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Polar Coordinate Systems
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples Of Polar Coordinates
Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, example.edu, second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (; ) is a district of Barcelona between the old city ( Ciutat Vella) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Motion
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mecca
Mecca (; officially Makkah al-Mukarramah, commonly shortened to Makkah ()) is a city and administrative center of the Mecca Province of Saudi Arabia, and the Holiest sites in Islam, holiest city in Islam. It is inland from Jeddah on the Red Sea, in a narrow valley above sea level. Its last recorded population was 1,578,722 in 2015. Its estimated metro population in 2020 is 2.042million, making it the List of cities in Saudi Arabia by population, third-most populated city in Saudi Arabia after Riyadh and Jeddah. Pilgrims more than triple this number every year during the Pilgrimage#Islam, pilgrimage, observed in the twelfth Islamic calendar, Hijri month of . Mecca is generally considered "the fountainhead and cradle of Islam". Mecca is revered in Islam as the birthplace of the Prophets and messengers in Islam, Islamic prophet Muhammad. The Hira cave atop the ("Mountain of Light"), just outside the city, is where Muslims believe the Quran was first revealed to Muhammad. Vis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b\cdot\theta with real numbers and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book '' On Spirals''. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon. Derivation of general equation of spiral A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Spirals
''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle. Contents Preface Archimedes begins ''On Spirals'' with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of ''On the Sphere and Cylinder'' (Περὶ σφαίρας καὶ κυλίνδρου) and ''On Conoids and Spheroids'' (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results of ''On Spirals''. Archimedean spiral The Archimedean spiral was first studied by Conon and was later studied by Archimedes in ''On Spirals''. Archimedes was able to find various tangents to the spiral. He defines the spiral as: Trisecting an angle The construction as to how Archimedes trisected the angle is as follows: Suppose the angle ABC is to be trisected. Trisec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chord (geometry)
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just ''secant''. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word ''chord'' is from the Latin ''chorda'' meaning '' bowstring''. In circles Among properties of chords of a circle are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem). In conics The midpoints of a set of parallel chords of a coni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hipparchus
Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 and 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens (fifth century BC), Timocharis, Aristyllus, Aristarchus of Samos, and Eratosthenes, among others. He developed trigonometry and constructed trigonometric tables, and he solved sever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astrologer
Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Different cultures have employed forms of astrology since at least the 2nd millennium BCE, these practices having originated in calendrical systems used to predict seasonal shifts and to interpret celestial cycles as signs of divine communications. Most, if not all, cultures have attached importance to what they observed in the sky, and some—such as the Hindus, Chinese, and the Maya—developed elaborate systems for predicting terrestrial events from celestial observations. Western astrology, one of the oldest astrological systems still in use, can trace its roots to 19th–17th century BCE Mesopotamia, from where it spread to Ancient Greece, Rome, the Islamic world, and eventually Central and Western Europe. Contemporary Western astrology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Astronomy
Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the Ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Museum and the Library of Alexandria in Ptolemaic Egypt. The development of astronomy by the Greek and notably Hellenistic astronomers is considered to be a major phase in the history of astronomy. Greek astronomy is characterized by seeking a geometrical model for celestial phenomena. Most of the names of the stars, planets, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Before Christ
The terms (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendars. The term is Medieval Latin and means 'in the year of the Lord', but is often presented using "our Lord" instead of "the Lord", taken from the full original phrase "''anno Domini nostri Jesu Christi''", which translates to 'in the year of our Lord Jesus Christ'. The form "BC" is specific to English and equivalent abbreviations are used in other languages: the Latin form is but is rarely seen. This calendar era is based on the traditionally reckoned year of the conception or birth of Jesus, ''AD'' counting years from the start of this epoch and ''BC'' denoting years before the start of the era. There is no year zero in this scheme; thus ''the year AD 1 immediately follows the year 1 BC''. This dating system was devised in 525 by Dionysius Exiguus, but was not widely used until the 9th century. Traditionally, English follows Latin usage by placing the "AD" abbr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hipparchos 1
Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 and 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens (fifth century BC), Timocharis, Aristyllus, Aristarchus of Samos, and Eratosthenes, among others. He developed trigonometry and constructed trigonometric tables, and he solved several ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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