"the Equator the only great circle. By contrast, all meridians of longitude, paired with their opposite meridian in the other hemisphere, form great circles. The diameter of the sphere which passes through the center of the circle is called its axis and the endpoints of this diameter are called its poles. A circle of a sphere can also be defined as the set of points at a given angular distance from a given pole. When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:"
"is to a perigon."" A spherical wedge of ""α"" = radians (180°) is called a ""hemisphere"", while a spherical wedge of ""α"" = 2 radians (360°) constitutes a complete ball. The volume of a spherical wedge can be intuitively related to the ""AB"" definition in that while the volume of a ball of radius ""r"" is given by ""r"", the volume a spherical wedge of the same radius ""r"" is given by Extrapolating the same principle and considering that the surface area of a sphere is given by 4""r"", it can be seen that the surface area of the lune"
"Sphere A sphere (from Greek σφαῖρα — ""sphaira"", ""globe, ball"") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a ""circle"" circumscribes its ""disk""). Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance from a given point, but in a three-dimensional space. This distance is the radius of the ball, which is made up from all points with a distance less than from the given point,"
"Semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term ""semicircle"" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each"
"sphere with most areas being recognizable In transverse aspect, one hemisphere becomes the Adams hemisphere-in-a-square projection (the pole is placed at the corner of the square). Its four singularities are at the North Pole, the South Pole, on the equator at 25°W, and on the equator at 155°E, in the Arctic, Atlantic, and Pacific oceans, and in Antarctica. That great circle divides the traditional Western and Eastern hemispheres. In oblique aspect (45 degrees) of one hemisphere becomes the Guyou hemisphere-in-a-square projection (the pole is placed in the middle of the edge of the square). Its four singularities are at 45"
"is also called tractricoid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane. The name ""pseudosphere"" comes about because it is a two-dimensional surface of constant negative curvature just like a sphere with positive Gauss curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively"
"Semiosphere Semiosphere is the sphere of semiosis in which sign processes operate in the set of all interconnected Umwelten. The concept was coined by Yuri Lotman in 1984 and is now applied to many fields, including cultural semiotics generally, biosemiotics, zoosemiotics, geosemiotics, etc. The concept is treated more fully in the collection of Lotman's writings published in English under the title ""Universe of the Mind: A Semiotic Theory of Culture""(1990) Juri Lotman, a semiotician at Tartu University, Estonia, was inspired by Vladimir Vernadsky's terms biosphere and noosphere to propose that a semiosphere comes into being when any two Umwelten are"
"Quasi-sphere In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case. This article uses the following notation and terminology: A quasi-sphere is a submanifold of a pseudo-Euclidean space consisting of the points for which the displacement vector from a reference point satisfies the equation where and . Since"
"are standard -spheres, and those with zero curvature are hyperplanes. Quasi-sphere In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case. This article uses the following notation and terminology: A quasi-sphere is a submanifold of a pseudo-Euclidean space consisting of the points for which the displacement vector from"
"Hemispheres of Earth In geography and cartography, the hemispheres of Earth refer to any division of the globe into two hemispheres (from Ancient Greek , meaning ""half of a sphere""). The most common such divisions are by latitudinal or longitudinal markers: The East–West division can also be seen in a cultural sense, as a division into two cultural hemispheres. However, other schemes have sought to divide the planet in a way that maximizes the preponderance of one geographic feature or another in each division: The Earth may also be split into hemispheres of day and night by the terrestrial terminator."
"Hemispheres of Earth In geography and cartography, the hemispheres of Earth refer to any division of the globe into two hemispheres (from Ancient Greek , meaning ""half of a sphere""). The most common such divisions are by latitudinal or longitudinal markers: The East–West division can also be seen in a cultural sense, as a division into two cultural hemispheres. However, other schemes have sought to divide the planet in a way that maximizes the preponderance of one geographic feature or another in each division: The Earth may also be split into hemispheres of day and night by the terrestrial terminator."
"Semi-square (astrological aspect) In astrology, a semi-square is an astrological aspect that two planets make when they are 45° apart from each other. The semi-square is considered to be the 8th harmonic of the chart because it is one-eighth of the 360° circle that the zodiac resides in (i.e., 360 / 8 = 45). The semi-square is considered to be a minor hard aspect because it is thought to cause friction in the native's life and prompt them to take some action to reduce that friction. For example, if the Sun is posited in 10° Aquarius and Venus is posited"
"sphere orthogonal to any two spheres of a pencil of spheres is orthogonal to all of them and its center lies in the radical plane of the pencil. Pairs of points on a sphere that lie on a straight line through the sphere's center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere and, consequently, divides it into two equal parts. The plane sections of a sphere are called ""spheric sections"". They are all circles and those that are not great circles are called ""small circles""."
"spheres. The radius of the midsphere is called the midradius. The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric. If is the midsphere of a polyhedron , then the intersection of with any face of is a circle. The circles formed in this way on all of the faces of form a system of circles on that are tangent exactly when the faces they lie in share an edge. Dually, if is a vertex of , then"
"Semi-dome In architecture, a semi-dome (or half-dome) is a half dome that covers a semi-circular area in a building. Semi-domes are a common feature of apses in Ancient Roman and traditional church architecture, and in mosques and iwans in Islamic architecture. A semi-dome, or the whole apse, may also be called a conch after the scallop shell often carved as decoration of the semi-dome (all shells were conches in Ancient Greek), though this is usually used for subsidiary semi-domes, rather than the one over the main apse. Small semi-domes have been often decorated in a shell shape from ancient times,"
"Spheroid 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. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, shaped like an American football or rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, shaped like a lentil. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and"
"Spherical cap In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a ""hemisphere"". The volume of the spherical cap and the area of the curved surface may be calculated using combinations of If formula_5 denotes the latitude in geographic coordinates, then formula_6. The relationship between formula_3 and formula_1 is irrelevant as long as formula_9. For"
"referred to as the Northern Hemisphere. For the purpose of celestial mapping, astronomers may conceive the sky as the inside of a sphere divided into two halves by the celestial equator. The Northern Sky or Northern Hemisphere is therefore the half of the celestial sphere that is north of the celestial equator. Even if this geocentric model is the ideal projection of the terrestrial equator onto the imaginary celestial sphere, the northern and southern celestial hemispheres is not to be confused with descriptions of the terrestrial hemispheres of Earth itself. Northern celestial hemisphere The northern celestial hemisphere, also called the"
"Circular sector A circular sector or circle sector (symbol: ⌔), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, formula_1 the radius of the circle, and formula_2 is the arc length of the minor sector. A sector with the central angle of 180° is called a half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants"
"Semidiameter In geometry, the semidiameter or semi-diameter of a set of points may be one half of its diameter; or, sometimes, one half of its extent along a particular direction. The semi-diameter of a sphere, circle, or interval is the same thing as its radius — namely, any line segment from the center to its boundary. The semi-diameters of a non-circular ellipse are the halves of its extents along the two axes of symmetry. They are the parameters ""a"", ""b"" of the implicit equation Likewise, the semi-diameters of an ellipsoid are the parameters ""a"", ""b"", and ""c"" of its implicit"
"Southern celestial hemisphere The southern celestial hemisphere, also called the Southern Sky, is the southern half of the celestial sphere; that is, it lies south of the celestial equator. This arbitrary sphere, on which seemingly fixed stars form constellations, appears to rotate westward around a polar axis due to Earth's rotation. At any given time, the entire Southern Sky is visible from the geographic South Pole, while less of this hemisphere is visible the further north the observer is located. The northern counterpart is the northern celestial hemisphere. In the context of astronomical discussions or writing about celestial mapping, it"
"Northern celestial hemisphere The northern celestial hemisphere, also called the Northern Sky, is the northern half of the celestial sphere; that is, it lies north of the celestial equator. This arbitrary sphere appears to rotate westward around a polar axis due to Earth's rotation. At any given time, the entire Northern Sky is visible from the geographic North Pole, while less of this hemisphere is visible the further south the observer is located. The southern counterpart is the southern celestial hemisphere. In the context of astronomical discussions or writing about celestial cartography, this celestial hemisphere may also simply then be"
"Half-space (geometry) In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either ""open"" or ""closed"". An open half-space is either of the two open sets produced by the subtraction"
"may also simply then be referred to as the Southern Hemisphere. For the purpose of celestial mapping, the sky is considered by astronomers as the inside of a sphere divided in two halves by the celestial equator. The Southern Sky or Southern Hemisphere is therefore that half of the celestial sphere that is south of the celestial equator. Even if this one is the ideal projection of the terrestrial equatorial onto the imaginary celestial sphere, the Northern and Southern celestial hemispheres must not be confused with descriptions of the terrestrial hemispheres of Earth itself. From the South Pole, in good"
"the naked eye) until the moment it aligns with the Sun and begins to wax, at which point it becomes new again. ""Half moon"" is often used to mean the first- and third-quarter moons, while the term ""quarter"" refers to the extent of the Moon's cycle around the Earth, not its shape. When an illuminated hemisphere is viewed from a certain angle, the portion of the illuminated area that is visible will have a two-dimensional shape as defined by the intersection of an ellipse and circle (in which the ellipse's major axis coincides with the circle's diameter). If the half-ellipse"
"surface called the real projective plane, which can also be thought of as the northern hemisphere with antipodal points of the equator identified. The hemisphere is conjectured to be the optimal (least area) isometric filling of the Riemannian circle. Spheres can be generalized to spaces of any number of dimensions. For any natural number , an ""-sphere,"" often written as , is the set of points in ()-dimensional Euclidean space that are at a fixed distance from a central point of that space, where is, as before, a positive real number. In particular: Spheres for are sometimes called hyperspheres. The"
"the great circle on the celestial sphere whose plane is normal to the local gravity vector. In practice, the horizon can be defined as the plane tangent to a still liquid surface, such as a pool of mercury. The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir. The following are two independent horizontal angular coordinates: The horizontal coordinate system is sometimes called other names, such as the az/el system, the alt/az system, or the alt-azimuth system, from the name of the mount used for telescopes, whose two axes follow"
"Half-pipe A half-pipe is a structure used in gravity extreme sports such as snowboarding, skateboarding, skiing, freestyle BMX, skating and scooter riding. The structure resembles a cross-section of a swimming pool, essentially two concave ramps (or quarter-pipes), topped by copings and decks, facing each other across a flat transition, also known as a ""tranny"". Originally half-pipes were half sections of a large diameter pipe. Since the 1980s, half-pipes contain an extended ""flat bottom"" between the quarter-pipes; the original style half-pipes are no longer built. Flat ground provides time to regain balance after landing and more time to prepare for the"
"angle is ""d""/""n"" of the half-circle. ""2"" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a ""non""-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects. A fundamental domain triangle (""p"" ""q"" ""r""), with vertex angles /""p"", /""q"", and /""r"", can exist in different spaces depending on the value of the sum"
"is not a perfect sphere), as well as on spheroidal celestial bodies. The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres. A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point. The Funk transform integrates a function along all great circles of the sphere. Great circle A great circle, also known as an orthodrome, of a sphere is the intersection"
"Half-graben A half-graben is a geological structure bounded by a fault along one side of its boundaries, unlike a full graben where a depressed block of land is bordered by parallel faults. A rift is a region where the lithosphere extends as two parts of the Earth's crust pull apart. Often a rift will form in an area of the crust that is already weakened by earlier geological activity. Extensional faults form parallel to the axis of the rift. An extensional fault may be seen as a crack in the crust that extends down at an angle to the vertical."
"Toliman (α Centauri), 4.2 light-years away, are also located in the Southern Sky, having declinations around –60° – too close to the south celestial pole that neither are visible from Central Europe. Southern celestial hemisphere The southern celestial hemisphere, also called the Southern Sky, is the southern half of the celestial sphere; that is, it lies south of the celestial equator. This arbitrary sphere, on which seemingly fixed stars form constellations, appears to rotate westward around a polar axis due to Earth's rotation. At any given time, the entire Southern Sky is visible from the geographic South Pole, while less"
"infinity) define a radius of a quasi-sphere, and thus specifies the quasi-sphere. Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. ) as the ""radius squared"", in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radius squared, those with negative radius squared, those with zero radius squared. In spaces with a positive-definite quadratic form (i.e. Euclidean space), the quasi-spheres with negative radius squared are the empty set, those with zero radius each consist of a single point, those with positive radius squared"