title: Coins of the pound sterling

text: Outside the United Kingdom, the British Crown Dependencies of Jersey and Guernsey use the pound sterling as their currencies. However, they produce local issues of coinage in the same denominations and specifications, but with different designs. These circulate freely alongside UK coinage and English, Northern Irish, and Scottish banknotes within these territories, but must be converted in order to be used in the UK. The island of Alderney also produces occasional commemorative coins. "(See coins of the Jersey pound, coins of the Guernsey pound, and Alderney pound for details.)". The Isle of Man is a unique case among the Crown Dependencies, issuing its own currency, the Manx pound. While the Isle of Man recognises the Pound Sterling as a secondary currency, coins of the Manx pound are not legal tender in the UK.


title: Coins of the pound sterling

text: The pound sterling is also the official currency of the British overseas territories of South Georgia and the South Sandwich Islands, British Antarctic Territory and Tristan da Cunha. South Georgia and the South Sandwich Islands produces occasional special collectors' sets of coins. In 2008, British Antarctic Territory issued a £2 coin commemorating the centenary of Britain's claim to the region.


title: Coins of the pound sterling

text: The currencies of the British overseas territories of Gibraltar, the Falkland Islands and Saint Helena/Ascension — namely the Gibraltar pound, Falkland Islands pound and Saint Helena pound — are pegged one-to-one to the pound sterling but are technically separate currencies. These territories issue their own coinage, again with the same denominations and specifications as the UK coinage but with local designs, as coins of the Gibraltar pound, coins of the Falkland Islands pound and coins of the Saint Helena pound.


title: Coins of the pound sterling

text: Before decimalisation in 1971, the pound was divided into 240 pence rather than 100, though it was rarely expressed in this way. Rather it was expressed in terms of pounds, shillings and pence, where:


title: Coins of the pound sterling

text: Thus: £1 = 240d. The penny was further subdivided at various times, though these divisions vanished as inflation made them irrelevant:


title: Coins of the pound sterling

text: Using the example of five shillings and sixpence, the standard ways of writing shillings and pence were:


title: Coins of the pound sterling

text: The abbreviation for the old penny, d, was derived from the Roman "denarius", and the abbreviation for the shilling, s, from the Roman "solidus". The shilling was also denoted by the slash symbol, also called a solidus for this reason, which was originally an adaptation of the long s. The symbol "£", for the pound, is derived from the first letter of the Latin word for pound, "libra".


title: Coins of the pound sterling

text: A similar pre-decimal system operated in France, also based on the Roman currency, consisting of the "livre" (L), "sol" or "sou" (s) and "denier" (d). Until 1816 another similar system was used in the Netherlands, consisting of the "gulden" (G), "stuiver" (s;  G) and "duit", (d;  s or  G).


title: Coins of the pound sterling

text: The farthing (d) had been demonetised on 1 January 1961, whilst the crown (5/-) was issued periodically as a commemorative coin but rarely found in circulation.


title: Coins of the pound sterling

text: The crown, half crown, florin, shilling, and sixpence were cupronickel coins (in historical times silver or silver alloy); the penny, halfpenny, and farthing were bronze; and the threepence was a twelve-sided nickel-brass coin (historically it was a small silver coin).


title: Coins of the pound sterling

text: Some of the pre-decimalisation coins with exact decimal equivalent values continued in use after 1971 alongside the new coins, albeit with new names (the shilling became equivalent to the 5p coin, with the florin equating to 10p), and the others were withdrawn almost immediately. The use of florins and shillings as legal tender in this way ended in 1991 and 1993 when the 5p and 10p coins were replaced with smaller versions. Indeed, while pre-decimalisation shillings were used as 5p coins, for a while after decimalisation many people continued to call the new 5p coin a shilling, since it remained of a pound, but was now counted as 5p (five new pence) instead of 12d (twelve old pennies). The pre-decimalisation sixpence, also known as a sixpenny bit or sixpenny piece, was equivalent to p, but was demonetised in 1980.


title: Coins of the pound sterling

text: Some pre-decimalisation coins or denominations became commonly known by colloquial and slang terms, perhaps the most well known being "bob" for a shilling, and "quid" for a pound. A farthing was a "mag", a silver threepence was a "joey" and the later nickel-brass threepence was called a "threepenny bit" ( or bit, i.e. thrup'ny or threp'ny bit – the apostrophe was pronounced on a scale from full "e" down to complete omission); a sixpence was a "tanner", the two-shilling coin or florin was a "two-bob bit". Bob is still used in phrases such as "earn/worth a bob or two", and "bob‐a‐job week". The two shillings and sixpence coin or half-crown was a "half-dollar", also sometimes referred to as "two and a kick". A value of two pence was universally pronounced "tuppence", a usage which is still heard today, especially among older people. The unaccented suffix "-pence", pronounced , was similarly appended to the other numbers up to twelve; thus "fourpence", "sixpence-three-farthings", "twelvepence-ha'penny", but "eighteen pence" would usually be said "one-and-six".


title: Coins of the pound sterling

text: "Quid" remains as popular slang for one or more pounds to this day in Britain in the form "a quid" and then "two quid", and so on. Similarly, in some parts of the country, "bob" continued to represent one-twentieth of a pound, that is five new pence, and "two bob" is 10p.


title: Coins of the pound sterling

text: The introduction of decimal currency caused a new casual usage to emerge, where any value in pence is spoken using the suffix "pee": e.g. "twenty-three pee" or, in the early years, "two-and-a-half pee" rather than the previous "tuppence-ha'penny". Amounts over a pound are normally spoken thus: "five pounds forty". A value with less than ten pence over the pound is sometimes spoken like this: "one pound and a penny", "three pounds and fourpence". The slang term "bit" has almost disappeared from use completely, although in Scotland a fifty pence is sometimes referred to as a "ten bob bit". Decimal denomination coins are generally described using the terms "piece" or coin, for example, "a fifty-pee piece", a "ten-pence coin".


title: Coins of the pound sterling

text: All coins since the 17th century have featured a profile of the current monarch's head. The direction in which they face changes with each successive monarch, a pattern that began with the Stuarts, as shown in the table below:


title: Coins of the pound sterling

text: For the Tudors and pre-Restoration Stuarts, both left- and right-facing portrait images were minted within the reign of a single monarch (left-facing images were more common). In the Middle Ages, portrait images tended to be full face.


title: Coins of the pound sterling

text: There was a small quirk in this alternating pattern when Edward VIII became king in January 1936 and was portrayed facing left, the same as his predecessor George V. This was because Edward thought his left side to be better than his right. However, Edward VIII abdicated in December 1936 and his coins were never put into general circulation. When George VI came to the throne, he had his coins struck with him facing the left, as if Edward VIII's coins had faced right (as they should have done according to tradition). Thus, in a timeline of circulating British coins, George V and VI's coins both feature left-facing portraits, although they follow directly chronologically.


title: Coins of the pound sterling

text: From a very early date, British coins have been inscribed with the name of the ruler of the kingdom in which they were produced, and a longer or shorter title, always in Latin; among the earliest distinctive English coins are the silver pennies of Offa of Mercia, which were inscribed with the legend "King Offa". As the legends became longer, words in the inscriptions were often abbreviated so that they could fit on the coin; identical legends have often been abbreviated in different ways depending upon the size and decoration of the coin. Inscriptions which go around the edge of the coin generally have started at the center of the top edge and proceeded in a clockwise direction. A very lengthy legend would be continued on the reverse side of the coin. All monarchs used Latinised names, save Edward III, both Elizabeths, and Charles III (which would have been EDWARDUS, ELIZABETHA, and CAROLUS respectively).


title: Coins of the pound sterling

text: Some coins made for circulation in the British colonies are considered part of British coinage because they have no indication of what country it was minted for and they were made in the same style as contemporary coins circulating in the United Kingdom.


title: Coins of the pound sterling

text: A three halfpence ( pence, of a pound) coin was circulated mainly in the West Indies and Ceylon in the starting in 1834. Jamaicans referred to the coin as a "quatty".


title: Coins of the pound sterling

text: The half farthing ( of a penny, of a pound) coin was initially minted in 1828 for use in Ceylon, but was declared legal tender in the United Kingdom in 1842.


title: Coins of the pound sterling

text: In addition to the title, a Latin or French motto might be included, generally on the reverse side of the coin. These varied between denominations and issues; some were personal to the monarch, others were more general. Some of the mottos were:


title: Coins of the pound sterling

text: Coins with errors in the minting process that reach circulation are often seen as valuable items by coin collectors.


title: Coins of the pound sterling

text: In 1983, the Royal Mint mistakenly produced some two pence pieces with the old wording "New Pence" on the reverse (tails) side, when the design had been changed from 1982 to "Two Pence".


title: Coins of the pound sterling

text: In 2016, a batch of double-dated £1 coins was released into circulation. These coins had the main date on the obverse stating '2017', but the micro-engraving having '2016' on it. it is not known how many exist and are in circulation, but the amount is fewer than half a million.


title: Coins of the pound sterling

text: In June 2009, the Royal Mint estimated that between 50,000 and 200,000 dateless 20 pence coins had entered circulation, the first undated British coin to enter circulation in more than 300 years. It resulted from the accidental combination of old and new face tooling in a production batch, creating what is known as a mule, following the 2008 redesign which moved the date from the reverse (tails) to the obverse (heads) side.


title: Zhang Yimou

text: Zhang Yimou (; born 2 April 1950) is a Chinese film director, producer, writer, actor and former cinematographer. He is a part of the Fifth Generation of Chinese filmmakers, Honorary Doctorate of Boston University and Yale University, Distinguished Professor of Beijing Film Academy. He made his directorial debut in 1988 with "Red Sorghum".


title: Zhang Yimou

text: Zhang has won numerous awards and recognitions, with three Academy Awards nominations for Best Foreign Language Film for "Ju Dou" in 1990, "Raise the Red Lantern" in 1991, and "Hero" in 2003; a Silver Lion, two Golden Lion prizes and the Glory to the Filmmaker Award at the Venice Film Festival; Grand Jury Prize, Prize of the Ecumenical Jury and Technical Grand Prize at the Cannes Film Festival; the Golden Bear, the Silver Bear Grand Jury Prize and the Prize of the Ecumenical Jury at the Berlin International Film Festival. In 1993, he was a member of the jury at the 43rd Berlin International Film Festival. Zhang directed the opening and closing ceremonies of the 2008 Beijing Summer Olympic Games as well as the opening and closing ceremonies of the 2022 Beijing Winter Olympic Games, which received considerable international acclaim.


title: Zhang Yimou

text: One of Zhang's recurrent themes is the resilience of Chinese people in the face of hardship and adversity, a theme which has been explored in such films as "To Live" (1994) and "Not One Less" (1999). His films are particularly noted for their rich use of colour, as can be seen in some of his early films, like "Raise the Red Lantern", and in his wuxia films like "Hero" and "House of Flying Daggers". His highest budgeted film to date is the 2016 monster film "The Great Wall", set in Imperial China and starring Matt Damon.


title: Zhang Yimou

text: Zhang Yimou () was born in Xi'an, the capital of Shaanxi province. Zhang's father, Zhang Bingjun (), a dermatologist, had been an officer in the National Revolutionary Army under Chiang Kai-shek during the Chinese Civil War; an uncle and an elder brother had followed the Nationalist forces to Taiwan after their 1949 defeat. Zhang's mother, Zhang Xiaoyou (), was a doctor at the 2nd Hospital affiliated Xi'an Jiao Tong University who graduated from Xi'an Medical University. He has two younger brothers, Zhang Weimou () and Zhang Qimou (). As a result of his family's ties to the Nationalist movement, Zhang faced difficulties in his early life.


title: Zhang Yimou

text: During the Cultural Revolution of the 1960s and 1970s, Zhang left his school studies and went to work, first as a farm labourer for 3 years, and later at a cotton textile mill for 7 years in the city of Xianyang. During this time he took up painting and amateur still photography, selling his own blood to buy his first camera. In 1978, he went to Beijing Film Academy and majored in photography. He has an Honorary Doctorate Degree from Boston University and also one from Yale University.


title: Zhang Yimou

text: When the Beijing Film Academy reopened its doors to new students in 1978, following the abandonment of policies adopted during the Cultural Revolution, Zhang, at 27, was over the regulation age for admission, and was without the prerequisite academic qualifications. After a personal appeal to the Ministry of Culture, and showing a portfolio of his personal photographic works, the authorities relented and admitted him to the Faculty of Cinematography. Zhang graduated with the class of 1982, which also included Chen Kaige, Tian Zhuangzhuang, and Zhang Junzhao. The class went on to form the core of the Fifth Generation, who were a part of an artistic reemergence in China after the end of the Cultural Revolution.


title: Zhang Yimou

text: Zhang and his co-graduates were assigned to small regional studios, and Zhang was sent to work for the Guangxi Film Studio as a cinematographer. Though originally intended to work as director's assistants, the graduates soon discovered there was a dearth of directors so soon after the Cultural Revolution, and gained permission to start making their own films. This led to the production of Zhang Junzhao's "One and Eight", on which Zhang Yimou worked as director of photography, and Chen Kaige's "Yellow Earth", in 1984. These two films were successes at the Hong Kong Film Festival and helped to bring the new Chinese cinema to the attention of worldwide audiences, signaling a departure from the earlier propagandist films of the Cultural Revolution. "Yellow Earth" is today widely considered the inaugural film of the Fifth Generation directors.


title: Zhang Yimou

text: In 1985, after moving back to his home town of Xi'an, Zhang was engaged as cinematographer and lead actor for director Wu Tianming's upcoming film "Old Well", which was subsequently released in 1987. The lead role won Zhang a Best Actor award at the Tokyo International Film Festival.


title: Zhang Yimou

text: 1988 saw the release of Zhang's directorial debut, "Red Sorghum", starring Chinese actress Gong Li in her first leading role. "Red Sorghum" was met with critical acclaim, bringing Zhang to the forefront of the world's art directors, and winning him a Golden Bear for Best Picture at the 38th Berlin International Film Festival in 1988.


title: Zhang Yimou

text: "Codename Cougar" (or "The Puma Action"), a minor experiment in the political thriller genre, was released in 1989, featuring Gong Li and eminent Chinese actor Ge You. However, it garnered less-than-positive reviews at home and Zhang himself later dismissed the film as his worst.


title: Zhang Yimou

text: In the same year, Zhang began work on his next project, the period drama "Ju Dou". Starring Gong Li in the eponymous lead role, along with Li Baotian as the male lead, "Ju Dou", garnered as much critical acclaim as had "Red Sorghum", and became China's first film to be nominated for an Academy Award for Best Foreign Language Film. "Ju Dou" highlighted the way in which the "gaze" can have different meanings, from voyeurism to ethical appeal.


title: Zhang Yimou

text: After the success of "Ju Dou", Zhang began work on "Raise the Red Lantern". Based on Su Tong's novel "Wives and Concubines", the film depicted the realities of life in a wealthy family compound during the 1920s. Gong Li was again featured in the lead role, her fourth collaboration with Zhang as director.


title: Zhang Yimou

text: "Raise the Red Lantern" received almost unanimous international acclaim. Film critic Roger Ebert of the "Chicago Sun-Times" noted its "voluptuous physical beauty" and sumptuous use of colours. Gong Li's acting was also praised as starkly contrasting with the roles she played in Zhang's earlier films. "Raise the Red Lantern" was nominated in the Best Foreign Language Film category at the 1992 Academy Awards, becoming the second Chinese film to earn this distinction (after Zhang's "Ju Dou"). It eventually lost out to Gabriele Salvatores's "Mediterraneo".


title: Zhang Yimou

text: Zhang's next directorial work, "The Story of Qiu Ju", in 1992, once again starring Gong Li in the lead role. The film, which tells the tale of a peasant woman seeking justice for her husband after he was beaten by a village official, was a hit at film festivals and won the Golden Lion award at the 1992 Venice Film Festival.


title: Zhang Yimou

text: Next, Zhang directed "To Live", an epic film based on the novel by Yu Hua of the same name. "To Live" highlighted the resilience of the ordinary Chinese people, personified by its two main characters, amidst three generations of upheavals throughout Chinese politics of the 20th century. It was banned in China, but released at the 1994 Cannes Film Festival and won the Grand Jury Prize, as well as earning a Best Actor prize for Ge You. "To Live" was banned in China by the Chinese State Administration of Radio, Film, and Television, due to its critical portrayal of various policies and campaigns of the Communist government.


title: Zhang Yimou

text: "Shanghai Triad" followed in 1995, featuring Gong Li in her seventh film under Zhang's direction. The two had developed a romantic as well as a professional relationship, but this would end during production of "Shanghai Triad". Zhang and Gong would not work together again until 2006's "Curse of the Golden Flower".


title: Zhang Yimou

text: 1997 saw the release of "Keep Cool", a black comedy film about life in modern China. "Keep Cool" marked only the second time Zhang had set a film in the modern era, after "The Story of Qiu Ju".


title: Zhang Yimou

text: As in "The Story of Qiu Ju", Zhang returned to the neorealist habit of employing non-professional actors and location shooting for "Not One Less" in 1999 which won him his second Golden Lion prize in Venice.


title: Zhang Yimou

text: Shot immediately after "Not One Less", Zhang's 1999 film "The Road Home" featured a new leading lady in the form of the young actress Zhang Ziyi, in her film debut. The film is based on a simple throw-back narrative centering on a love story between the narrator's parents.


title: Zhang Yimou

text: "Happy Times", a relatively unknown film by Zhang, was based loosely on the short story "Shifu: You'll Do Anything for a Laugh", by Mo Yan. Starring popular Chinese actor Zhao Benshan and actress Dong Jie, it was an official selection for the Berlin International Film Festival in 2002.


title: Zhang Yimou

text: Zhang's next major project was the ambitious wuxia drama "Hero", released in China in 2002. With an impressive lineup of Asian stars, including Jet Li, Maggie Cheung, Tony Leung Chiu-Wai, Zhang Ziyi, and Donnie Yen, "Hero" told a fictional tale about Ying Zheng, the King of the State of Qin (later to become the first Emperor of China), and his would-be assassins. The film was released in North America in 2004, two years after its Chinese release, by American distributor Miramax Films, and became a huge international hit. "Hero" was one of the few foreign-language films to debut at number 1 at the U.S. box office, and was one of the nominees for Best Foreign Language Film at the 2003 Academy Awards.


title: Zhang Yimou

text: Zhang followed up the huge success of "Hero" with another martial arts epic, "House of Flying Daggers", in 2004. Set in the Tang Dynasty, it starred Zhang Ziyi, Andy Lau, and Takeshi Kaneshiro as characters caught in a dangerous love triangle. "House of Flying Daggers" received acclaim from critics, who noted the use of colour that harked back to some of Zhang's earlier works.


title: Zhang Yimou

text: Released in China in 2005, "Riding Alone for Thousands of Miles" was a return to the more low-key drama that characterized much of Zhang's middle period pieces. The film stars Japanese actor Ken Takakura, as a father who wishes to repair relations with his alienated son, and is eventually led by circumstance to set out on a journey to China. Zhang had been an admirer of Takakura for over thirty years.


title: Zhang Yimou

text: 2006's "Curse of the Golden Flower" saw him reunited with leading actress Gong Li. Taiwanese singer Jay Chou and Hong Kong star Chow Yun-fat also starred in the period epic based on a play by Cao Yu.


title: Zhang Yimou

text: Zhang's recent films, and his involvement with the 2008 Olympic ceremonies, have not been without controversy. Some critics claim that his recent works, contrary to his earlier films, have received approval from the Chinese government. However, in interviews, Zhang has said that he is not interested in politics, and that it was an honour for him to direct the Olympic ceremonies because it was "a once-in-a-lifetime opportunity."


title: Zhang Yimou

text: In 2008, he won a Peabody Award "for creating a spell-binding, unforgettable celebration of the Olympic promise, featuring a cast of thousands" at the opening ceremony of the Beijing Olympics.


title: Zhang Yimou

text: On 24 May 2010, Zhang was awarded an honorary Doctor of Fine Arts degree by Yale University, and was described as "a genius with camera and choreography."


title: Zhang Yimou

text: Zhang's 2011 "The Flowers of War" was his most expensive film to date, budgeting for $90.2 million, until his 2016 "The Great Wall" surpassed it with a budget of $150 million.


title: Zhang Yimou

text: After the mixed reception and financial disappointment of "The Great Wall," Zhang returned in 2018 with the critically acclaimed "Shadow", which received 12 nominations at the 55th Golden Horse Awards and eventually won four, including Best Director.


title: Zhang Yimou

text: Reception of Zhang Yimou's films has been mixed. While some critics praise his striking aesthetics and ability to break into the Western art market, some Chinese-based critics have attacked Zhang for pandering to Western audiences and portraying China as weak, exotic, and vulnerable.


title: Zhang Yimou

text: Starting in the 1990s, Zhang Yimou has been directing stage productions in parallel with his film career.


title: Zhang Yimou

text: In 1998, he directed an acclaimed version of Puccini's opera "Turandot", firstly in Florence and then later Turandot at the Forbidden City, Beijing, with Zubin Mehta conducting, the latter documented in the film "The Turandot Project (2000)". He reprised his version of "Turandot" in October 2009, at the Bird's Nest Stadium in Beijing, and plans to tour with the production in Europe, Asia and Australia in 2010.


title: Zhang Yimou

text: In 2001, Zhang adapted his 1991 film "Raise the Red Lantern" for the stage, directing a ballet version.


title: Zhang Yimou

text: Zhang has co-directed a number of outdoor folk musicals under the title "Impression". These include "Impression, Liu Sanjie", which opened in August 2003 at the Li River, Guangxi province; "Impression Lijiang", in June 2006 at the foot of Jade Dragon Snow Mountain in Lijiang, Yunnan province; "Impression West Lake", in late 2007 at the West Lake in Hangzhou, Zhejiang province; "Impression Hainan" in late 2009, set in Hainan Island; and "Impression Dahongpao" set on Mount Wuyi, in Fujian province. All five performances were co-directed by Wang Chaoge and Fan Yue.


title: Zhang Yimou

text: Zhang also led the production of Tan Dun's opera, "The First Emperor", which had its world premiere at the Metropolitan Opera on 21 December 2006.


title: Zhang Yimou

text: In 2017 he directed an innovative ballet titled ‘’2047 Apologue’’, where the 12 minute solo finale "The Weaving Machine" was choreographed by Rose Alice Larkings and including hundreds of LED lamps. Onstage as Rose Alice danced the 12 minute solo was an elderly Chinese weaver at her loom, highlighting the old crafts and industries which remain so important in a world of new technology.


title: Zhang Yimou

text: Zhang Yimou was chosen to direct the Beijing portion of the closing ceremonies of the 2004 Summer Olympics in Athens, Greece, as well as the opening and closing ceremonies of the 2008 Summer Olympics in Beijing, China, alongside co-director and choreographer Zhang Jigang.


title: Zhang Yimou

text: Zhang was a runner-up for the Time Magazine Person of the Year award in 2008. Steven Spielberg, who withdrew as an adviser to the Olympic ceremonies to pressure China into helping with the conflict in Darfur, described Zhang's works in the Olympic ceremonies in "Time magazine", saying "At the heart of Zhang's Olympic ceremonies was the idea that the conflict of man foretells the desire for inner peace. This theme is one he's explored and perfected in his films, whether they are about the lives of humble peasants or exalted royalty. This year he captured this prevalent theme of harmony and peace, which is the spirit of the Olympic Games. In one evening of visual and emotional splendor, he educated, enlightened, and entertained us all."


title: Zhang Yimou

text: On 7 January 2022, it was reported that Zhang Yimou has once again chosen to direct the opening and closing ceremonies of the 2022 Winter Olympics and 2022 Winter Paralympics, which would be once again held in Beijing, China.


title: Zhang Yimou

text: Associated Press reported on 9 May 2013 that Zhang was being investigated for violating China's one-child policy. AP reported that he had allegedly fathered 7 children with 4 women, and faced large potential fines.


title: Zhang Yimou

text: According to the mainstream media in China, Zhang married Chen Ting, who is a dancer in December 2011; she had three children with him. However, when the news came out, Zhang had no immediate response. On 29 November 2013, under pressure from the public and criticism on the Internet, Zhang's studio released a statement that acknowledged Chen Ting and their three children. On 9 January 2014, the Lake District Family Planning Bureau, in accord with China's one-child policy, said Zhang was required to pay an unplanned birth and social maintenance fee totaling RMB 7.48 million (roughly US$1.2 million). On 7 February 2014, it was reported that Zhang had paid the fee.


title: Peng Shuai

text: Peng Shuai (; born 8 January 1986) is a Chinese retired professional tennis player. In February 2014, she was ranked world No. 1 doubles player by the WTA, becoming the first Chinese tennis player to achieve that ranking (in either singles or doubles). She peaked at No. 14 of the singles rankings in August 2011, and won two singles and 23 doubles titles on the WTA Tour.


title: Peng Shuai

text: Peng won a gold medal at the 2010 Asian Games, defeating Akgul Amanmuradova in the singles final. At the 2013 Wimbledon Championships, Peng won her first major title in doubles, with Hsieh Su-wei of Taiwan. She also won the women’s doubles title at the 2014 French Open with Hsieh. Her best performance at a major in singles came at the 2014 US Open where she reached the semifinals, becoming the third Chinese tennis player in history to make a Grand Slam semifinal after Zheng Jie and Li Na.


title: Peng Shuai

text: Peng is known for playing with two hands on both sides and hits very flat. She has defeated many top-10 and top-5 players, including Kim Clijsters, Martina Hingis, Amélie Mauresmo, Anastasia Myskina, Elena Dementieva, Francesca Schiavone, Jelena Janković, Agnieszka Radwańska, Marion Bartoli, and Vera Zvonareva.


title: Peng Shuai

text: On November 2, 2021, Peng posted on Weibo accusing Zhang Gaoli, a retired Vice Premier of China, of sexually assaulting her three years earlier. Peng and Zhang had purportedly been involved in an extramarital affair both before and after the incident, and their relationship ended shortly before her post. Information about her story was censored by the Chinese government. Afterwards, Peng disappeared from public view in what was suspected to be a forced disappearance. She showed up in state media two weeks later for appearances that observers believed to have been staged in response to foreign inquiries. E-mails and interview publications have depicted her denying that she made the accusation of sexual assault. The incident elicited international concern over her safety, whereabouts, and ability to communicate freely. The WTA suspended all events in China. In September 2022, the WTA announced that events would return to China in 2023, including the end of year finals.


title: Peng Shuai

text: Peng Shuai was born in Xiangtan. She began playing at age eight when an uncle, a famous tennis coach in China and the only other family member who plays tennis, introduced her to the game. She favors hardcourts and two-handed forehand (though her backhand is the better side). At age 13, Peng was admitted to a hospital for heart surgery to repair a defect, a situation which she explained in the 2008 "Impossible is Nothing" campaign from Adidas. Following surgery, in 1999 she joined the state training program in Tianjin aimed at producing internationally competitive athletes, especially Olympians.


title: Peng Shuai

text: In June 2001, the 15-year-old Peng won her first singles title at Baotou, a $10k tournament, defeating countrywoman Sun Tiantian in the semifinal. In October, she debuted on the WTA Tour at Shanghai as a wildcard.


title: Peng Shuai

text: In February 2002, Peng won her third $10k singles title. The following week, she extended her winning streak to twelve and reached the final of a $25k tournament at New Delhi before losing the championship match to Eva Birnerová. Further success proved elusive, and she took nearly seven months off from competition at singles events, ending the year with a world ranking of 358.


title: Peng Shuai

text: In 2003, she won her first $25k title at Jackson, Mississippi and her first $50k event at Changsha, improving her year-end ranking to 226.


title: Peng Shuai

text: In 2004, she won her first $75k tournament at Dothan, Alabama. She rose up to No. 107 in the rankings and gained her direct entry into her first Grand Slam main draw at Wimbledon. She was defeated there by 14th seed Silvia Farina Elia. Peng reached her first WTA Tour quarterfinal at Cincinnati in August, where she lost to second seed Vera Zvonareva. By the end of 2004, Peng had won her eighth ITF singles title at Shenzhen-2. She ended the year ranking 73 in the world and was able to focus solely on WTA events thereafter.


title: Peng Shuai

text: In mid-2000s, Peng decided to "fly solo" and no longer give over half of her earnings to the state training program. She and three other Chinese players broke out of the state's control by effectively threatening to stop playing.


title: Peng Shuai

text: At the Australian Open, Peng won her first Grand Slam singles match, defeating Maria Elena Camerin before being overcome by Venus Williams in the second round. In September 2005, she reached two further WTA quarterfinals at Beijing and Guangzhou, ultimately finishing the year ranking 37th.


title: Peng Shuai

text: In 2006, Peng lost her first-round ties at the Australian Open, withdrew from subsequent tournaments, and dropped out of the top 60 after losing at Indian Wells and Miami. At Wimbledon, however, she reached the third round of a Grand Slam for the first time, defeating 20th seed Shahar Pe'er before losing to 16th seed Flavia Pennetta. She reached the semifinals of the China Open and represented her country for the first time in her Fed Cup career, winning both her ties against Indonesia.


title: Peng Shuai

text: In 2007, Peng failed to get past the second round of the Australian Open and the first round of the US Open but reached the semifinals of the Tier-III Pattaya Open, losing to Sybille Bammer. At the China Open, Peng beat former world No. 1 and five-time Grand Slam champion Martina Hingis in the final match of Hingis's professional career. She finished the year with a 26–21 record in singles, a doubles title in Guangzhou with Yan Zi, and one top-ten win against Amélie Mauresmo.


title: Peng Shuai

text: After a string of losses in 2008, Peng recorded her first win of the year at the Tier-II Bangalore Open against Anne Kremer before losing to Venus Williams; she also won the doubles title with Sun Tiantian. At Strasbourg, she recorded a top-ten win when top seed Marion Bartoli was forced to retire in round one while trailing. Peng reached the second round of the US Open for the first time in her career, defeating Eleni Daniilidou before losing to Flavia Pennetta.


title: Peng Shuai

text: At the Summer Olympics, Peng competed in the women's singles, and the women's doubles with Sun Tiantian. The doubles pair were knocked out in the first round, and Peng lost to Alizé Cornet in the second round of the singles.


title: Peng Shuai

text: In early 2009, Peng announced that she will be coached full-time for the 2009 season by Tarik Benhabiles. She won the Sydney International doubles title with Hsieh Su-wei, defeating Nathalie Dechy and Casey Dellacqua in the final. At the Italian Open in May, Peng partnered with Hsieh again and won the doubles title by defeating Daniela Hantuchová and Ai Sugiyama. At the French Open, Peng was knocked out of the singles event but partnered with Hsieh to reach the semifinals in doubles, defeating Hantuchová & Sugiyama in the third round and the Radwańska sisters in the quarterfinals before losing to Victoria Azarenka and Elena Vesnina.


title: Peng Shuai

text: At the 2009 Wimbledon Championships, Peng fell to No. 11 Agnieszka Radwańska in the second round, after a mammoth battle of three and a half hours. Despite saving five match points, she eventually lost in three sets. Her 'never say die' attitude won her the admiration of many English fans. During the China Open, Peng defeated the 2008 champion and former world No. 1 Jelena Janković, who picked up a wrist injury in the latter stages of the match. In the third round, Peng won against former world No. 1 Maria Sharapova. In the quarterfinals, she lost to Nadia Petrova. In doubles, Peng partnered with Hsieh and won the title, bringing her ranking to a career high of 13.


title: Peng Shuai

text: At both the 2010 Moorilla Hobart International and the Australian Open, Peng was stopped by Zheng Jie. Peng reached the second round of the Indian Wells Open and the semifinals at the Estoril Open. Afterwards, she suffered a series of injuries. In the first round of the Premier Mandatory Madrid Open, she had to retire due to a right adductor muscle strain. In the second round, she was defeated by Arantxa Parra Santonja. Peng then withdrew from the French Open and missed the whole of the grass-court season due to illness. In the US Open, Peng advanced to the third round before withdrawing from the tournament with injury. She then suffered two first-round exits at the Pan Pacific Open and the China Open, lowering her world ranking to No. 95.


title: Peng Shuai

text: In the ITF Circuit, Peng won a trophy at the $100k event in Taipei and ended the season representing China in the 2010 Asian Games in Guangzhou. She won gold in the team event alongside Li Na, Yan Zi and Zhang Shuai. In the doubles event, she gained a bronze with partner Yan Zi, and in the singles event, she won another gold, defeating Akgul Amanmuradova of Uzbekistan.


title: Peng Shuai

text: At the 2011 Auckland Open, Peng caused a big upset by defeating No. 3 seed Kuznetsova in the second round. At the Australian Open, she defeated Kateryna Bondarenko, Jelena Janković, and Ayumi Morita to reach the fourth round of a Grand Slam for the first time in her career, before losing to Agnieszka Radwańska. Her ranking rose to No. 40. In the third round of the Indian Wells Open, she defeated No. 7 seed Li Na for the first time in her career. Peng reached the semifinals at the Family Circle Cup in Charleston, rising to a new career high of 29 in the world rankings. Peng also won the doubles title alongside Zheng Jie at the Italian Open.


title: Peng Shuai

text: Peng reached the finals at the Brussels Open but fell to world No. 1, Caroline Wozniacki. At the French Open, Peng retired due to illness in the third round. At Wimbledon, she reached the round of 16 before losing to the fifth seed and eventual runner-up Maria Sharapova. Posting good results at Cincinnati, Peng reached her career-high ranking of world No. 14. After withdrawing due to an injury from the US Open Series, she reached the round of 16 at the US Open before losing to Flavia Pennetta.


title: Peng Shuai

text: At both the 2012 Australian Open and the Dubai Tennis Championships, Peng fell in the second round. She took a month's break after her Malaysian Open and returned to play in the Madrid Open. However, she lost in early rounds in Madrid, Rome, and Brussels. At the Wimbledon Championships, she defeated Sandra Zaniewska, Ayumi Morita and Arantxa Rus to reach her second consecutive round of 16 before losing to Maria Kirilenko. At the 2012 Summer Olympics, Peng reached the second round in the women's singles and the quarterfinals in the women's doubles, partnering with Zheng Jie.


title: Peng Shuai

text: Peng started her 2013 season by reaching the semifinals in the new Shenzhen Open. She was not able to progress beyond the first or the second round, however, at Hobart International, the Australian Open, Madrid Open, Italian Open, and reached only the third round at Indian Wells. She reached the final in the Brussels Open, defeating Sofia Arvidsson, Olga Govortsova, Sloane Stephens, and Romina Oprandi before losing to Kaia Kanepi. Peng again lost in either the first or the second round at Roland Garros, Wimbledon, the US Open, and China Open. She ended the year with a ranking of No. 45 in the world.


title: Peng Shuai

text: On the other hand, Peng began her huge success as a doubles player in 2013. Paired with long-time childhood friend Hsieh Su-wei, Peng clinched five double's titles in 2013, including two Premier-5 events (Rome and Cincinnati), Wimbledon, and WTA Championships. Peng became the first Chinese player to win the WTA Tour Championships, and the fifth to win a Grand Slam title, after Zheng Jie, Yan Zi, Sun Tiantian and Li Na.


title: Peng Shuai

text: In 2014 Peng reached the final in the Shenzhen Open and lost to Li Na. At the Australian Open, she lost to Kurumi Nara in the opening round. She also lost in the second round in the doubles event with Hsieh against Shahar Pe'er and Sílvia Soler Espinosa. Peng then won two consecutive doubles titles, winning the Pattaya Open with Zhang Shuai defeating Alla Kudryavtseva and Anastasia Rodionova in the final, and winning the Qatar Open with Hsieh Su-wei defeating Květa Peschke and Katarina Srebotnik in the final. On 17 February, Peng became the world No. 1 in doubles, making her the first Chinese tennis player (man or woman, in singles or doubles) to reach world No. 1. Peng and Hsieh continued their success by winning three more titles in the season, including two Premiere Mandatory events (Indian Wells and Beijing) and French Open. In the Wimbledon Championships, they failed in defending their title and also lost their No. 1 ranking.


title: Peng Shuai

text: However, since Wimbledon, Peng found her pace in the singles events. She reached the last 16 at the Wimbledon Championships. She also clinched title in the 125K event in Nanchang. At the US Open, Peng made her first Grand Slam singles quarterfinal and semifinal appearances, defeating compatriot Zheng Jie, fourth-seeded Agnieszka Radwańska, 28th-seeded Roberta Vinci, 14th-seeded Lucie Šafářová, and rising star Belinda Bencic en route, all in straight sets. In the semi-final, she had to retire against 10th-seeded Caroline Wozniacki in the semifinal at 6–7, 3–4 down, when she suffered severe cramps due to heat illness and had to be taken off court in a wheelchair. She skipped playing the following Hong Kong Open to recover, but made promotional appearances there. She came back to the game in the Wuhan Open, China Open, and Tianjin Open.


title: Peng Shuai

text: Peng and Hsieh entered the WTA Finals as the second seed. They beat Garbiñe Muguruza/Carla Suárez Navarro and Alla Kudryavtseva/Anastasia Rodionova in straight sets to reach the final. However, they lost disastrously to Cara Black/Sania Mirza. They pairing then came to conclusion as Peng had previously announced during the US Open.


title: Peng Shuai

text: Peng had a difficult 2015 season. Due to injuries, she had a lot of first round or second round losses. Her best performance of the season was fourth round of the Australian Open, which tied her best performance, although she lost it to second seed and the eventual runner-up Maria Sharapova.


title: Peng Shuai

text: Peng failed to win even one doubles match in 2015. She started the season with her Tianjin teammate Xu Yifan in preparation for the 2016 Olympics, but ended up losing in the first round matches in the Shenzhen Open and Australian Open. In their first-round match in the Australian Open against Kimiko Date-Krumm and Casey Dellacqua, they wasted a 6–4, 5–0 lead, two match points in the second set and three more in the twelfth game of the final set and eventually lost to their opponents. Peng was particularly frustrated by Xu, who was constantly attacked by their opponents and making a lot of unforced errors, and decided to split the partnership. She played two more tournaments, with Květa Peschke at Dubai and Zarina Diyas at Madrid, but was not able to win a set.


title: Peng Shuai

text: After the first round retirement to Polona Hercog in the first round of French Open, Peng announced the end of her 2015 season due to injuries in her back and waist.


title: Peng Shuai

text: Peng returned to the tour in 2016 in Indian Wells. She suffered from a number of early exits in several events, including the first-round loss in the Rio Summer Olympics. In the China Open, Peng defeated Venus Williams in the first round but lost to Caroline Garcia in the second round. In the Tianjin Open, she clinched her first ever WTA singles title. As a resident in Tianjin, she received a withdraw from seventh seed and compatriot Zhang Shuai in the beginning round. In the second round, she defeated qualifier Chang Kai-chen. She benefited from the withdraw from Agnieszka Radwańska in the quarterfinal and beat Danka Kovinić from Montenegro in the semifinal, which lasted for two days due to rain delays. Peng had to play the final several hours after the semifinal against the 2014 champion Alison Riske and won in two sets. She also won the doubles final match with Christina McHale.


title: Peng Shuai

text: Peng started the season in the Shenzhen Open, where she lost to the eventual champion, Kateřina Siniaková, in the opening round. In the doubles event, she clinched her 21st title with Andrea Hlaváčková without losing a set. In the singles event of the Australian Open, she lost to Eugenie Bouchard in the second round. In the doubles event, Peng and Hlaváčková reached the final without dropping a set, defeating third seed and Olympic champion Ekaterina Makarova/Elena Vesnina and top-seed Caroline Garcia/Kristina Mladenovic. In the final, they lost a tight match against the 2015 champion Mattek-Sands/Šafářová in three sets.


title: Peng Shuai

text: Peng then competed in the Taiwan Open, where she reached her eighth WTA final without losing a set. In the final, she lost to the top seed and world No. 13 Elina Svitolina. She also reached the final in the doubles event with Hlaváčková and lost to Vesnina/Makarova with a match tie-break. She reached the fourth round in Indian Wells but fell in the first round in the Madrid Open against Carla Suárez Navarro.


title: Peng Shuai

text: She won her second WTA singles title at the Jiangxi International Open in Nanchan, China defeating Nao Hibino in the final.


title: Peng Shuai

text: In August 2018, Peng was banned for six months and fined $10,000 by the Tennis Integrity Unit (TIU) for coercion and offering possible financial reward so that her main partner would agree to withdraw from the ladies doubles, even after the sign-in deadline at the 2017 Wimbledon Championships. Her partner, Alison Van Uytvanck, refused and Peng withdrew from the tournament instead. Three months of her ban and $5,000 of the fine were suspended. Her former coach Bertrand Perret of France was also banned for three months.


title: Peng Shuai

text: Peng began her 2019 season at the Shenzhen Open. She retired during the third set of her first-round match against Kristýna Plíšková due to a thigh injury. In doubles, she and compatriot, Yang Zhaoxuan, won the title beating Duan Yingying/Renata Voráčová in the final. Peng was defeated in the first rounds at the Australian Open, Roland Garros, and Wimbledon, and in the second round at the US Open.


title: Peng Shuai

text: At the Jiangxi International Open, Peng made it to the semifinals where she was defeated by Elena Rybakina. In doubles, she and Zhang Shuai reached the final but lost to compatriots Wang Xinyu/Zhu Lin. Peng lost in the second round at Guangzhou in singles but won the doubles title with Laura Siegemund, defeating Alexa Guarachi/Giuliana Olmos in the final. She was defeated in the first or second rounds in Wuhan, Beijing, and Tianjin but won against Zhu Lin in the final of the Suzhou Ladies Open.


title: Peng Shuai

text: Peng started her 2020 season at the Shenzhen Open, where she lost in the second round to fifth seed and eventual champion Ekaterina Alexandrova. In Hobart, she and Zhang Shuai reached the doubles final but were defeated by Nadiia Kichenok/Sania Mirza. At the Australian Open, Thailand Open, and Qatar Total Open, she was defeated in either the first or the second round.


title: Peng Shuai

text: In February 2022, Peng announced her retirement during an interview with the French magazine "L'Équipe", where she also mentioned her 2021 social post and subsequent events.


title: Peng Shuai

text: In November 2021, Peng accused Zhang Gaoli, a retired Chinese vice-premier and high-ranking CCP member, reportedly of sexual assault in 2018. The revelation came from her Weibo post, which disclosed that the two had gotten together even before 2018. Zhang officially retired that year, however, and after inviting Peng to visit him and his wife, he wanted to have sex with her again. Peng refused at first, but Zhang continued to try to persuade her. Scared and panicking, she finally agreed due to previous feelings for him. They renewed their semi-private relationship until a dispute on 30 October. Peng's post drew attention to the MeToo movement in China, where activist Zhou Xiaoxuan expressed her sympathies, but it was removed within 20 minutes of being uploaded, and related discussions became widely censored in China.


title: Peng Shuai

text: Peng did not communicate on social media afterwards and was not reachable by the WTA. Chinese officials gave vague responses to inquiries, while the Chinese Tennis Association said she was safe. WTA chief executive Steve Simon called on Chinese authorities to investigate her allegations and stop censoring the subject, and multiple tennis personalities expressed their concerns. Depictions of Peng emerged via Chinese state media, but various China watchers and human rights advocates worried that they might be government propaganda and she might not be free. The International Tennis Federation, Amnesty International, the EU, and the UNHCHR called for proof of her safety, while the WTA decided to suspend all tournaments in China. The IOC said it held two video calls with her. In a later video from "Lianhe Zaobao" and an interview with the French magazine "L'Équipe", Peng said she had not accused anyone of sexual assault, but some skepticism persists.


title: Peng Shuai

text: Peng attended a meeting with IOC officials and several events at the 2022 Winter Olympics. Her revelations and subsequent events remain censored within China. She was included in "Time" magazine's "100 Most Influential People of 2022", where feminist activist Lü Pin highlighted Peng's Weibo post, disappearance, and impact on the defense of women's rights against authoritarianism.


title: Peng Shuai

text: In August 2022, WTA president David Haggerty said he had spoken to Peng and she appeared to be safe. In September 2022, the WTA announced that events would return to China in 2023, including the end of year finals.


title: Chrysler PT Cruiser

text: The Chrysler PT Cruiser is a retro-styled compact car manufactured and marketed internationally by Chrysler in 5-door hatchback wagon (2001–2010) and 2-door convertible (2005–2008) body styles—over a single generation, with an intermediate facelift for model year 2006.


title: Chrysler PT Cruiser

text: Evoking 1930s styling, the exterior of the PT Cruiser was designed by Bryan Nesbitt. Interior packaging was noted for its high-roof, high h-point seating, and flexible cargo and passenger configurations enabled by a multi-level rear cargo shelf and rear seats a user could fold, tumble, or remove.


title: Chrysler PT Cruiser

text: By the end of production in July 2010, worldwide production had reached 1.35 million. Originally planned as a Plymouth model, The PT Cruiser was ultimately marketed as a Chrysler when Plymouth was discontinued. In its nameplate, "PT" stands for "Personal Transport" or "Personal Transportation". "PT" was the PT Cruiser's product code.


title: Chrysler PT Cruiser

text: The PT Cruiser is a front-wheel drive 5-passenger vehicle, classified as a truck in the US by the NHTSA for CAFE fuel economy calculations but as a car by most other metrics. Chrysler specifically designed the PT Cruiser to fit the NHTSA criteria for a light truck in order to bring the average fuel efficiency of the company's truck fleet into compliance with CAFE standards. A turbocharged GT model was introduced for the 2003 model year. A convertible was introduced for the 2005 model year.


title: Chrysler PT Cruiser

text: Manufactured at Toluca Car Assembly in Toluca, Mexico. On March 8, 2006, DaimlerChrysler said it had built 1,000,000 PT Cruisers. The final PT Cruiser was built on July 9, 2010. A total of 1,050,281 PT Cruisers were marketed in the United States.


title: Chrysler PT Cruiser

text: The PT Cruiser was also assembled at the Eurostar Automobilwerk in Graz, Austria, for markets outside North America, using the production code PG — for the model year 2002. European PT Cruisers built in 2001 or from 2003 on were manufactured in Mexico under the PT production code. The US version used a 2.4 L four-cylinder gasoline engine. In addition to this standard model, a 2.2 L four-cylinder diesel engine built by Mercedes-Benz was also available in Europe, Asia, and South Africa. A 2.0 L engine (D4RE) was also available outside the US. It produced SAE at 6500 rpm with of torque at 4800 rpm.


title: Chrysler PT Cruiser

text: In July 2000, the PT Cruiser replaced the Neon in Japan as their small car offering. Chrysler sold over 10,000 PT Cruisers in Japan. Both the five-door hatchback and the two-door convertible, and the GT turbo engine package were sold in Japan in right-hand drive configurations. Japanese versions were manufactured at the Austrian factory and were equipped very similarly to European specifications.


title: Chrysler PT Cruiser

text: From 2002 until 2007, the PT Cruiser was used in the city of Urayasu, Chiba Prefecture by the Maihama Resort Cab company to serve as a taxi in the vicinity of Tokyo Disneyland because of its fun, retro appearance. The PT Cruiser taxis augmented the Maihama Resort Line service to Tokyo Disneyland. The ten Chrysler PT Cruisers were to help customers "hold on to that lingering Disney feeling all the way home." The cars were offered in Japan until April 2010, when it was replaced by the Dodge Caliber.


title: Chrysler PT Cruiser

text: In 2001 "Motor Trend" named the PT Cruiser as its Car of the Year and "Car and Driver" put it on its Ten Best list. the PT Cruiser also won the North American Car of the Year. The PT Cruiser also won the North American Car of the Year, based on a solid majority of top automotive journalists charged with "picking a vehicle that has set new standards or benchmarks in its class while considering aspects such as general design, safety, fuel economy, handling, general roadworthiness, performance, functionality, technical innovation, driver satisfaction and price." It also held the distinction of being listed by "Consumer Guide" automotive editors as "Best Buy" continuously from 2001 through 2006.


title: Chrysler PT Cruiser

text: The PT Cruiser was part of the nostalgia wave that included models such as the Volkswagen New Beetle and the Mini Cooper. Rather than recalling previous namesake models, the PT Cruiser recalled 1930s and 1940s styling, influenced by the Chrysler Airflow.


title: Chrysler PT Cruiser

text: The 2000 PT Cruiser grew out of a collaboration with Robert A. Lutz, who was an executive at Chrysler at the time, Dr. Clotaire Rapaille, and Bryan Nesbitt. Nesbitt later went on to design the Chevrolet HHR.


title: Chrysler PT Cruiser

text: The PT Cruiser's design was loosely inspired by the Chrysler Pronto Cruizer concept car, while recalling the Chrysler and Desoto Airflow. Aspects of the PT Cruiser's rear styling also resembled the Chrysler CCV, a retro-styled compact vehicle recalling the Citroen 2CV, while its high roof evoked the Chevrolet Advance Design trucks.


title: Chrysler PT Cruiser

text: On launch, the PT Cruiser was described as a market segment-buster. Chrysler's Dieter Zetsche described it as a continuing example of the automaker's innovation for new segments, demonstrating "you can have head-turning style, practicality, and value all in one package."


title: Chrysler PT Cruiser

text: "Car and Driver" described the design as "highly distinctive, appeals to a broad segment of the public, and is characteristically American." The public's reaction to the PT Cruiser was polarized, but inspired a strong following among owners. Overall, car "was a valuable vehicle for many consumers."


title: Chrysler PT Cruiser

text: PT Cruiser models included the Classic edition, Limited edition, Touring edition, Couture edition (2010 only), "Dream Cruiser", "Street Cruiser", "Pacific Coast Highway" edition", and PT Cruiser GT. The non-GT Turbo () edition models (introduced in 2004) were identified by a "2.4L Turbo" badge on the lower right-hand corner of the rear lift-gate. The GT model (introduced in 2003) has a "2.4L Turbo High Output" badge on the right-hand corner of the lift-gate indicating the 215–230 hp engine version.


title: Chrysler PT Cruiser

text: Updates for 2006 included scalloped headlights, a revised grille no longer extending below the "bumper", a new lower front fascia that eliminated the patented brake cooling ducts, and redesigned taillights as well as available round fog lamps. The changes reduced the "retro vibe" that did not satisfy some customers as well as "exposing the main pitfall of retro design: How do you update old?"


title: Chrysler PT Cruiser

text: Interior updates included a revised interior with an updated dash with an analog clock in the center stack. The audio system featured a line-in jack for MP3 players integrated into the dash. Features such as satellite radio, a premium sound system by Boston Acoustics with external amplifier and subwoofer, and UConnect hands-free Bluetooth for compatible cell phones also became available in 2006. The standard audio system now included an AM/FM stereo with single-disc CD player and a 3.5-millimeter auxiliary audio input jack with six speakers, replacing a cassette player and four speakers. The turbocharged 2.4 L I4 was available in or "High-Output" versions. A "Mopar" cruise control unit became available as an aftermarket unit on 2007 models. Also in 2007, Chrysler dealers were permitted to order vehicles with separate options (unbundled options from option packages) such as anti-lock brakes and Side Impact Airbags. Sirius Satellite Radio also became an option that could be installed as a dealer option with a factory appearance (i.e. roof mount satellite antenna).


title: Chrysler PT Cruiser

text: Original plans called for the PT Cruiser to receive a complete redesign for 2007, but that was pushed back and little was changed as Chrysler was sorting out powertrains, styling, and even positioning as a luxury compact SUV. The convertible and the High Output engine were discontinued for the 2008 model year and sales dropped as information spread that Chrysler was going to discontinue the PT Cruiser in the near future.


title: Chrysler PT Cruiser

text: For 2010, the last year of production, one trim level was offered, the PT Cruiser Classic. It included a 2.4 L I4 engine producing , standard four-speed automatic transmission with overdrive, AM/FM stereo with single-disc CD player, auxiliary audio input jack and six speakers, rear-mounted spoiler, air conditioning, cruise control, tire pressure monitoring system (TPMS), front and rear side SRS airbags, keyless entry with two remotes and panic alarm, ABS anti-lock brakes, and sixteen-inch alloy wheels. Optional features included leather seating surfaces, a power front driver's bucket seat, an engine block heater, a power tilt-and-sliding sunroof, and heated dual front bucket seats. Exterior color options were Bright Silver Metallic, Brilliant Black Crystal Pearl Coat, Deep Water Blue Pearl Coat, Inferno Red Crystal Tinted Pearl Coat, Silver Steel Metallic, and Stone White Clear Coat. The only interior color option was Pastel Slate Gray.


title: Chrysler PT Cruiser

text: In 2002, Euro NCAP gave the PT Cruiser a three (out of five) stars rating. The car received bad results in the frontal impact test (6 out of 16 possible points). The height of the seats and side airbags helped to attain a maximum score of 16 points in the side impact test. The low frontal scores are in part explained by the cushioning near the knees, designed to protect unbelted occupants, which is not a factor in the EU where passengers must wear seat belts.


title: Chrysler PT Cruiser

text: For 2008, the Insurance Institute for Highway Safety gave the PT Cruiser the highest rating of "Good" overall for occupant protection in frontal crashes and the lowest overall rating of "Poor" for side crashes. It was the only small car to not offer electronic stability control.


title: Chrysler PT Cruiser

text: Two-door convertible (same features as 5-door hatchback models with power-retractable cloth convertible top)


title: Chrysler PT Cruiser

text: The PT Cruiser GT (also known as the GT Cruiser) is a high output turbocharged hot hatch variant of the PT Cruiser introduced as a 2003 model. The engine was upgraded over the standard PT. This included stronger crankcase webbing, a thicker deck with 11 mm head bolts (vs 10 mm), an oil drain back for the turbo, a cast aluminum structural oil pan, a higher capacity oil pump, a crankshaft of higher hardness steel, improved machining of bearing journals, oil squirters (to cool underside of pistons), eutectic aluminum alloy pistons made specially by Mahle, and forged connecting rods with cracked caps and 9 mm bolts. The cylinder head was also different for turbo engines, from naturally aspirated. The turbo version was used by both the PT Cruiser GT and Dodge Neon SRT-4 with larger diameter valves and seats, exhaust valves made of Inconel, improved cooling as well as larger oil drain back passages, and different camshafts. The PT Cruiser Turbo engine package differs from the SRT-4 because the intake manifold, turbocharger plumbing, and intercooler are different. Upgrades such as MOPAR Stage 1 and supporting turbo components were available from the manufacturer. Many of the performance parts from the SRT-4 are compatible with the PT Cruiser GT.


title: Chrysler PT Cruiser

text: Factory Flame package – Flame-inspired vinyl decals of a 'tone-on-tone' type applied on the hood, front fenders, and front doors starting with 2002 models. Four flame designs: fading orange-to-red on cars painted red, fading blue to cranberry flames on cranberry finished cars, a dark silver that fades into bright silver on bright silver paint, and fading deep magenta to black flames highlighted with a blue border on black cars.


title: Chrysler PT Cruiser

text: Woodie package – Available on all Chrysler PT Cruiser models from 2002 to 2004 models, the simulated wood panels were on the sides of the vehicle and on the rear hatch. The vinyl graphic featured a linear Medium Oak woodgrain framed with Light Ash surround moldings. Chrysler described the designers were inspired by the automaker's wood-body heritage from the early Town & Country models.


title: Chrysler PT Cruiser

text: Available in 2006 was a Route 66 edition in either Solar Yellow or Black. This version featured Solar Yellow brake calipers, body-color grille, solar-tinted sunscreen glass, liftgate mounted spoiler, 17-inch all-season performance tires, four-wheel anti-lock brakes with low-speed traction control, speed control, and a sport suspension. Brightwork accents included chrome body-side moldings, a chrome exhaust tip and , chrome-plated, and five-spoke Empire aluminum wheels. Exterior identification included a 'Street Cruiser' badge with Solar Yellow accents on the liftgate and 'Route 66' badges on the front doors.


title: Chrysler PT Cruiser

text: A total of 1,443 Route 66 models were produced in North America, with 142 of those equipped with 5-speed manual transmissions.


title: Chrysler PT Cruiser

text: Announced during the 2006 Woodward Dream Cruise and named for one of the most picturesque driving routes (officially called California State Route 1) in the United States.


title: Chrysler PT Cruiser

text: Based on the 2007 Chrysler PT Cruiser Touring model, this version featured a Pacific Blue Pearl body, 4-wheel disc anti-lock brakes, bright front door sill scuff pads, front bright accent ring cupholders, silver shift knob, chrome lock knobs, chrome bodyside molding, "Street Cruiser" badge, Pacific Coast Highway edition badge, rear body-color spoiler, bright exhaust tip, leather steering wheel with bright spokes, sport suspension, supplemental side airbags, 6-way power driver's seat, Sirius satellite radio, 17x6 inch aluminum platinum clad wheels with 205/50R17XL 93H BSW all-season performance tires, and cloth seats with blue insert.


title: Chrysler PT Cruiser

text: A version of the 2008 PT Cruiser LX was named for a road (Sunset Boulevard in Los Angeles, California) that has been described as an influence for many movies, songs, and TV shows. Street Cruiser Sunset Boulevard edition version was based on the 2008 PT Cruiser LX and was limited to 500 units. It included "Sunset Crystal" paint accented with various chrome body-trim pieces, deep-tint glass, 16-inch chrome wheels with all-season touring tires.


title: Chrysler PT Cruiser

text: The Dream Cruiser Series 5 production totaled 1,750 units for the US market during the 2009 model year. It included 4-speed automatic transmission, pearl-white body (however one was made in inferno red and one finished in bright silver) with a black roof and spoiler, gray interior with leather and chrome trim, upper- and lower-billet aluminum cross-hatch pattern grilles, chromed door handles and body-side moldings, chromed stainless-steel exhaust tip (on turbo), SRT Design wheels with all-season performance tires, unique 'PT' door badging, PT Dream Cruiser Series 5 tailgate badge, fog lamps, and a Brilliant Black Crystal Pearl hoop spoiler.


title: Chrysler PT Cruiser

text: Based on the Classic trim, the Sport version included a roof-mounted body-colored spoiler, 16-inch alloy wheels, PT Cruiser Sport badge, and an exclusive graphite metallic paint.


title: Chrysler PT Cruiser

text: The Couture edition was exhibited at the 2010 Geneva Motor Show as part of the expansion to the European market It featured a contrasting two-tone paint scheme: black above the beltline and silver metallic on the bottom section, with a red pinstripe dividing the two. The interior included Radar Red leather buckets (dark gray leather optional) with black piping, a red or black shifter knob, and chrome appliques. Outside features included 16-inch chrome-clad wheels with chromed bodyside moldings and accents. Production was limited to 500.


title: Chrysler PT Cruiser

text: The Chrysler Panel Cruiser was a design study that appeared at the 2000 Detroit Auto Show. Based on the PT Cruiser, it featured panels in place of the rear doors and a wooden floor in place of the rear seats. The Panel Cruiser was designed to mix elements of classic American panel trucks with those of a sports sedan. It rolled on 17-inch chrome wheels with 215/45 tires. The rear interior space featured a full wood floor with bright skid strips and wood bars along with the side quarter panels with cargo straps providing a multi-use cargo area.


title: Area of a circle

text: In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.


title: Area of a circle

text: One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula–that the area is half the perimeter times the radius–namely, , holds in the limit for a circle.


title: Area of a circle

text: Although often referred to as the area of a circle in informal contexts, strictly speaking the term "disk" refers to the interior region of the circle, while "circle" is reserved for the boundary only, which is a curve and covers no area itself. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle.


title: Area of a circle

text: Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis. However, the area of a disk was studied by the Ancient Greeks. Eudoxus of Cnidus in the fifth century B.C. had found that the area of a disk is proportional to its radius squared. Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book "Measurement of a Circle". The circumference is 2"r", and the area of a triangle is half the base times the height, yielding the area  "r" for the disk. Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality.


title: Area of a circle

text: A variety of arguments have been advanced historically to establish the equation formula_1 to varying degrees of mathematical rigor. The most famous of these is Archimedes' method of exhaustion, one of the earliest uses of the mathematical concept of a limit, as well as the origin of Archimedes' axiom which remains part of the standard analytical treatment of the real number system. The original proof of Archimedes is not rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident.


title: Area of a circle

text: The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius.


title: Area of a circle

text: Following Archimedes' argument in "The Measurement of a Circle" (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. We eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way.


title: Area of a circle

text: Suppose that the area "C" enclosed by the circle is greater than the area "T" = ⁄"cr" of the triangle. Let "E" denote the excess amount. Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, "G", is greater than "E", split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, "G". Continue splitting until the total gap area, "G", is less than "E". Now the area of the inscribed polygon, "P" = "C" − "G", must be greater than that of the triangle.


title: Area of a circle

text: But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon; its length, "h", is less than the circle radius. Also, let each side of the polygon have length "s"; then the sum of the sides, "ns", is less than the circle circumference. The polygon area consists of "n" equal triangles with height "h" and base "s", thus equals ⁄"nhs". But since "h" < "r" and "ns" < "c", the polygon area must be less than the triangle area, ⁄"cr", a contradiction. Therefore, our supposition that "C" might be greater than "T" must be wrong.


title: Area of a circle

text: Suppose that the area enclosed by the circle is less than the area "T" of the triangle. Let "D" denote the deficit amount. Circumscribe a square, so that the midpoint of each edge lies on the circle. If the total area gap between the square and the circle, "G", is greater than "D", slice off the corners with circle tangents to make a circumscribed octagon, and continue slicing until the gap area is less than "D". The area of the polygon, "P", must be less than "T".


title: Area of a circle

text: This, too, forces a contradiction. For, a perpendicular to the midpoint of each polygon side is a radius, of length "r". And since the total side length is greater than the circumference, the polygon consists of "n" identical triangles with total area greater than "T". Again we have a contradiction, so our supposition that "C" might be less than "T" must be wrong as well.


title: Area of a circle

text: Therefore, it must be the case that the area enclosed by the circle is precisely the same as the area of the triangle. This concludes the proof.


title: Area of a circle

text: Following Satō Moshun and Leonardo da Vinci , we can use inscribed regular polygons in a different way. Suppose we inscribe a hexagon. Cut the hexagon into six triangles by splitting it from the center. Two opposite triangles both touch two common diameters; slide them along one so the radial edges are adjacent. They now form a parallelogram, with the hexagon sides making two opposite edges, one of which is the base, "s". Two radial edges form slanted sides, and the height, "h" is equal to its apothem (as in the Archimedes proof). In fact, we can also assemble all the triangles into one big parallelogram by putting successive pairs next to each other. The same is true if we increase it to eight sides and so on. For a polygon with 2"n" sides, the parallelogram will have a base of length "ns", and a height "h". As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius. In the limit, the parallelogram becomes a rectangle with width "r" and height "r".


title: Area of a circle

text: There are various equivalent definitions of the constant π. The conventional definition in pre-calculus geometry is the ratio of the circumference of a circle to its diameter:


title: Area of a circle

text: However, because the circumference of a circle is not a primitive analytical concept, this definition is not suitable in modern rigorous treatments. A standard modern definition is that is equal to twice the least positive root of the cosine function or, equivalently, the half-period of the sine (or cosine) function. The cosine function can be defined either as a power series, or as the solution of a certain differential equation. This avoids any reference to circles in the definition of , so that statements about the relation of to the circumference and area of circles are actually theorems, rather than definitions, that follow from the analytical definitions of concepts like "area" and "circumference".


title: Area of a circle

text: The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral


title: Area of a circle

text: The integral appearing on the right is an abelian integral whose value is a half-period of the sine function, equal to . Thus formula_6 is seen to be true as a theorem.


title: Area of a circle

text: Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula formula_1, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can develop trigonometric functions and the fundamental constant in a way that is totally independent of their relation to geometry. We have indicated where appropriate how each of these proofs can be made totally independent of all trigonometry, but in some cases that requires more sophisticated mathematical ideas than those afforded by elementary calculus.


title: Area of a circle

text: Using calculus, we can sum the area incrementally, partitioning the disk into thin concentric rings like the layers of an onion. This is the method of shell integration in two dimensions. For an infinitesimally thin ring of the "onion" of radius "t", the accumulated area is 2"t dt", the circumferential length of the ring times its infinitesimal width (one can approximate this ring by a rectangle with width=2"t" and height="dt"). This gives an elementary integral for a disk of radius "r".


title: Area of a circle

text: It is rigorously justified by the multivariate substitution rule in polar coordinates. Namely, the area is given by a double integral of the constant function 1 over the disk itself. If "D" denotes the disk, then the double integral can be computed in polar coordinates as follows:


title: Area of a circle

text: An equivalent rigorous justification, without relying on the special coordinates of trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula,


title: Area of a circle

text: Similar to the onion proof outlined above, we could exploit calculus in a different way in order to arrive at the formula for the area of a disk. Consider unwrapping the concentric circles to straight strips. This will form a right angled triangle with r as its height and 2r (being the outer slice of onion) as its base.


title: Area of a circle

text: The opposite and adjacent angles for this triangle are respectively in degrees 9.0430611..., 80.956939... and in radians 0.1578311... , 1.4129651...


title: Area of a circle

text: Explicitly, we imagine dividing up a circle into triangles, each with a height equal to the circle's radius and a base that is infinitesimally small. The area of each of these triangles is equal to formula_22. By summing up (integrating) all of the areas of these triangles, we arrive at the formula for the circle's area:


title: Area of a circle

text: It too can be justified by a double integral of the constant function 1 over the disk by reversing the order of integration and using a change of variables in the above iterated integral:


title: Area of a circle

text: The triangle proof can be reformulated as an application of Green's theorem in flux-divergence form (i.e. a two-dimensional version of the divergence theorem), in a way that avoids all mention of trigonometry and the constant . Consider the vector field formula_27 in the plane. So the divergence of r is equal to two, and hence the area of a disc "D" is equal to


title: Area of a circle

text: where n is the unit normal and "ds" is the arc length measure. For a circle of radius "R" centered at the origin, we have formula_30 and formula_31, so the above equality is


title: Area of a circle

text: The integral of "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle.


title: Area of a circle

text: Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of at the centre of the circle), each with an area of (derived from the expression for the area of a triangle: ). Note that due to small angle approximation. Through summing the areas of the triangles, the expression for the area of the circle can therefore be found:


title: Area of a circle

text: The last step follows since the trigonometric identity formula_40 implies that formula_41 and formula_42 have equal integrals over the interval formula_43, using integration by substitution. But on the other hand, since formula_44, the sum of the two integrals is the length of that interval, which is formula_45. Consequently, the integral of formula_46 is equal to half the length of that interval, which is formula_47.


title: Area of a circle

text: Therefore, the area of a circle of radius "r", which is twice the area of the semi-circle, is equal to formula_48.


title: Area of a circle

text: This particular proof may appear to beg the question, if the sine and cosine functions involved in the trigonometric substitution are regarded as being defined in relation to circles. However, as noted earlier, it is possible to define sine, cosine, and in a way that is totally independent of trigonometry, in which case the proof is valid by the change of variables formula and Fubini's theorem, assuming the basic properties of sine and cosine (which can also be proved without assuming anything about their relation to circles).


title: Area of a circle

text: The circle is the closed curve of least perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean plane has perimeter "C" and encloses an area "A" (by the Jordan curve theorem) then


title: Area of a circle

text: Moreover, equality holds in this inequality if and only if the curve is a circle, in which case formula_1 and formula_51.


title: Area of a circle

text: The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of Willebrord Snell ("Cyclometricus", 1621), further developed by Christiaan Huygens ("De Circuli Magnitudine Inventa", 1654), described in .


title: Area of a circle

text: Given a circle, let "u" be the perimeter of an inscribed regular "n-"gon, and let "U" be the perimeter of a circumscribed regular "n-"gon. Then "u" and "U" are lower and upper bounds for the circumference of the circle that become sharper and sharper as "n" increases, and their average ("u" + "U")/2 is an especially good approximation to the circumference. To compute "u" and "U" for large "n", Archimedes derived the following doubling formulae:


title: Area of a circle

text: Starting from a hexagon, Archimedes doubled "n" four times to get a 96-gon, which gave him a good approximation to the circumference of the circle.


title: Area of a circle

text: The number ⁄ is also an excellent approximation to , better than any other rational number with denominator less than 16604.


title: Area of a circle

text: This for "n" = 48 gives a better approximation (about 3.14159292) than Archimedes' method for "n" = 768.


title: Area of a circle

text: Let one side of an inscribed regular "n-"gon have length "s" and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle on a diameter. By Thales' theorem, this is a right triangle with right angle at B. Let the length of A′B be "c", which we call the complement of "s"; thus "c"+"s" = (2"r"). Let C bisect the arc from A to B, and let C′ be the point opposite C on the circle. Thus the length of CA is "s", the length of C′A is "c", and C′CA is itself a right triangle on diameter C′C. Because C bisects the arc from A to B, C′C perpendicularly bisects the chord from A to B, say at P. Triangle C′AP is thus a right triangle, and is similar to C′CA since they share the angle at C′. Thus all three corresponding sides are in the same proportion; in particular, we have C′A : C′C = C′P : C′A and AP : C′A = CA : C′C. The center of the circle, O, bisects A′A, so we also have triangle OAP similar to A′AB, with OP half the length of A′B. In terms of side lengths, this gives us


title: Area of a circle

text: In the first equation C′P is C′O+OP, length "r"+⁄"c", and C′C is the diameter, 2"r". For a unit circle we have the famous doubling equation of Ludolph van Ceulen,


title: Area of a circle

text: If we now circumscribe a regular "n-"gon, with side A″B″ parallel to AB, then OAB and OA″B″ are similar triangles, with A″B″ : AB = OC : OP. Call the circumscribed side "S"; then this is "S" : "s" = 1 : ⁄"c". (We have again used that OP is half the length of A′B.) Thus we obtain


title: Area of a circle

text: Call the inscribed perimeter "u" = "ns", and the circumscribed perimeter "U" = "nS". Then combining equations, we have


title: Area of a circle

text: When more efficient methods of finding areas are not available, we can resort to "throwing darts". This Monte Carlo method uses the fact that if random samples are taken uniformly scattered across the surface of a square in which a disk resides, the proportion of samples that hit the disk approximates the ratio of the area of the disk to the area of the square. This should be considered a method of last resort for computing the area of a disk (or any shape), as it requires an enormous number of samples to get useful accuracy; an estimate good to 10 requires about 100 random samples .


title: Area of a circle

text: We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently is that we can dissect the disk into a large but "finite" number of pieces and then reassemble the pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition.


title: Area of a circle

text: Circles can be defined in non-Euclidean geometry, and in particular in the hyperbolic and elliptic planes.


title: Area of a circle

text: For example, the unit sphere formula_62 is a model for the two-dimensional elliptic plane. It carries an intrinsic metric that arises by measuring geodesic length. The geodesic circles are the parallels in a geodesic coordinate system.


title: Area of a circle

text: More precisely, fix a point formula_63 that we place at the zenith. Associated to that zenith is a geodesic polar coordinate system formula_64, formula_65, formula_66, where z is the point formula_67. In these coordinates, the geodesic distance from z to any other point formula_68 having coordinates formula_64 is the value of formula_70 at x. A spherical circle is the set of points a geodesic distance "R" from the zenith point z. Equivalently, with a fixed embedding into formula_71, the spherical circle of radius formula_72 centered at z is the set of x in formula_62 such that formula_74.


title: Area of a circle

text: We can also measure the area of the spherical disk enclosed within a spherical circle, using the intrinsic surface area measure on the sphere. The area of the disk of radius "R" is then given by


title: Area of a circle

text: More generally, if a sphere formula_76 has radius of curvature formula_77, then the area of the disk of radius "R" is given by


title: Area of a circle

text: Observe that, as an application of L'Hôpital's rule, this tends to the Euclidean area formula_79 in the flat limit formula_80.


title: Area of a circle

text: The hyperbolic case is similar, with the area of a disk of intrinsic radius "R" in the (constant curvature formula_81) hyperbolic plane given by


title: Area of a circle

text: where cosh is the hyperbolic cosine. More generally, for the constant curvature formula_83 hyperbolic plane, the answer is


title: Area of a circle

text: These identities are important for comparison inequalities in geometry. For example, the area enclosed by a circle of radius "R" in a flat space is always greater than the area of a spherical circle and smaller than a hyperbolic circle, provided all three circles have the same (intrinsic) radius. That is,


title: Area of a circle

text: for all formula_86. Intuitively, this is because the sphere tends to curve back on itself, yielding circles of smaller area than those in the plane, whilst the hyperbolic plane, when immersed into space, develops fringes that produce additional area. It is more generally true that the area of the circle of a fixed radius "R" is a strictly decreasing function of the curvature.


title: Area of a circle

text: In all cases, if formula_87 is the curvature (constant, positive or negative), then the isoperimetric inequality for a domain with area "A" and perimeter "L" is


title: Area of a circle

text: We can stretch a disk to form an ellipse. Because this stretch is a linear transformation of the plane, it has a distortion factor which will change the area but preserve "ratios" of areas. This observation can be used to compute the area of an arbitrary ellipse from the area of a unit circle.


title: Area of a circle

text: Consider the unit circle circumscribed by a square of side length 2. The transformation sends the circle to an ellipse by stretching or shrinking the horizontal and vertical diameters to the major and minor axes of the ellipse. The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is /4, which means the ratio of the ellipse to the rectangle is also /4. Suppose "a" and "b" are the lengths of the major and minor axes of the ellipse. Since the area of the rectangle is "ab", the area of the ellipse is "ab"/4.


title: Area of a circle

text: We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.


title: WebP

text: WebP is an image file format developed by Google intended as a replacement for JPEG, PNG, and GIF file formats. It supports both lossy and lossless compression, as well as animation and alpha transparency.


title: WebP

text: Google announced the WebP format in September 2010, and released the first stable version of its supporting library in April 2018.


title: WebP

text: WebP was first announced by Google on 30 September in 2010 as a new open format for lossy compressed true-color graphics on the web, producing files that were smaller than JPEG files for comparable image quality. It was based on technology which Google had acquired with the purchase of On2 Technologies. As a derivative of the VP8 video format, it is a sister project to the WebM multimedia container format. WebP-related software is released under a BSD free software license.


title: WebP

text: On 3 October 2011, Google added an "Extended File Format" allowing WebP support for animation, ICC profile, XMP and Exif metadata, and tiling (compositing very large images from maximum 16384×16384 tiles). Older animated GIF files can be converted to animated WebP.


title: WebP

text: On 18 November 2011, Google announced a new lossless compression mode, and support for transparency (alpha channel) in both lossless and lossy modes; support was enabled by default in libwebp 0.2.0 (16 August 2012). According to Google's measurements in November 2011, a conversion from PNG to WebP resulted in a 45% reduction in file size when starting with PNGs found on the web, and a 28% reduction compared to PNGs that are recompressed with pngcrush and PNGOUT.


title: WebP

text: In July 2016, Apple added WebP support to early beta versions of macOS Sierra and iOS 10, but support was later removed in the GM seed versions of iOS 10 and macOS Sierra released in September 2016. In September 2020, WebP support was added in Safari version 14.


title: WebP

text: WebP's lossy compression algorithm is based on the intra-frame coding of the VP8 video format and the Resource Interchange File Format (RIFF) as a container format. As such, it is a block-based transformation scheme with eight bits of color depth and a luminance–chrominance model with chroma subsampling by a ratio of 1:2 (YCbCr 4:2:0). Without further content, the mandatory RIFF container has an overhead of only twenty bytes, though it can also hold additional metadata. The side length of WebP images is limited to pixels.


title: WebP

text: WebP is based on block prediction. Each block is predicted on the values from the three blocks above it and from one block to the left of it (block decoding is done in raster-scan order: left to right and top to bottom). There are four basic modes of block prediction: horizontal, vertical, DC (one color), and TrueMotion. Mispredicted data and non-predicted blocks are compressed in a 4×4 pixel sub-block with a discrete cosine transform or a Walsh–Hadamard transform. Both transforms are done with fixed-point arithmetic to avoid rounding errors. The output is compressed with entropy encoding. WebP also has explicit support for parallel decoding.


title: WebP

text: The reference implementation consists of converter software in the form of a command-line program for Linux (cwebp) and a programming library for the decoding, the same as for WebM. The open-source community ported the converter to other platforms, such as Windows.


title: WebP

text: The WebP container (i.e., RIFF container for WebP) allows feature support over and above the basic use case of WebP (i.e., a file containing a single image encoded as a VP8 key frame). The WebP container provides additional support for:


title: WebP

text: WebP's lossless compression, a newer algorithm unrelated to VP8, was designed by Google software engineer Jyrki Alakuijala. It uses advanced techniques such as dedicated entropy codes for different color channels, exploiting 2D locality of backward reference distances and a color cache of recently used colors. This complements basic techniques such as dictionary coding, Huffman coding and color indexing transform. This format uses a recursive definition: all of the control images, such as the local entropy code selection, are encoded the same way as the whole image itself.


title: WebP

text: Google has proposed using WebP for animated images as an alternative to the popular GIF format, citing the advantages of 24-bit color with transparency, combining frames with lossy and lossless compression in the same animation, and support for seeking to specific frames. Google reports a 64% reduction in file size for images converted from animated GIFs to lossy WebP, however, with a very noticeable visual impact, both at default settings, and optimised settings. When converting using lossless WebP, a 19% reduction is achieved as reported by Google, although real world performance is nearer to 10%.


title: WebP

text: Google actively promotes WebP, and Google Chrome and all Chromium-based browsers support the format. The proprietary PageSpeed Insights tool suggests that webmasters switch from JPEG and PNG to WebP in order to improve their website speed score.


title: WebP

text: Microsoft Edge versions released after January 2020 are based on the Chromium browser, and have native WebP support. EdgeHTML-based versions of Microsoft Edge support WebP through a platform extension (installed by default) (unless running in the security-hardened "Application Guard" mode, which does not support platform extensions).


title: WebP

text: Mozilla Firefox (and its forks Pale Moon and Waterfox) officially supports WebP since January 2019. It was initially considered for implementation in 2013.


title: WebP

text: WebP can also be displayed in all major browsers using the WebPJS JavaScript library, although support in Internet Explorer 6 and above is achieved using Flash.


title: WebP

text: Acorn (from version 7.0), Picasa (from version 3.9), PhotoLine, Pixelmator, ImageMagick, XnView, IrfanView, GDAL, Aseprite, Paint.NET (from version 4.2.5), GIMP (from version 2.10), gThumb (from version 3.1.1), Xara Designer Pro (from version 18.0), Adobe Photoshop (from version 23.2) and GraphicConverter all natively support WebP.


title: WebP

text: In 2019, Google released a free plug-in that enables WebP support in earlier versions of Adobe Photoshop. Free Photoshop plug-ins had been released by Telegraphics and fnordware before that. GIMP up to version 2.8 also supported WebP via a plugin; later, this plugin was shipped in GIMP 2.9 branch, and received multiple improvements. Google has also released a plug-in for Microsoft Windows that enables WebP support in Windows Photo Viewer, Microsoft Office 2010, FastPictureViewer, and any other application that uses Windows Imaging Component. Blender supports WebP since version 3.2.


title: WebP

text: FFmpeg linked with the VP8/VP9 reference codec library "libvpx" can extract VP8 key frames from WebM media and a script can then add the WebP RIFF header and the NUL pad byte for odd frame lengths. Meanwhile, FFmpeg supports "libwebp" directly.


title: WebP

text: Gmail and Google Photos both support WebP. Support for WebP is also planned for Google App Engine. The "Instant Previews" feature of Google Search uses WebP internally to reduce disk space used by previews. Android 4.0 supports encoding and decoding WebP images (via bitmap and Skia). SDL_image supports the format since 1.2.11.


title: WebP

text: Telegram Messenger uses WebP for their Stickers, claiming they are displayed 5 times faster compared to the other formats usually used in messaging apps.


title: WebP

text: LibreOffice supports the import of WebP images since version 7.4, so does the LibreOffice technology based online office Collabora Online.


title: WebP

text: Content management systems (CMS) usually do not support WebP natively or by default. However, for most popular CMS, extensions are available for automated conversion from other image formats to WebP and delivering WebP images to compatible browsers. Since June 2021, WordPress supports WebP natively.


title: WebP

text: Like VP8 on which it is based, former lossy WebP supports only 8-bit YUV format, which may cause color loss on images with thin contrast elements (such as in pixel art and computer graphics) and ghosting in anaglyph. To overcome this restriction, new lossless WebP supports VP8L encoding that works exclusively with 8-bit RGBA (red, green, blue, alpha) color space.


title: WebP

text: In September 2010, Fiona Glaser, a developer of the x264 encoder, wrote a very early critique of WebP. Comparing different encodings (JPEG, x264, and WebP) of a reference image, she stated that the quality of the WebP-encoded result was the worst of the three, mostly because of blurriness on the image. Her main remark was that "libvpx, a much more powerful encoder than ffmpeg's jpeg encoder, loses because it tries too hard to optimize for PSNR" (peak signal-to-noise ratio), arguing instead that "good psycho-visual optimizations are more important than anything else for compression".


title: WebP

text: In October 2013, Josh Aas from Mozilla Research published a comprehensive study of current lossy encoding techniques and was not able to conclude that WebP outperformed mozjpeg by any significant margin.


title: WebP

text: Google has been developing the second version of WebP since June 2021. Its reference implementation is . The main goal of this new format is to reach similar compression ratios as AVIF while remaining faster to encode and decode.


title: WebP

text: On October 12, 2022, Google changed WebP 2's development repository's README file to state that "WebP 2 will not be released as an image format" and began describing WebP 2 as a "playground for image compression experiments".