1.
Celestial spheres
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The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others. Since it was believed that the stars did not change their positions relative to one another. In modern thought, the orbits of the planets are viewed as the paths of those planets through mostly empty space, when scholars applied Ptolemys epicycles, they presumed that each planetary sphere was exactly thick enough to accommodate them. In Greek antiquity the ideas of celestial spheres and rings first appeared in the cosmology of Anaximander in the early 6th century BC, all these wheel rims had originally been formed out of an original sphere of fire wholly encompassing the Earth, which had disintegrated into many individual rings. Hence, in Anaximanderss cosmogony, in the beginning was the sphere, out of which celestial rings were formed, as viewed from the Earth, the ring of the Sun was highest, that of the Moon was lower, and the sphere of the stars was lowest. Following Anaximander, his pupil Anaximenes held that the stars, Sun, Moon, and planets are all made of fire. But whilst the stars are fastened on a crystal sphere like nails or studs, the Sun, Moon, and planets. And unlike Anaximander, he relegated the fixed stars to the region most distant from the Earth, after Anaximenes, Pythagoras, Xenophanes and Parmenides all held that the universe was spherical. And much later in the fourth century BC Platos Timaeus proposed that the body of the cosmos was made in the most perfect and uniform shape, but it posited that the planets were spherical bodies set in rotating bands or rings rather than wheel rims as in Anaximanders cosmology. Each planet is attached to the innermost of its own set of spheres. In his Metaphysics, Aristotle developed a cosmology of spheres. Aristotle considers that these spheres are made of a fifth element. Each of these concentric spheres is moved by its own god—an unchanging divine unmoved mover, by using eccentrics and epicycles, his geometrical model achieved greater mathematical detail and predictive accuracy than had been exhibited by earlier concentric spherical models of the cosmos. In Ptolemys physical model, each planet is contained in two or more spheres, but in Book 2 of his Planetary Hypotheses Ptolemy depicted thick circular slices rather than spheres as in its Book 1. The planetary spheres were arranged outwards from the spherical, stationary Earth at the centre of the universe in order, the spheres of the Moon, Mercury, Venus, Sun, Mars, Jupiter. In more detailed models the seven planetary spheres contained other secondary spheres within them, in antiquity the order of the lower planets was not universally agreed. Plato and his followers ordered them Moon, Sun, Mercury, Venus, a series of astronomers, beginning with the Muslim astronomer al-Farghānī, used the Ptolemaic model of nesting spheres to compute distances to the stars and planetary spheres. Al-Farghānīs distance to the stars was 20,110 Earth radii which, on the assumption that the radius of the Earth was 3,250 miles, came to 65,357,500 miles
2.
Earth's rotation
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Earths rotation is the rotation of the planet Earth around its own axis. The Earth rotates from the west towards east, as viewed from North Star or polestar Polaris, the Earth turns counter-clockwise. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earths axis of rotation meets its surface and this point is distinct from the Earths North Magnetic Pole. The South Pole is the point where the Earths axis of rotation intersects its surface. The Earth rotates once in about 24 hours with respect to the sun, Earths rotation is slowing slightly with time, thus, a day was shorter in the past. This is due to the effects the Moon has on Earths rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago, analysis of historical astronomical records shows a slowing trend of 2.3 milliseconds per century since the 8th century BCE. Among the ancient Greeks, several of the Pythagorean school believed in the rotation of the rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus, though his system was complicated, in the third century BCE, Aristarchus of Samos suggested the suns central place. However, Aristotle in the fourth century criticized the ideas of Philolaus as being based on rather than observation. He established the idea of a sphere of fixed stars that rotated about the earth and this was accepted by most of those who came after, in particular Claudius Ptolemy, who thought the earth would be devastated by gales if it rotated. In the 10th century, some Muslim astronomers accepted that the Earth rotates around its axis, treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemys arguments against it. At the Maragha and Samarkand observatories, the Earths rotation was discussed by Tusi and Qushji, in medieval Europe, Thomas Aquinas accepted Aristotles view and so, reluctantly, did John Buridan and Nicole Oresme in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the earths rotation begin to be established, Copernicus pointed out that if the movement of the earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion, for Copernicus this was the first step in establishing the simpler pattern of planets circling a central sun. Tycho Brahe, who produced accurate observations on which Kepler based his laws, in 1600, William Gilbert strongly supported the earths rotation in his treatise on the earths magnetism and thereby influenced many of his contemporaries. Those like Gilbert who did not openly support or reject the motion of the earth about the sun are often called semi-Copernicans, however, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of the Earth. The earths rotation implies that the bulges and the poles are flattened
3.
Geocentric model
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In astronomy, the geocentric model is a superseded description of the universe with the Earth at the center. Under the geocentric model, the Sun, Moon, stars, the geocentric model served as the predominant description of the cosmos in many ancient civilizations, such as those of Aristotle and Ptolemy. Two observations supported the idea that the Earth was the center of the Universe, first, the Sun appears to revolve around the Earth once per day. While the Moon and the planets have their own motions, they appear to revolve around the Earth about once per day. The stars appeared to be on a sphere, rotating once each day along an axis through the north and south geographic poles of the Earth. Second, the Earth does not seem to move from the perspective of an Earth-bound observer, it appears to be solid, stable, Ancient Greek, ancient Roman and medieval philosophers usually combined the geocentric model with a spherical Earth. It is not the same as the older flat Earth model implied in some mythology, the ancient Jewish Babylonian uranography pictured a flat Earth with a dome-shaped rigid canopy named firmament placed over it. The astronomical predictions of Ptolemys geocentric model were used to prepare astrological and astronomical charts for over 1500 years. The geocentric model held sway into the modern age, but from the late 16th century onward, it was gradually superseded by the Heliocentric model of Copernicus, Galileo. There was much resistance to the transition between these two theories, christian theologians were reluctant to reject a theory that agreed with Bible passages. Others felt a new, unknown theory could not subvert an accepted consensus for geocentrism, the geocentric model entered Greek astronomy and philosophy at an early point, it can be found in Pre-Socratic philosophy. In the 6th century BC, Anaximander proposed a cosmology with the Earth shaped like a section of a pillar, the Sun, Moon, and planets were holes in invisible wheels surrounding the Earth, through the holes, humans could see concealed fire. About the same time, the Pythagoreans thought that the Earth was a sphere, later these views were combined, so most educated Greeks from the 4th century BC on thought that the Earth was a sphere at the center of the universe. In the 4th century BC, two influential Greek philosophers, Plato and his student Aristotle, wrote works based on the geocentric model, according to Plato, the Earth was a sphere, stationary at the center of the universe. In his Myth of Er, a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and these spheres, known as crystalline spheres, all moved at different uniform speeds to create the revolution of bodies around the Earth. They were composed of a substance called aether. Aristotle believed that the moon was in the innermost sphere and therefore touches the realm of Earth, causing the dark spots and he further described his system by explaining the natural tendencies of the terrestrial elements, Earth, water, fire, air, as well as celestial aether. His system held that Earth was the heaviest element, with the strongest movement towards the center, the tendency of air and fire, on the other hand, was to move upwards, away from the center, with fire being lighter than air
4.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included
5.
Right ascension
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Right ascension is the angular distance measured eastward along the celestial equator from the vernal equinox to the hour circle of the point in question. When combined with declination, these astronomical coordinates specify the direction of a point on the sphere in the equatorial coordinate system. Right ascension is the equivalent of terrestrial longitude. Both right ascension and longitude measure an angle from a direction on an equator. Right ascension is measured continuously in a circle from that equinox towards the east. Any units of measure could have been chosen for right ascension, but it is customarily measured in hours, minutes. Astronomers have chosen this unit to measure right ascension because they measure a stars location by timing its passage through the highest point in the sky as the Earth rotates. The highest point in the sky, called meridian, is the projection of a line onto the celestial sphere. A full circle, measured in units, contains 24 × 60 × 60 = 86 400s, or 24 × 60 = 1 440m. Because right ascensions are measured in hours, they can be used to time the positions of objects in the sky. For example, if a star with RA = 01h 30m 00s is on the meridian, sidereal hour angle, used in celestial navigation, is similar to right ascension, but increases westward rather than eastward. Usually measured in degrees, it is the complement of right ascension with respect to 24h and it is important not to confuse sidereal hour angle with the astronomical concept of hour angle, which measures angular distance of an object westward from the local meridian. The Earths axis rotates slowly westward about the poles of the ecliptic and this effect, known as precession, causes the coordinates of stationary celestial objects to change continuously, if rather slowly. Therefore, equatorial coordinates are inherently relative to the year of their observation, coordinates from different epochs must be mathematically rotated to match each other, or to match a standard epoch. The right ascension of Polaris is increasing quickly, the North Ecliptic Pole in Draco and the South Ecliptic Pole in Dorado are always at right ascension 18h and 6h respectively. The currently used standard epoch is J2000.0, which is January 1,2000 at 12,00 TT, the prefix J indicates that it is a Julian epoch. Prior to J2000.0, astronomers used the successive Besselian Epochs B1875.0, B1900.0, the concept of right ascension has been known at least as far back as Hipparchus who measured stars in equatorial coordinates in the 2nd century BC. But Hipparchus and his successors made their star catalogs in ecliptic coordinates, the easiest way to do that is to use an equatorial mount, which allows the telescope to be aligned with one of its two pivots parallel to the Earths axis
6.
Declination
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In astronomy, declination is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declinations angle is measured north or south of the celestial equator, the root of the word declination means a bending away or a bending down. It comes from the root as the words incline and recline. Declination in astronomy is comparable to geographic latitude, projected onto the celestial sphere, points north of the celestial equator have positive declinations, while those south have negative declinations. Any units of measure can be used for declination, but it is customarily measured in the degrees, minutes. Declinations with magnitudes greater than 90° do not occur, because the poles are the northernmost and southernmost points of the celestial sphere, the Earths axis rotates slowly westward about the poles of the ecliptic, completing one circuit in about 26,000 years. This effect, known as precession, causes the coordinates of stationary celestial objects to change continuously, therefore, equatorial coordinates are inherently relative to the year of their observation, and astronomers specify them with reference to a particular year, known as an epoch. Coordinates from different epochs must be rotated to match each other. The currently used standard epoch is J2000.0, which is January 1,2000 at 12,00 TT, the prefix J indicates that it is a Julian epoch. Prior to J2000.0, astronomers used the successive Besselian Epochs B1875.0, B1900.0, the declinations of Solar System objects change very rapidly compared to those of stars, due to orbital motion and close proximity. This similarly occurs in the Southern Hemisphere for objects with less than −90° − φ. An extreme example is the star which has a declination near to +90°. Circumpolar stars never dip below the horizon, conversely, there are other stars that never rise above the horizon, as seen from any given point on the Earths surface. Generally, if a star whose declination is δ is circumpolar for some observer, then a star whose declination is −δ never rises above the horizon, as seen by the same observer. Likewise, if a star is circumpolar for an observer at latitude φ, neglecting atmospheric refraction, declination is always 0° at east and west points of the horizon. At the north point, it is 90° − |φ|, and at the south point, from the poles, declination is uniform around the entire horizon, approximately 0°. Non-circumpolar stars are visible only during certain days or seasons of the year, the Suns declination varies with the seasons. As seen from arctic or antarctic latitudes, the Sun is circumpolar near the summer solstice, leading to the phenomenon of it being above the horizon at midnight
7.
Equatorial coordinate system
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The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. The origin at the center of the Earth means the coordinates are geocentric, a right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane. This description of the orientation of the frame is somewhat simplified. A slow motion of Earths axis, precession, causes a slow, continuous turning of the system westward about the poles of the ecliptic. Superimposed on this is a motion of the ecliptic. In order to fix the primary direction, these motions necessitate the specification of the equinox of a particular date, known as an epoch. The three most commonly used are, Mean equinox of an epoch is a fixed standard direction. Mean equinox of date is the intersection of the ecliptic of date with the mean equator, commonly used in planetary orbit calculation. True equinox of date is the intersection of the ecliptic of date with the true equator and this is the actual intersection of the two planes at any particular moment, with all motions accounted for. A position in the coordinate system is thus typically specified true equinox and equator of date, mean equinox and equator of J2000.0. Note that there is no mean ecliptic, as the ecliptic is not subject to small periodic oscillations, a stars spherical coordinates are often expressed as a pair, right ascension and declination, without a distance coordinate. The direction of distant objects is the same for all observers. Telescopes equipped with equatorial mounts and setting circles employ the equatorial coordinate system to find objects, setting circles in conjunction with a star chart or ephemeris allow the telescope to be easily pointed at known objects on the celestial sphere. The declination symbol δ, measures the distance of an object perpendicular to the celestial equator, positive to the north. For example, the celestial pole has a declination of +90°. The origin for declination is the equator, which is the projection of the Earths equator onto the celestial sphere. Declination is analogous to terrestrial latitude, the right ascension symbol α, measures the angular distance of an object eastward along the celestial equator from the vernal equinox to the hour circle passing through the object. The vernal equinox point is one of the two where the ecliptic intersects the celestial equator, there are = 15° in one hour of right ascension, 24h of right ascension around the entire celestial equator
8.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
9.
Navigation
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Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude
10.
Abstraction
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An abstraction is the product of this process — a concept that acts as a super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, in a type–token distinction, a type is more abstract than its tokens. Abstraction in its use is a material process, discussed in the themes below. Its development is likely to have been connected with the development of human language. Abstraction involves induction of ideas or the synthesis of particular facts into one theory about something. It is the opposite of specification, which is the analysis or breaking-down of an idea or abstraction into concrete facts. Thales believed that everything in the universe comes from one main substance and he deduced or specified from a general idea, everything is water, to the specific forms of water such as ice, snow, fog, and rivers. Modern scientists can use the opposite approach of abstraction, or going from particular facts collected into one general idea. This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between abstract and concrete, in this sense the process of abstraction entails the identification of similarities between objects, and the process of associating these objects with an abstraction. For example, picture 1 below illustrates the concrete relationship Cat sits on Mat, for example, graph 1 below expresses the abstraction agent sits on location. This conceptual scheme entails no specific hierarchical taxonomy, only a progressive exclusion of detail, things that do not exist at any particular place and time are often considered abstract. By contrast, instances, or members, of such a thing might exist in many different places and times. Those abstract things are said to be multiply instantiated, in the sense of picture 1, picture 2. It is not sufficient, however, to abstract ideas as those that can be instantiated. Although the concepts cat and telephone are abstractions, they are not abstract in the sense of the objects in graph 1 below, perhaps confusingly, some philosophies refer to tropes as abstract particulars — e. g. the particular redness of a particular apple is an abstract particular. This is similar to qualia and sumbebekos, karl Marxs writing on the commodity abstraction recognizes a parallel process. The state as both concept and material practice exemplifies the two sides of this process of abstraction, conceptually, the current concept of the state is an abstraction from the much more concrete early-modern use as the standing or status of the prince, his visible estates. At the same time, materially, the practice of statehood is now constitutively and materially more abstract than at the time when princes ruled as the embodiment of extended power and that difference accounts for the ontological usefulness of the word abstract
11.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
12.
Concentric objects
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In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another, in the Euclidean plane, two circles that are concentric necessarily have different radii from each other. However, circles in three-dimensional space may be concentric, and have the radius as each other. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth, more generally, every two great circles on a sphere are concentric with each other and with the sphere. The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric, for the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The region of the plane between two circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell. For a given point c in the plane, the set of all circles having c as their forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii, every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of circles by a Möbius transformation. The ripples formed by dropping an object into still water naturally form an expanding system of concentric circles. Evenly spaced circles on the used in target archery or similar sports provide another familiar example of concentric circles. Coaxial cable is a type of cable in which the combined neutral. Johannes Keplers Mysterium Cosmographicum envisioned a system formed by concentric regular polyhedra. Concentric circles are found in diopter sights, a type of mechanic sights commonly found on target rifles. They usually feature a disk with a small-diametre hole near the shooters eye. When these sights are aligned, the point of impact will be in the middle of the front sight circle. Centered cube number Homoeoid Focaloid Circular symmetry Magic circle Geometry, Concentric circles demonstration With interactive animation
13.
Sky
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The sky is everything that lies above the surface of the Earth, including the atmosphere and outer space. In the field of astronomy, the sky is called the celestial sphere. This is viewed from Earths surface as a dome where the Sun, stars, planets. The celestial sphere is divided into regions called constellations. Usually, the sky is used informally as the point of view from the Earths surface, however. In some cases, such as in discussing the weather, the sky refers to only the lower, during daylight, the sky appears to be blue because air scatters blue sunlight more than it scatters red. At night, the sky appears to be a dark surface or region scattered with stars. During the day, the Sun can be seen in the sky obscured by clouds. In the night sky the Moon, planets and stars are visible in the sky, some of the natural phenomena seen in the sky are clouds, rainbows, and aurorae. Lightning and precipitation can also be seen in the sky during storms, birds, insects, aircraft, and kites are often considered to fly in the sky. Due to human activities, smog during the day and light pollution during the night are often seen above large cities. Except for light that directly from the sun, most of the light in the day sky is caused by scattering. The scattering due to molecule sized particles is greater in the forward and backward directions than it is in the lateral direction. Scattering is significant for light at all visible wavelengths but is stronger at the end of the visible spectrum, meaning that the scattered light is bluer than its source. The remaining sunlight, having lost some of its wavelength components. Scattering also occurs more strongly in clouds. Individual water droplets exposed to light will create a set of colored rings. If a cloud is thick enough, scattering from water droplets will wash out the set of colored rings
14.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
15.
Projection screen
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A projection screen is an installation consisting of a surface and a support structure used for displaying a projected image for the view of an audience. Projection screens may be installed, as in a movie theater, painted on the wall. As in a room or other non-dedicated viewing space. Another popular type of portable screens are inflatable screens for movie screening. Flat or curved screens may be used depending on the used to project the image. Screens can be designed for front or back projection, the more common being front projection systems. In commercial movie theaters, the screen is a surface that may be either aluminized or a white surface with small glass beads. The screen also has hundreds of small, evenly spaced holes to allow air to and from the speakers and subwoofer, rigid wall-mounted screens maintain their geometry perfectly just like the big movie screens, which makes them suitable for applications that demand exact reproduction of image geometry. Such screens are used in home theaters, along with the pull-down screens. Pull-down screens are used in spaces where a permanently installed screen would require too much space. These commonly use painted fabric that is rolled in the case when not used. Fixed-frame screens provide the greatest level of tension on the screens surface. They are often used in theater and professional environments where the screen does not need to be recessed into the case. Electric screens can be mounted, ceiling mounted or ceiling recessed. These are often larger screens, though electric screens are available for theater use as well. Electric screens are similar to pull-down screens, but instead of the screen being pulled down manually, switchable projection screens can be switched between opaque and clear. In the opaque state, projected image on the screen can be viewed from both sides and it is very good for advertising on store windows. Mobile screens usually use either a pull-down screen on a free stand and these can be used when it is impossible or impractical to mount the screen to a wall or a ceiling
16.
Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
17.
Celestial coordinate system
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In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects, satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, the coordinate systems are implemented in either spherical coordinates or rectangular coordinates. Spherical coordinates, projected on the sphere, are analogous to the geographic coordinate system used on the surface of the Earth. These differ in their choice of fundamental plane, which divides the sphere into two equal hemispheres along a great circle. Rectangular coordinates, in units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental plane. Each coordinate system is named after its choice of fundamental plane, the following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane, the primary direction is the starting point of the longitudinal coordinates. The origin is the distance point, the center of the celestial sphere. The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, the positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of relative to an observers ideal horizon. The equatorial coordinate system is centered at Earths center, but fixed relative to the celestial poles, the coordinates are based on the location of stars relative to Earths equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the solar system, the equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescopes or other instruments scales so that they match the equatorial coordinates of the object to observe. There are also subdivisions into mean of date coordinates, which average out or ignore nutation, and true of date, the fundamental plane is the plane of the Earths orbit, called the ecliptic plane. The geocentric ecliptic system was the coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon. The heliocentric ecliptic system describes the orbital movement around the Sun. The system is used for computing the positions of planets and other solar system bodies
18.
Astronomical object
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An astronomical object or celestial object is a naturally occurring physical entity, association, or structure that current astronomy has demonstrated to exist in the observable universe. In astronomy, the object and body are often used interchangeably. Examples for astronomical objects include planetary systems, star clusters, nebulae and galaxies, while asteroids, moons, planets, and stars are astronomical bodies. A comet may be identified as both body and object, It is a body when referring to the nucleus of ice and dust. The universe can be viewed as having a hierarchical structure, at the largest scales, the fundamental component of assembly is the galaxy. Galaxies are organized groups and clusters, often within larger superclusters. Disc galaxies encompass lenticular and spiral galaxies with features, such as spiral arms, at the core, most galaxies have a supermassive black hole, which may result in an active galactic nucleus. Galaxies can also have satellites in the form of dwarf galaxies, the constituents of a galaxy are formed out of gaseous matter that assembles through gravitational self-attraction in a hierarchical manner. At this level, the fundamental components are the stars. The great variety of forms are determined almost entirely by the mass, composition. Stars may be found in systems that orbit about each other in a hierarchical organization. A planetary system and various objects such as asteroids, comets and debris. The various distinctive types of stars are shown by the Hertzsprung–Russell diagram —a plot of stellar luminosity versus surface temperature. Each star follows a track across this diagram. If this track takes the star through a region containing a variable type. An example of this is the instability strip, a region of the H-R diagram that includes Delta Scuti, RR Lyrae, the table below lists the general categories of bodies and objects by their location or structure. International Astronomical Naming Commission List of light sources List of Solar System objects Lists of astronomical objects SkyChart, Sky & Telescope Monthly skymaps for every location on Earth
19.
Brooklyn Museum
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The Brooklyn Museum is an art museum located in the New York City borough of Brooklyn. At 560,000 square feet, the museum is New York Citys third largest in physical size, the museum initially struggled to maintain its building and collection, only to be revitalized in the late 20th century, thanks to major renovations. Significant areas of the collection include antiquities, specifically their collection of Egyptian antiquities spanning over 3,000 years, African, Oceanic, and Japanese art make for notable antiquities collections as well. American art is represented, starting at the Colonial period. Artists represented in the collection include Mark Rothko, Edward Hopper, Norman Rockwell, Winslow Homer, Edgar Degas, Georgia OKeeffe, the museum also has a Memorial Sculpture Garden which features salvaged architectural elements from throughout New York City. The roots of the Brooklyn Museum extend back to the 1823 founding by Augustus Graham of the Brooklyn Apprentices’ Library in Brooklyn Heights, in 1890, under its director Franklin Hooper, Institute leaders reorganized as the Brooklyn Institute of Arts and Sciences and began planning the Brooklyn Museum. The initial design for the Brooklyn Museum was four times as large as the actualized version, Daniel Chester French, the noted sculptor of the Lincoln Memorial, was the principal designer of the pediment sculptures and the monolithic 12. 5-foot figures along the cornice. The figures were created by 11 sculptors and carved by the Piccirilli Brothers, by 1920, the New York City Subway reached the museum with a subway station, this greatly improved access to the once-isolated museum from Manhattan and other outer boroughs. The Brooklyn Institutes director Franklin Hooper was the museums first director and he was followed by Philip Newell Youtz, Laurance Page Roberts, Isabel Spaulding Roberts, Charles Nagel, Jr. and Edgar Craig Schenck. Thomas S. Buechner became the director in 1960, making him one of the youngest directors in the country. Buechner oversaw a major transformation in the way the museum displayed art and brought some one thousand works that had languished in the museums archives and put them on display. Buechner played a role in rescuing the Daniel Chester French sculptures from destruction due to an expansion project at the Manhattan Bridge in the 1960s. The Brooklyn Museum changed its name to Brooklyn Museum of Art in 1997, on March 12,2004, the museum announced that it would revert to its previous name. In April 2004, the museum opened the James Polshek-designed entrance pavilion on the Eastern Parkway façade, in September 2014, Lehman announced that he was planning to retire around June 2015. In May 2015, Creative Time president and artistic director Anne Pasternak was named the Museums next director, member institutions occupy land or buildings owned by the City of New York and derive part of their yearly funding from the City. The Brooklyn Museum also supplements its earned income with funding from Federal and State governments, as well as donations by individuals. Major benefactors include Frank Lusk Babbott, the museum is the site of the annual Brooklyn Artists Ball which has included celebrity hosts such as Sarah Jessica Parker and Liv Tyler. The Brooklyn Museum exhibits collections that seek to embody the rich heritage of world cultures
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Equidistant
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A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry the locus of points equidistant from two points is their perpendicular bisector. For a triangle the circumcentre is a point equidistant from each of the three vertices, every non-degenerate triangle has such a point. This result can be generalised to cyclic polygons, the circumcentre is equidistant from each of the vertices, likewise, the incentre of a triangle or any other tangential polygon is equidistant from the points of tangency of the polygons sides with the circle. Every point on a perpendicular bisector of the side of a triangle or other polygon is equidistant from the two vertices at the ends of that side, every point on the bisector of an angle of any polygon is equidistant from the two sides that emanate from that angle. The center of a rectangle is equidistant from all four vertices, a point on the axis of symmetry of a kite is equidistant between two sides. The center of a circle is equidistant from every point on the circle, likewise the center of a sphere is equidistant from every point on the sphere. A parabola is the set of points in a plane equidistant from a point and a fixed line. In shape analysis, the skeleton or medial axis of a shape is a thin version of that shape that is equidistant from its boundaries. In Euclidean geometry, parallel lines are equidistant in the sense that the distance of any point on one line from the nearest point on the line is the same for all points. In hyperbolic geometry the set of points that are equidistant from and on one side of a given line form an hypercycle
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Earth
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Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
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Relative direction
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Direction is the information vested in the relative position of a destination point with respect to a starting point, absent the distance information. Directions may be relative to some indicated reference, or absolute according to some previously agreed upon frame of reference. Direction is often indicated manually by an index finger or written as an arrow. On a vertically oriented sign representing a plane, such as a road sign. Mathematically, direction may be specified by a direction vector, or equivalently by the angles made by the most direct path with respect to a specified set of axes. Examples of 2D and 3D directions represented numerically by direction vectors are shown, the same unit vector construct is used to specify spatial directions in 2D, 3D or higher dimensions. As illustrated, each unique 2D direction is equivalent numerically to a point on the circle, while each unique 3D direction may be represented numerically as a point on the unit sphere. For example, in software where 2D direction is represented by angle theta, 2D direction vectors vary continuously while offering 1,1 representation, thus eliminating a singularity exception to be handled. Representational simplicity is even greater for direction vectors in 3D, as compared to polar angle-pair, in Euclidean and affine geometry, the direction of a subspace is the vector space associated to this subspace. In the case of a subspace of one the direction vectors are the nonzero vectors of this vector space. Cardinal direction Compass Direction vector Navigation Radio direction finder Relative direction Affine space
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Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
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Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
25.
Centre (geometry)
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In geometry, a centre of an object is a point in some sense in the middle of the object. According to the definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of groups then a centre is a fixed point of all the isometries which move the object onto itself. The centre of a circle is the point equidistant from the points on the edge, similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends. For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions, so the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this being the fixed point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect. For an equilateral triangle, these are the point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex. e. F=thf for some real power h, thus the position of a centre is independent of scale. F is symmetric in its last two arguments, i. e. f= f, thus position of a centre in a triangle is the mirror-image of its position in the original triangle. This strict definition excludes pairs of points such as the Brocard points. The Encyclopedia of Triangle Centers lists over 9,000 different triangle centres, a tangential polygon has each of its sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon, a cyclic polygon has each of its vertices on a particular circle, called the circumcircle or circumscribed circle. The centre of the circumcircle, called the circumcentre, can be considered a centre of the polygon, if a polygon is both tangential and cyclic, it is called bicentric. The incentre and circumcentre of a polygon are not in general the same point. The centre of a general polygon can be defined in different ways. The vertex centroid comes from considering the polygon as being empty, the side centroid comes from considering the sides to have constant mass per unit length. The usual centre, called just the centroid comes from considering the surface of the polygon as having constant density and these three points are in general not all the same point. Centrepoint Centre of mass Chebyshev centre Fixed points of isometry groups in Euclidean space
26.
Parallel (geometry)
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
27.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
28.
Solar System
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The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets, of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury. The Solar System formed 4.6 billion years ago from the collapse of a giant interstellar molecular cloud. The vast majority of the mass is in the Sun. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being composed of rock. The four outer planets are giant planets, being more massive than the terrestrials. All planets have almost circular orbits that lie within a flat disc called the ecliptic. The Solar System also contains smaller objects, the asteroid belt, which lies between the orbits of Mars and Jupiter, mostly contains objects composed, like the terrestrial planets, of rock and metal. Beyond Neptunes orbit lie the Kuiper belt and scattered disc, which are populations of trans-Neptunian objects composed mostly of ices, within these populations are several dozen to possibly tens of thousands of objects large enough that they have been rounded by their own gravity. Such objects are categorized as dwarf planets, identified dwarf planets include the asteroid Ceres and the trans-Neptunian objects Pluto and Eris. In addition to two regions, various other small-body populations, including comets, centaurs and interplanetary dust clouds. Six of the planets, at least four of the dwarf planets, each of the outer planets is encircled by planetary rings of dust and other small objects. The solar wind, a stream of charged particles flowing outwards from the Sun, the heliopause is the point at which pressure from the solar wind is equal to the opposing pressure of the interstellar medium, it extends out to the edge of the scattered disc. The Oort cloud, which is thought to be the source for long-period comets, the Solar System is located in the Orion Arm,26,000 light-years from the center of the Milky Way. For most of history, humanity did not recognize or understand the concept of the Solar System, the invention of the telescope led to the discovery of further planets and moons. The principal component of the Solar System is the Sun, a G2 main-sequence star that contains 99. 86% of the known mass. The Suns four largest orbiting bodies, the giant planets, account for 99% of the mass, with Jupiter. The remaining objects of the Solar System together comprise less than 0. 002% of the Solar Systems total mass, most large objects in orbit around the Sun lie near the plane of Earths orbit, known as the ecliptic
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Vanishing point
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In graphical perspective, a vanishing point is a point in the image plane where the projections of a set of parallel lines in space intersect. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. The vanishing point may also be referred to as the point, as lines having the same directional vector, say D. Mathematically, let q ≡ be a point lying on the plane, where f is the focal length, and let vq ≡ be the unit vector associated with q. When the image plane is parallel to two axes, lines parallel to the axis which is cut by this image plane will meet at infinity i. e. at the vanishing point. Lines parallel to the two axes will not form vanishing points as they are parallel to the image plane. Similarly, when the plane intersects two world-coordinate axes, lines parallel to those planes will meet at infinity and form two vanishing points. In three-point perspective the image plane intersects the x, y, the vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π, some authors have used the phrase, the image of a line includes its vanishing point. Guidobaldo del Monte gave several verifications, and Humphry Ditton called the result the main and Great Proposition. She notes, in terms of geometry, the vanishing point is the image of the point at infinity associated with L. As a vanishing point originates in a line, so a vanishing line originates in a plane α that is not parallel to the picture π. Given the eye point O, and β the plane parallel to α and lying on O, to put it simply, the vanishing line is obtained by the intersection of the image plane with a plane parallel to the ground plane, passing through the camera center. For different sets of line, their respective vanishing points will lie on this line. The horizon line is a line that represents the eye level of the observer. If the object is below the line, its vanishing lines angle up to the horizon line. If the object is above, they slope down, all vanishing lines end at the horizon line. Proof, Consider the ground plane π, as y = c which is, for the sake of simplicity, also, consider a line L that lies in the plane π, which is defined by the equation ax + bz = d
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Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition
31.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
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Great circle
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A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant
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Moon
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The Moon is an astronomical body that orbits planet Earth, being Earths only permanent natural satellite. It is the fifth-largest natural satellite in the Solar System, following Jupiters satellite Io, the Moon is second-densest satellite among those whose densities are known. The average distance of the Moon from the Earth is 384,400 km, the Moon is thought to have formed about 4.51 billion years ago, not long after Earth. It is the second-brightest regularly visible celestial object in Earths sky, after the Sun and its surface is actually dark, although compared to the night sky it appears very bright, with a reflectance just slightly higher than that of worn asphalt. Its prominence in the sky and its cycle of phases have made the Moon an important cultural influence since ancient times on language, calendars, art. The Moons gravitational influence produces the ocean tides, body tides, and this matching of apparent visual size will not continue in the far future. The Moons linear distance from Earth is currently increasing at a rate of 3.82 ±0.07 centimetres per year, since the Apollo 17 mission in 1972, the Moon has been visited only by uncrewed spacecraft. The usual English proper name for Earths natural satellite is the Moon, the noun moon is derived from moone, which developed from mone, which is derived from Old English mōna, which ultimately stems from Proto-Germanic *mǣnōn, like all Germanic language cognates. Occasionally, the name Luna is used, in literature, especially science fiction, Luna is used to distinguish it from other moons, while in poetry, the name has been used to denote personification of our moon. The principal modern English adjective pertaining to the Moon is lunar, a less common adjective is selenic, derived from the Ancient Greek Selene, from which is derived the prefix seleno-. Both the Greek Selene and the Roman goddess Diana were alternatively called Cynthia, the names Luna, Cynthia, and Selene are reflected in terminology for lunar orbits in words such as apolune, pericynthion, and selenocentric. The name Diana is connected to dies meaning day, several mechanisms have been proposed for the Moons formation 4.51 billion years ago, and some 60 million years after the origin of the Solar System. These hypotheses also cannot account for the angular momentum of the Earth–Moon system. This hypothesis, although not perfect, perhaps best explains the evidence, eighteen months prior to an October 1984 conference on lunar origins, Bill Hartmann, Roger Phillips, and Jeff Taylor challenged fellow lunar scientists, You have eighteen months. Go back to your Apollo data, go back to computer, do whatever you have to. Dont come to our conference unless you have something to say about the Moons birth, at the 1984 conference at Kona, Hawaii, the giant impact hypothesis emerged as the most popular. Afterward there were only two groups, the giant impact camp and the agnostics. Giant impacts are thought to have been common in the early Solar System, computer simulations of a giant impact have produced results that are consistent with the mass of the lunar core and the present angular momentum of the Earth–Moon system
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Parallax
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The term is derived from the Greek word παράλλαξις, meaning alternation. Due to foreshortening, nearby objects have a larger parallax than more distant objects when observed from different positions, astronomers use the principle of parallax to measure distances to the closer stars. Here, the parallax is the semi-angle of inclination between two sight-lines to the star, as observed when the Earth is on opposite sides of the Sun in its orbit. Parallax also affects optical instruments such as rifle scopes, binoculars, microscopes, many animals, including humans, have two eyes with overlapping visual fields that use parallax to gain depth perception, this process is known as stereopsis. In computer vision the effect is used for stereo vision, and there is a device called a parallax rangefinder that uses it to find range. A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a needle-style speedometer gauge. When viewed from directly in front, the speed may show exactly 60, as the eyes of humans and other animals are in different positions on the head, they present different views simultaneously. This is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception, animals also use motion parallax, in which the animals move to gain different viewpoints. For example, pigeons bob their heads up and down to see depth, the motion parallax is exploited also in wiggle stereoscopy, computer graphics which provide depth cues through viewpoint-shifting animation rather than through binocular vision. Parallax arises due to change in viewpoint occurring due to motion of the observer, of the observed, what is essential is relative motion. By observing parallax, measuring angles, and using geometry, one can determine distance, astronomers also use the word parallax as a synonym for distance measurent by other methods, see parallax #Astronomy. In a geostatic model, the movement of the star would have to be taken as real with the star oscillating across the sky with respect to the background stars, the parsec is defined as the distance for which the annual parallax is 1 arcsecond. Annual parallax is measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars, the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Stellar parallax remains the standard for calibrating other measurement methods, accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets. The angles involved in these calculations are very small and thus difficult to measure, the nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ±0.0003 arcsec. This angle is approximately that subtended by an object 2 centimeters in diameter located 5.3 kilometers away, the fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age. In 1989, the satellite Hipparcos was launched primarily for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold
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Heliocentrism
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Heliocentrism is the astronomical model in which the Earth and planets revolve around the Sun at the center of the Solar System. Historically, Heliocentrism was opposed to geocentrism, which placed the Earth at the center, in the following century, Johannes Kepler elaborated upon and expanded this model to include elliptical orbits, and Galileo Galilei presented supporting observations made using a telescope. With the observations of William Herschel, Friedrich Bessel, and others, astronomers realized that the sun, the Ptolemaic system was a sophisticated astronomical system that managed to calculate the positions for the planets to a fair degree of accuracy. However, he rejected the idea of a spinning earth as absurd as he believed it would create huge winds and his planetary hypotheses were sufficiently real that the distances of moon, sun, planets and stars could be determined by treating orbits celestial spheres as contiguous realities. This system postulated the existence of a counter-earth collinear with the Earth and central fire, the Sun revolved around the central fire once a year, and the stars were stationary. The Earth maintained the same face towards the central fire. Kepler gave an explanation of the Pythagoreans central fire as the Sun. Heraclides of Pontus said that the rotation of the Earth explained the apparent daily motion of the celestial sphere and it used to be thought that he believed Mercury and Venus to revolve around the Sun, which in turn revolves around the Earth. Macrobius Ambrosius Theodosius later described this as the Egyptian System, stating that it did not escape the skill of the Egyptians, the first person known to have proposed a heliocentric system, however, was Aristarchus of Samos. Like Eratosthenes, Aristarchus calculated the size of the Earth, and measured the size and distance of the Moon and Sun, from his estimates, he concluded that the Sun was six to seven times wider than the Earth and thus hundreds of times more voluminous. His writings on the system are lost, but some information is known from surviving descriptions and critical commentary by his contemporaries. This is the account as you have heard from astronomers. The stars are in fact much farther away than the distance that was assumed in ancient times. Archimedes says that Aristarchus made the stars distance larger, suggesting that he was answering the natural objection that Heliocentrism requires stellar parallactic oscillations and he apparently agreed to the point but placed the stars so distant as to make the parallactic motion invisibly minuscule. Thus Heliocentrism opened the way for realization that the universe was larger than the geocentrists taught, according to one of Plutarchs characters in the dialogue, the philosopher Cleanthes had held that Aristarchus should be charged with impiety for moving the hearth of the world. Since Plutarch mentions the followers of Aristarchus in passing, it is likely there were other astronomers in the Classical period who also espoused Heliocentrism. Seleucus adopted the system of Aristarchus and is said to have proved the heliocentric theory. He may have used trigonometric methods that were available in his time
36.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
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Shorthand
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Shorthand is an abbreviated symbolic writing method that increases speed and brevity of writing as compared to longhand, a more common method of writing a language. The process of writing in shorthand is called stenography, from the Greek stenos and it has also been called brachygraphy, from Greek brachys and tachygraphy, from Greek tachys, depending on whether compression or speed of writing is the goal. A typical shorthand system provides symbols or abbreviations for words and common phrases, abbreviation methods are alphabet-based and use different abbreviating approaches. Several autocomplete programs, standalone or integrated in text editors, based on word lists, many journalists use shorthand writing to quickly take notes at press conferences or other similar scenarios. Shorthand was used widely in the past, before the invention of recording. Shorthand was considered a part of secretarial training and police work. Although the primary use of shorthand has been to record oral dictation or discourse, for example, healthcare professionals may use shorthand notes in medical charts and correspondence. The earliest known indication of shorthand systems is from the Parthenon in Ancient Greece and this shows a writing system primarily based on vowels, using certain modifications to indicate consonants. Hellenistic tachygraphy is reported from the 2nd century BC onwards, though there are indications that it might be older, the oldest datable reference is a contract from Middle Egypt, stating that Oxyrhynchos gives the semeiographer Apollonios for two years to be taught shorthand writing. Hellenistic tachygraphy consisted of word stem signs and word ending signs, over time, many syllabic signs were developed. In Ancient Rome, Marcus Tullius Tiro, a slave and later a freedman of Cicero, the Tironian notes consisted of Latin word stem abbreviations and of word ending abbreviations. The original Tironian notes consisted of about 4000 signs, but new signs were introduced, in order to have a less complex writing system, a syllabic shorthand script was sometimes used. After the decline of the Roman Empire, the Tironian notes were no longer used to transcribe speeches, though they were known and taught. After the 11th century, however, they were mostly forgotten, when many monastery libraries were secularized in the course of the 16th-century Protestant Reformation, long-forgotten manuscripts of Tironian notes were rediscovered. In imperial China, clerks used an abbreviated, highly cursive form of Chinese characters to record court proceedings and these records were used to create more formal transcripts. One cornerstone of imperial court proceedings was that all confessions had to be acknowledged by the signature, personal seal, or thumbprint. Versions of this technique survived in clerical professions into the modern day, an interest in shorthand or short-writing developed towards the end of the 16th century in England. In 1588 Timothy Bright published his Characterie, An Arte of Shorte, Swifte and Secrete Writing by Character which introduced a system with 500 arbitrary symbols each representing one word
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Eclipse
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An eclipse occurs during a syzygy. The term eclipse is most often used to either a solar eclipse, when the Moons shadow crosses the Earths surface, or a lunar eclipse. A binary star system can also produce eclipses if the plane of the orbit of its constituent stars intersects the observers position, for any two objects in space, a line can be extended from the first through the second. The latter object will block some amount of light being emitted by the former, as viewed from such a location, this shadowing event is known as an eclipse. Typically the cross-section of the involved in an astronomical eclipse are roughly disk shaped. The region of an objects shadow during an eclipse is divided into three parts, The umbra, within which the object completely covers the light source, for the Sun, this light source is the photosphere. The antumbra, extending beyond the tip of the umbra, within which the object is completely in front of the light source, the penumbra, within which the object is only partially in front of the light source. A total eclipse occurs when the observer is within the umbra, an annular eclipse when the observer is within the antumbra, during a lunar eclipse only the umbra and penumbra are applicable. This is because Earths apparent diameter from the viewpoint of the Moon is nearly four times that of the Sun, the same terms may be used analogously in describing other eclipses, e. g. the antumbra of Deimos crossing Mars, or Phobos entering Marss penumbra. For Earth, on average L is equal to 1. 384×106 km, hence the umbral cone of the Earth can completely envelop the Moon during a lunar eclipse. If the occulting object has an atmosphere, however, some of the luminosity of the star can be refracted into the volume of the umbra. This occurs, for example, during an eclipse of the Moon by the Earth—producing a faint, on Earth, the shadow cast during an eclipse moves very approximately at 1 km per sec. This depends on the location of the shadow on the Earth, http, //www. sciforums. com/threads/speed-of-eclipse-shadow. 53722/ An eclipse cycle takes place when a series of eclipses are separated by a certain interval of time. This happens when the orbital motions of the bodies form repeating harmonic patterns, a particular instance is the saros, which results in a repetition of a solar or lunar eclipse every 6,585.3 days, or a little over 18 years. Because this is not a number of days, successive eclipses will be visible from different parts of the world. An eclipse involving the Sun, Earth, and Moon can occur only when they are nearly in a line, allowing one to be hidden behind another. Because the orbital plane of the Moon is tilted with respect to the plane of the Earth. The Sun, Earth and nodes are aligned twice a year, there can be from four to seven eclipses in a calendar year, which repeat according to various eclipse cycles, such as a saros
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Equator
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The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north
