administered on a level by the Rugby League International Federation. International Rugby League is dominated by Australia, England and New Zealand, in Papua New Guinea it is the national sport. Other nations from the South Pacific and Europe also play in the Pacific Cup, distinctive features common to both rugby codes include the oval ball and throwing the ball forward is not allowed, so that players can gain ground only by running with the ball or by kicking it. As the sport of rugby league moved further away from its counterpart, rule changes were implemented with the aim of making a faster-paced. League players may not contest possession after making a tackle, play is continued with a play-the-ball, in league, if the team in possession fails to score before a set of six tackles, it surrenders possession