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Polar coordinate system

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 is called the pole, 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 radius, the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is denoted by r or ρ, the angular coordinate by φ, θ, or t. Angles in polar notation are expressed in either degrees or radians. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth 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 and orbital motion. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point.

The Archimedean spiral for example is much simpler to express using polar forms. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates; the polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system. Alexis Clairaut was the first to think of polar coordinates in three dimensions, Leonhard Euler was the first to develop them; the concepts of angle and radius were used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus created a table of chord functions giving the length of the chord for each angle, there are references to his using polar coordinates in establishing stellar positions. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle; the Greek work, did not extend to a full coordinate system.

From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately; the calculation is the conversion of the equatorial polar coordinates of Mecca to its polar coordinates relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, whose polar axis is the line through the location and its antipodal point. There are various accounts of the introduction of polar coordinates as part of a formal coordinate system; the full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653.

Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used. In Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates; the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, Leonhard Euler was the first to develop them; the radial coordinate is denoted by r or ρ, the angular coordinate by φ, θ, or t.

The angular coordinate is specified as φ by ISO standard 31-11. However, in mathematical literature the angle is denoted by θ instead of φ. Angles in polar notation are expressed in either degrees or radians. Degrees are traditionally used in navigation and many applied disciplines, while radians are more common in mathematics and mathematical physics; the angle φ is defined to start at 0° from a reference direction, to increase for rotations in either counterclockwise or clockwise orientation. In mathematics, e.g. the reference direction is drawn as a ray from the pole horizontally to the right, the polar angle increases to positive angles for ccw rotations, whereas in navigation the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the opposite orientations. Adding any number of full turns to the angular coordinate does not change the corresponding direction. Any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angl

Lloyd White (diplomat)

George David Lloyd White was a New Zealand diplomat and public servant, who served as New Zealand's ambassador to the United States from 1972 to 1978. Born in Nelson, New Zealand in 1918, White was educated at Timaru Boys' High School from 1930 to 1935, he attended Canterbury University College, graduating Master of Arts with first-class honours in economics in 1941. He served with the 2nd NZEF from 1941 to 1945, was commissioned as a second lieutenant in 1943. White joined the New Zealand public service in 1945, posted to the Economic Stabilisation Commission, he joined the Department of External Affairs in 1949, had diplomatic postings to London—where he was economic counsellor at the New Zealand High Commission from 1954 to 1955 and deputy high commissioner from 1961 to 1964—and to Washington, where he was chargé d'affaires from 1958 to 1961 and ambassador from 1972 to 1978. Domestically he served as deputy secretary of foreign affairs between 1964 and 1972. Following his retirement in 1978, White was chief executive of the Queen Elizabeth II National Trust.

In 1957, White was appointed a Member of the Royal Victorian Order. He died in Lower Hutt in 1981 and was buried at Akatarawa Cemetery


In Greek mythology, Pheme known as Ossa, was the personification of fame and renown, her favour being notability, her wrath being scandalous rumors. She was a daughter either of Gaia or of Elpis, was described as "she who initiates and furthers communication" and had an altar at Athens. A tremendous gossip, Pheme was said to have pried into the affairs of mortals and gods repeated what she learned, starting off at first with just a dull whisper, but repeating it louder each time, until everyone knew. In art, she was depicted with wings and a trumpet. In Roman mythology, Fama was described as having multiple tongues, eyes and feathers by Virgil and other authors. Virgil wrote that she "had her feet on the ground, her head in the clouds, making the small seem great and the great seem greater". In Homer Pheme is called the messenger of Zeus. In English Renaissance theatre, Rumour was a stock personification, best known from William Shakespeare's Henry IV, Part 2. James C. Bulman's Arden Shakespeare edition notes numerous lesser known theatrical examples.

The Greek word pheme is related to ϕάναι "to speak" and can mean "fame", "report", or "rumor". The Latin word fama, with the same range of meanings, is related to the Latin fari, is, through French, the etymon of the English "fame". Polychronion Iris Gná Smith, William. "Ossa" Gianni Guastella, "La Fama degli antichi e le sue trasformazioni tra Medioevo e Rinascimento," in Sergio Audano, Giovanni Cipriani, Aspetti della Fortuna dell'Antico nella Cultura Europea: atti della settima giornata di studi, Sestri Levante, 19 marzo 2010, 35-74. Theoi Greek Mythology -- Pheme "Fama". Encyclopædia Britannica. 1911

The Carpenters

The Carpenters were an American vocal and instrumental duo consisting of siblings Karen and Richard Carpenter. They produced a distinct soft musical style, combining Karen's contralto vocals with Richard's arranging and composition skills. During their 14-year career, the Carpenters recorded ten albums, along with numerous singles and several television specials; the siblings were born in New Haven and moved to Downey, California, in 1963. Richard took piano lessons as a child, progressing to California State University, Long Beach, while Karen learned the drums, they first performed together as a duo in 1965 and formed the jazz-oriented Richard Carpenter Trio followed by the middle-of-the-road group Spectrum. Signing as Carpenters to A&M Records in 1969, they achieved major success the following year with the hit singles " Close to You" and "We've Only Just Begun". Subsequently, the duo's brand of melodic pop produced a record-breaking run of hit recordings on the American Top 40 and Adult Contemporary charts, they became leading sellers in the soft rock, easy listening and adult contemporary music genres.

The Carpenters had three number-one singles and five number-two singles on the Billboard Hot 100 and fifteen number-one hits on the Adult Contemporary chart, in addition to twelve top-10 singles. They have sold more than 90 million records worldwide, making them one of the best-selling music artists of all time; the duo toured continually during the 1970s. Their career together ended in 1983 when Karen died from heart failure brought on by complications of anorexia. Extensive news coverage surrounding these circumstances increased public awareness of eating disorders. Though the Carpenters were criticized for their clean-cut and wholesome conservative image in the 1970s, their music has since been re-evaluated, attracting critical acclaim and continued commercial success; the Carpenter siblings were both born at Grace–New Haven Hospital in New Haven, Connecticut, to Harold Bertram and Agnes Reuwer. Harold was born in Wuzhou, moving to Britain in 1917, the US in 1921, while Agnes was born and grew up in Baltimore, Maryland.

They married on April 9, 1935. Richard was a quiet child who spent most of his time at home listening to Rachmaninoff, Red Nichols and Spike Jones, playing the piano. Karen was outgoing, she began ballet and tap classes aged four. Karen and Richard were close, shared a common interest in music. In particular, they became fans of Les Paul and Mary Ford, whose music featured multiple overdubbed voices and instruments. Richard began piano lessons aged eight, but grew frustrated with the formal direction of the lessons and quit after a year, he had begun to teach himself how to play by ear by 11, resumed studying with a different teacher. He took a greater interest in playing this time, would practice at home. By age 14, he was interested in performing professionally, started lessons at Yale School of Music. In June 1963, the Carpenter family moved to the Los Angeles suburb of Downey hoping that it would mean better musical opportunities for Richard, he was asked to be the organist for services at the local Methodist church.

In late 1964, Richard enrolled at California State College at Long Beach where he met future songwriting partner John Bettis, Wesley Jacobs, a friend who played the bass and tuba for the Richard Carpenter Trio, choral director Frank Pooler, with whom Richard would collaborate to create the Christmas standard "Merry Christmas Darling" in 1966. That same year, Karen enrolled at Downey High School, where she found she had a knack for playing the drums, she had tried playing the glockenspiel, but had been inspired by her friend Frankie Chavez, drumming since he was three. She became enthusiastic about the drums, began to learn complex pieces, such as Dave Brubeck's "Take Five". Chavez persuaded her parents to buy a Ludwig drum kit in late 1964, she began lessons with local jazz players, including how to read concert music, she replaced the entry-level kit with a large Ludwig set, a similar set-up to Brubeck's drummer, Joe Morello. Richard and Karen gave their first public performance together in 1965, as part of the pit band for a local production of Guys and Dolls.

By 1965, Karen had been practicing the drums for a year, Richard was refining his piano techniques under Pooler's tuition. Late that year, Richard teamed up with Jacobs, who played stand-up bass. With Karen drumming, the three formed the jazz-oriented Richard Carpenter Trio. Richard led the band and wrote all the arrangements, they began to rehearse daily, he bought a tape recorder, began to make recordings of the group. Neither Karen nor Richard sang. Karen subsequently became more confident in singing, began to take lessons with Frank Pooler, he taught her a mixture of classical and pop singing, but realised she most enjoyed performing Richard's new material. Pooler said, "Karen was a born pop singer". In early 1966, Karen tagged along at a late-night session in the g

Ruggero Luigi Emidio Antici Mattei

Ruggero Luigi Emidio Antici Mattei was an Italian Cardinal of the Roman Catholic Church. He served as Latin Patriarch of Constantinople from 1866 to 1875, was elevated to the cardinalate by Pope Pius IX in 1875. Antici Mattei was born in Recanati to Carlo Teodoro Antici and baron of Pescia, Anna Maria Mattei. A member of the house of Mattei, he was related to Cardinals Girolamo Mattei, Gaspare Mattei, Alessandro Mattei, Mario Mattei, Lorenzo Girolamo Mattei, he was confirmed on July 4, 1813. In 1818 he entered Collegio Nazareno, studied at Collegio Romano from 1826 to 1832, he received the insignias of the clerical character on May 12, 1831, followed by minor orders and diaconate. He was ordained a priest in Rome on September 7, 1834, he served as examiner of the clergy of the patriarchal Vatican basilica, curate in the abbey of Forlimpopoli, canon of the chapter of the Lateran Basilica. In 1837 he was named canon of the chapter of the Vatican basilica becoming its dean, he was appointed referendary prelate on July 13, 1843, served as a judge of the Reverend Fabric of St. Peter's from 1843 to 1847.

He was secretary of the Sacred Consistorial Congregation and of the Sacred College of Cardinals from 1850 to 1875, became prelate adjunct of the Sacred Congregation of the Tridentine Council in 1851. On January 8, 1866, he was appointed Latin Patriarch of Constantinople by Pope Pius IX and Dean of the Assistants to the Pontifical Throne, he received his episcopal consecration on the following February 25 from Cardinal Costantino Patrizi Naro. He was named auditor general of the Apostolic Chamber on March 31, 1875. Http://

2018 Allan Cup

The 2018 Allan Cup was the 2018 Canadian Grand National Championship of Senior ice hockey and the 110th year the trophy was awarded. The tournament played in Rosetown, Saskatchewan from April 9 to 14, 2018; the Stoney Creek Generals defeated the Lacombe Generals 7-4 to win the national championship. Rosetown, Saskatchewan was named the host community in November 2016. Notable players competing in this tournament include Ian White, Ryan O'Marra, Lukas Sutter. Rosetown Red Wings 20-2-0-2 record, 1st in ACHW Defeated by Lacombe Generals 5-3 in league playoffs Defeated Bethune AG Bulldogs 3-games-to-1 to win Saskatchewan championship Lacombe Generals 2009, 2013, 2016 Allan Cup champions 18-4-0-2 record, 2nd in ACHW. Defeated Stony Plain Eagles 7-1. Stoney Creek Generals 15-7-2 record, 1st in ACH Defeated Dundas Real McCoys 4-games-to-0, defeated Whitby Dunlops 4-games-to-1 to win league. Automatically advanced. Elsipogtog Hawks 7-5 record, 1st in NESHL 3-3 record, 2nd in playoff round robin. Official Allan Cup Site Allen Cup Site at