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Basques

The Basques ( or  ; euskaldunak ; vascos ; basques ) are a Southwestern European ethnic group,  characterised by the Basque language, a common culture and shared genetic ancestry to the ancient Vascones and Aquitanians. Basques are indigenous to, and primarily inhabit, an area traditionally known as the Basque Country (Euskal Herria)—a region that is located around the western end of the Pyrenees on the coast of the Bay of Biscay and straddles parts of north-central Spain and south-western France.

Etymology

The English word Basque may be pronounced or  and derives from the French Basque, itself derived from Gascon Basco (pronounced ), cognate with Spanish Vasco (pronounced ). Those, in turn, come from Latin Vascō (pronounced ; plural Vascōnēs—see history section below). The Latin generally evolved into the bilabials  and  in Gascon and Spanish, probably under the influence of Basque and the related Aquitanian (the Latin /w/ instead evolved into  in French, Italian and other Romance languages).

Several coins from the 2nd and the 1st centuries BC found in the Basque Country bear the inscription barscunes. The place in which they were minted is not certain but is thought to be somewhere near Pamplona, in the heartland of the area that historians believe was inhabited by the Vascones. Some scholars have suggested a Celtic etymology based on bhar-s-, meaning "summit", "point" or "leaves", according to which barscunes may have meant "the mountain people", "the tall ones" or "the proud ones", and others have posited a relationship to a Proto-Indo-European root *bar- meaning "border", "frontier", "march".

In Basque, people call themselves the euskaldunak, singular euskaldun, formed from euskal- (i.e. "Basque (language)") and -dun (i.e. "one who has"); euskaldun literally means a Basque-speaker. Not all Basques are Basque-speakers. Therefore, the neologism euskotar, plural euskotarrak, was coined in the 19th century to mean a Basque person, whether Basque-speaking or not. Alfonso Irigoyen posits that the word euskara is derived from an ancient Basque verb enautsi "to say" (compare modern Basque esan) and the suffix -(k)ara ("way (of doing something)"). Thus, euskara would mean literally "way of saying" or "way of speaking". One item of evidence in favour of that hypothesis is found in the Spanish book Compendio Historial, written in 1571 by the Basque writer Esteban de Garibay. He records the name of the Basque language as enusquera. That may, however, be a writing mistake.

In the 19th century, the Basque nationalist activist Sabino Arana posited an original root euzko, which he thought came from eguzkiko ("of the sun", related to the assumption of an original solar religion). On the basis of that putative root, Arana proposed the name Euzkadi for an independent Basque nation, composed of seven Basque historical territories. Arana's neologism Euzkadi (in the regularized spelling Euskadi) is still widely used in both Basque and Spanish since it is now the official name of the Autonomous Community of the Basque Country.

Genetic origins

The distinctiveness noted by studies of 'classical' genetic markers (such as blood groups) and the apparently "pre-Indo-European" nature of the Basque language has resulted in a popular and long-held misleading view that Basques are "living fossils" of the earliest modern humans who colonised Europe. Partly for these reasons, anthropological and genetic studies from the beginning and the end of the 20th century theorized that the Basques are the descendants of the original Cro-Magnons.

But although they are genetically distinctive in some ways due to isolation, the Basques are still very typically European in their Y-DNA and mtDNA sequences, and in some other genetic loci. These same sequences are widespread throughout the Western half of Europe, especially along the Western fringe of the continent.

Autosomal genetic studies have confirmed that Basques share close genetic ties to other Europeans, especially with Spaniards, who have a common genetic identity of over 70 % with Basques, a homogeneity amongst both their Spanish and French populations, according to high-density SNP genotyping study done in May 2010, and a genomic distinctiveness, relative to other European populations.

Studies of the Y-DNA haplogroups found that on their direct male lineages, the vast majority of modern Basques have a common ancestry with other Western Europeans, namely a marked predominance of Haplogroup R1b-DF27 (70 % according to some ). This had initially been theorised to be a Palaeolithic marker. But this theory encountered inconsistencies even prior to the most recent chronological re-evaluations. More recent studies suggest that R1b spread up to Western Europe from southwestern Eurasia in the Neolithic period or later, between 8,000 and 4,000 years ago. The age of the subclade which Basques carry, R1b-DF27, "is estimated at ~4,200 years ago, at the transition between the Neolithic and the Bronze Age, when the Y chromosome landscape of Western Europe was thoroughly remodeled. In spite of its high frequency in Basques, Y-STR internal diversity of R1b-DF27 is lower there, and results in more recent age estimates", implying it was brought to the region from elsewhere. Next to the main lineage R1b, high frequencies of E-V65, associated with the Maghreb, Italy and Spain, were found among Basque autochthonous inhabitants of Álava province (17.3 %), Biscay province (10.9 %), and Gipuzkoa province (3.3 %).

Several ancient DNA samples have been recovered and amplified from the Iberian and Basque region. The collection of mtDNA and Y-DNA haplogroups sampled there differed significantly compared to their modern frequencies. The authors concluded that there is "discontinuity" between ancient locals and modern Basques. While Basques harbour some very archaic mtDNA lineages, they are not of "undiluted Palaeolithic ancestry" but of significantly early Neolithic origin with a connection to the isolate Sardinian people. It appears that some 4,500 years ago almost all Y-DNA heritage from Iberian admixture of Mesolithic hunter-gatherers and Neolithic farmers was replaced by the R1b lineage of herders from the steppe, and the Basque genetic distinctiveness is a result of centuries of low population size, genetic drift, and endogamy.

In 2015, a new scientific study of Basque DNA was published which seems to indicate that Basques are descendants of Neolithic farmers who mixed with local Mesolithic hunters before becoming genetically isolated from the rest of Europe for millennia. Mattias Jakobsson from Uppsala University in Sweden analysed genetic material from eight Stone Age human skeletons found in El Portalón Cavern in Atapuerca, northern Spain. These individuals lived between 3,500 and 5,500 years ago, after the transition to farming in southwest Europe. The results show that these early Iberian farmers are the closest ancestors to present-day Basques. The findings were published in Proceedings of the National Academy of Sciences of the United States of America. According to the study, the "results show that the Basques trace their ancestry to early farming groups from Iberia, which contradicts previous views of them being a remnant population that trace their ancestry to Mesolithic hunter-gatherer groups." Still these early Neolithic farmer ancestors of the Basques did mix with local southwestern hunter-gatherers, and "the proportion of hunter gatherer-related admixture into early farmers also increased over the course of two millennia." This admixed group was also found to be ancestral to other modern-day Iberian peoples, but while the Basques remained relatively isolated for millennia after this time, later migrations into Iberia led to distinct and additional admixture in all other Iberian groups.

In 2019, a study was published in Science in which a more modern time-transect of Iberian ancient populations including the Basque were analyzed. From their abstract, it says: "and we reveal that present-day Basques are best described as a typical Iron Age population without the admixture events that later affected the rest of Iberia." This indicates Basques were isolated from admixture with outside groups since at least 1000 BC, or 3000 years before the present. In Iberia, these later admixture (interbreeding) events were with central European (Celtic), eastern Mediterranean (Phoenician, Greek and Roman), northern African (Carthaginian and Mauritanian) and northern European (Gothic and Frankish) populations, and genomic ancestry from them are found in varying degrees in all or most present-day Iberian populations, except – albeit to a limited extent even there – for the Basque.

History

Basque tribes were mentioned in Roman times by Strabo and Pliny, including the Vascones, Aquitani, and others. There is enough evidence to support the hypothesis that at that time and later they spoke old varieties of the Basque language (see: Aquitanian language).

In the Early Middle Ages, the territory between the Ebro and Garonne rivers was known as Vasconia, a vaguely defined ethnic area and political entity struggling to fend off pressure from the Iberian Visigothic kingdom and Arab rule to the south, as well as the Frankish push from the north. By the turn of the first millennium, the territory of Vasconia had fragmented into different feudal regions, such as Soule and Labourd, while south of the Pyrenees the Castile, Pamplona and the Pyrenean counties of Aragon, Sobrarbe, Ribagorça (later Kingdom of Aragon), and Pallars emerged as the main regional entities with Basque population in the 9th and 10th centuries.

The Kingdom of Pamplona, a central Basque realm, later known as Navarre, underwent a process of feudalization and was subject to the influence of its much larger Aragonese, Castilian and French neighbours. Castile deprived Navarre of its coastline by conquering key western territories (1199–1201), leaving the kingdom landlocked. The Basques were ravaged by the War of the Bands, bitter partisan wars between local ruling families. Weakened by the Navarrese civil war, the bulk of the realm eventually fell before the onslaught of the Spanish armies (1512–1524). However, the Navarrese territory north of the Pyrenees remained beyond the reach of an increasingly powerful Spain. Lower Navarre became a province of France in 1620.

Nevertheless, the Basques enjoyed a great deal of self-government until the French Revolution (1790) and the Carlist Wars (1839, 1876), when the Basques supported heir apparent Carlos V and his descendants. On either side of the Pyrenees, the Basques lost their native institutions and laws held during the Ancien régime. Since then, despite the current limited self-governing status of the Basque Autonomous Community and Navarre as settled by the Spanish Constitution, many Basques have attempted higher degrees of self-empowerment (see Basque nationalism), sometimes by acts of violence. Labourd, Lower Navarre, and Soule were integrated into the French department system (starting 1790), with Basque efforts to establish a region-specific political-administrative entity failing to take off to date. However, in January 2017, a single agglomeration community was established for the Basque Country in France.

Political and administrative divisions

The Basque region is divided into at least three administrative units, namely the Basque Autonomous Community and Navarre in Spain, and the arrondissement of Bayonne and the cantons of Mauléon-Licharre and Tardets-Sorholus in the département of Pyrénées Atlantiques, France.

The autonomous community (a concept established in the Spanish Constitution of 1978) known as Euskal Autonomia Erkidegoa or EAE in Basque and as Comunidad Autónoma Vasca or CAV in Spanish (in English: Basque Autonomous Community or BAC), is made up of the three Spanish provinces of Álava, Biscay and Gipuzkoa. The corresponding Basque names of these territories are Araba, Bizkaia and Gipuzkoa, and their Spanish names are Álava, Vizcaya and Guipúzcoa.

The BAC only includes three of the seven provinces of the currently called historical territories. It is sometimes referred to simply as "the Basque Country" (or Euskadi) by writers and public agencies only considering those three western provinces, but also on occasions merely as a convenient abbreviation when this does not lead to confusion in the context. Others reject this usage as inaccurate and are careful to specify the BAC (or an equivalent expression such as "the three provinces", up to 1978 referred to as "Provincias Vascongadas" in Spanish) when referring to this entity or region. Likewise, terms such as "the Basque Government" for "the government of the BAC" are commonly though not universally employed. In particular in common usage the French term Pays Basque ("Basque Country"), in the absence of further qualification, refers either to the whole Basque Country ("Euskal Herria" in Basque), or not infrequently to the northern (or "French") Basque Country specifically.

Under Spain's present constitution, Navarre (Nafarroa in present-day Basque, Navarra historically in Spanish) constitutes a separate entity, called in present-day Basque Nafarroako Foru Erkidegoa, in Spanish Comunidad Foral de Navarra (the autonomous community of Navarre). The government of this autonomous community is the Government of Navarre. In historical contexts Navarre may refer to a wider area, and that the present-day northern Basque province of Lower Navarre may also be referred to as (part of) Nafarroa, while the term "High Navarre" (Nafarroa Garaia in Basque, Alta Navarra in Spanish) is also encountered as a way of referring to the territory of the present-day autonomous community.

There are three other historic provinces parts of the Basque Country: Labourd, Lower Navarre and Soule (Lapurdi, Nafarroa Beherea and Zuberoa in Basque; Labourd, Basse-Navarre and Soule in French), devoid of official status within France's present-day political and administrative territorial organization, and only minor political support to the Basque nationalists. A large number of regional and local nationalist and non-nationalist representatives have waged a campaign for years advocating for the creation of a separate Basque département, while these demands have gone unheard by the French administration.

Population, main cities, and languages

There are 2,123,000 people living in the Basque Autonomous Community (279,000 in Alava, 1,160,000 in Biscay and 684,000 in Gipuzkoa). The most important cities in this region, which serve as the provinces' administrative centers, are Bilbao (in Biscay), San Sebastián (in Gipuzkoa), and Vitoria-Gasteiz (in Álava). The official languages are Basque and Spanish. Knowledge of Spanish is compulsory under the Spanish constitution (article no. 3), and knowledge and usage of Basque is a right under the Statute of Autonomy (article no. 6), so only knowledge of Spanish is virtually universal. Knowledge of Basque, after declining for many years during Franco's dictatorship owing to official persecution, is again on the rise due to favorable official language policies and popular support. Currently about 33 percent of the population in the Basque Autonomous Community speaks Basque.

Navarre has a population of 601,000; its administrative capital and main city, also regarded by many nationalist Basques as the Basques' historical capital, is Pamplona (Iruñea in modern Basque). Only Spanish is an official language of Navarre, and the Basque language is only co-official in the province's northern region, where most Basque-speaking Navarrese are concentrated.

About a quarter of a million people live in the French Basque Country. Nowadays Basque-speakers refer to this region as Iparralde (Basque for North), and to the Spanish provinces as Hegoalde (South). Much of this population lives in or near the Bayonne-Anglet-Biarritz (BAB) urban belt on the coast (in Basque these are Baiona, Angelu and Miarritze). The Basque language, which was traditionally spoken by most of the region's population outside the BAB urban zone, is today rapidly losing ground to French. The French Basque Country's lack of self-government within the French state is coupled with the absence of official status for the Basque language in the region. Attempts to introduce bilingualism in local administration have so far met direct refusal from French officials.

Basque diaspora

Large numbers of Basques have left the Basque Country to settle in the rest of Spain, France or other parts of the world in different historical periods, often for economic or political reasons. Historically the Basques abroad were often employed in shepherding and ranching and by maritime fisheries and merchants. Millions of Basque descendants (see Basque American and Basque Canadian) live in North America (the United States; Canada, mainly in the provinces of Newfoundland and Quebec), all over Latin America, South Africa, and Australia.

Latin America
Miguel de Unamuno said: "There are at least two things that clearly can be attributed to Basques: the Society of Jesus and the Republic of Chile." Chilean historian Luis Thayer Ojeda estimated that 48 percent of immigrants to Chile in the 17th and 18th centuries were Basque. Estimates range between 2.5 and 5 million Basque descendants live in Chile; the Basque have been a major if not the strongest influence in the country's cultural and economic development.

Basque place names are to be found, such as Nueva Vizcaya (now Chihuahua and Durango, Mexico), New Navarre (now Sonora and Sinaloa, Mexico), Biscayne Bay (United States), and Aguereberry Point (United States). Nueva Vizcaya was the first province in the north of the Viceroyalty of New Spain (Mexico) to be explored and settled by the Spanish. It consisted mostly of the area which is today the states of Chihuahua and Durango (the original Durango is a known city in Biscay).

In Mexico most descendants of Basque emigrees are concentrated in the cities of Monterrey, Saltillo, Reynosa, Camargo, and the states of Jalisco, Durango, Nuevo León, Tamaulipas, Coahuila, and Sonora. The Basques were important in the mining industry; many were ranchers and vaqueros (cowboys), and the rest opened small shops in major cities such as Mexico City, Guadalajara and Puebla. In Guatemala, most Basques have been concentrated in Sacatepequez Department, Antigua Guatemala, Jalapa for six generations now, while some have migrated to Guatemala City.

In Colombia, a large number of Basques settled mainly in Antioquia and the Coffee Axis. In 1955, Joaquín Ospina said: "Is there something more similar to the Basque people than the "antioqueños". Also, writer Arturo Escobar Uribe said in his book "Mitos de Antioquia" (Myths of Antioquia) (1950): "Antioquia, which in its clean ascendance predominates the peninsular farmer of the Basque provinces, inherited the virtues of its ancestors. ... Despite the predominance of the white race, its extension in the mountains ... has projected over Colombia's map the prototype of its race; in Medellín with the industrial paisa, entrepreneur, strong and steady ... in its towns, the adventurer, arrogant, world-explorer. ... Its myths, which are an evidence of their deep credulity and an indubitable proof of their Iberian ancestor, are the sequel of the conqueror's blood which runs through their veins". Bambuco, a Colombian folk music, has Basque roots.

United States
The largest of several important Basque communities in the United States is in the area around Boise, Idaho, home to the Basque Museum and Cultural Center, host to an annual Basque festival, as well as a festival for the Basque diaspora every five years. Reno, Nevada, where the Center for Basque Studies and the Basque Studies Library are located at the University of Nevada, is another significant nucleus of Basque population. Elko, Nevada, sponsors an annual Basque festival that celebrates the dance, cuisine and cultures of the Basque peoples of Spanish, French and Mexican nationalities who have arrived in Nevada since the late 19th century.

Texas has a large percentage of Hispanics descended from Basques who participated in the conquest of New Spain. Many of the original Tejanos had Basque blood, including those who fought in the Battle of the Alamo alongside many of the other Texans. Along the Mexican/Texan border, many Basque surnames can be found. The largest concentration of Basques who settled on Mexico's north-eastern "frontera", including the states of Chihuahua, Durango, Coahuila, Nuevo León, and Tamaulipas, also settled along Texas' Rio Grande from South Texas to West Texas. Many of the historic hidalgos, or noble families from this area, had gained their titles and land grants from Spain and Mexico; they still value their land. Some of North America's largest ranches, which were founded under these colonial land grants, can be found in this region.

California has a major concentration of Basques, most notably in the San Joaquin Valley between Stockton, Fresno and Bakersfield. The city of Bakersfield has a large Basque community and the city has several Basque restaurants, including Noriega's which won the 2011 James Beard Foundation America's Classic Award. There is a history of Basque culture in Chino, California. In Chino, two annual Basque festivals celebrate the dance, cuisine, and culture of the peoples. The surrounding area of San Bernardino County has many Basque descendants as residents. They are mostly descendants of settlers from Spain and Mexico. These Basques in California are grouped in the group known as Californios.

Basques of European Spanish-French and Latin American nationalities also settled throughout the western U.S. in states like Louisiana, New Mexico, Arizona, Utah, Colorado, Wyoming, Montana, Oregon, and Washington.

Language
The identifying language of the Basques is called Basque or Euskara, spoken today by 25%-30% of the region's population. An idea of the central place the language has in cultural terms is given by the fact that Basques identify themselves by the term euskaldun and their country as Euskal Herria, literally "Basque speaker" and "Country of the Basque Language" respectively. The language has been made a political issue by official Spanish and French policies restricting its use either historically or currently; however, this has not stopped the teaching, speaking, writing, and cultivating of this increasingly vibrant minority language. This sense of Basque identity tied to the local language does not only exist in isolation. For many Basques, it is juxtaposed with a sense of either Spanish or French identity tied with the use of the Spanish and French languages among other Basques, especially in the French Basque Country. Regarding the Spanish Basque Country, Basques that don't have a sense of Spanish identity make up an important part of the population. As with many European states, a regional identity, be it linguistically derived or otherwise, is not mutually exclusive with the broader national one. For example, Basque rugby union player for France, Imanol Harinordoquy, has said about his national identity:"'I am French and Basque. There is no conflict, I am proud of both. ... I have friends who are involved in the political side of things but that is not for me. My only interest is the culture, the Euskera language, the people, our history and ways.'"

As a result of state language promotion, school policies, the effects of mass media and migration, today virtually all Basques (except for some children below school age) speak the official language of their state (Spanish or French). There are extremely few Basque monolingual speakers: essentially all Basque speakers are bilingual on both sides of the border. Spanish or French is typically the first language of citizens from other regions (who often feel no need to learn Basque), and Spanish or French is also the first language of many Basques, all of which maintains the dominance of the state tongues of both France and Spain. Recent Basque Government policies aim to change this pattern, as they are viewed as potential threats against mainstream usage of the minority tongue.

The Basque language is thought to be a genetic language isolate in contrast with other European languages, vast majority of which belong to the broad Indo-European language family. Another peculiarity of Basque is that it has probably been spoken continuously in situ, in and around its present territorial location, for longer than most other modern European languages, which are typically thought to have been introduced in historic or prehistoric times through population migrations or other processes of cultural transmission.

However, popular stereotypes characterizing Basque as "the oldest language in Europe" and "unique among the world's languages" may be misunderstood and lead to erroneous assumptions. Over the centuries, Basque has remained in continuous contact with neighboring western European languages with which it has come to share numerous lexical properties and typological features; it is therefore misleading to exaggerate the "outlandish" character of Basque. Basque is also a modern language, and is established as a written and printed one used in present-day forms of publication and communication, as well as a language spoken and used in a very wide range of social and cultural contexts, styles, and registers.

Land and inheritance

Basques have a close attachment to their home (etxe(a) 'house, home'), especially when this consists of the traditional self-sufficient, family-run farm or baserri(a). Home in this context is synonymous with family roots. Some Basque surnames were adapted from old baserri or habitation names. They typically related to a geographical orientation or other locally meaningful identifying features. Such surnames provide even those Basques whose families may have left the land generations ago with an important link to their rural family origins: Bengoetxea "the house of further down", Goikoetxea "the house above", Landaburu "top of the field", Errekondo "next to the stream", Elizalde "by the church", Mendizabal "wide hill", Usetxe "house of birds" Ibarretxe "house in the valley", Etxeberria "the new house", and so on.

In contrast to surrounding regions, ancient Basque inheritance patterns, recognised in the fueros, favoured survival of the unity of inherited land holdings. In a kind of primogeniture, these usually were inherited by the eldest male or female child. As in other cultures, the fate of other family members depended on the assets of a family: wealthy Basque families tended to provide for all children in some way, while less-affluent families may have had only one asset to provide to one child. However, this heir often provided for the rest of the family (unlike in England, with strict primogeniture, where the eldest son inherited everything and often did not provide for others). Even though they were provided for in some way, younger siblings had to make much of their living by other means. Mostly after the advent of industrialisation, this system resulted in the emigration of many rural Basques to Spain, France or the Americas. Harsh by modern standards, this custom resulted in a great many enterprising figures of Basque origin who went into the world to earn their way, from Spanish conquistadors such as Lope de Aguirre and Francisco Vásquez de Coronado, to explorers, missionaries and saints of the Catholic Church, such as Francis Xavier.

A widespread belief that Basque society was originally matriarchal is at odds with the current, clearly patrilineal kinship system and inheritance structures. Some scholars and commentators have attempted to reconcile these points by assuming that patrilineal kinship represents an innovation. In any case, the social position of women in both traditional and modern Basque society is somewhat better than in neighbouring cultures, and women have a substantial influence in decisions about the domestic economy. In the past, some women participated in collective magical ceremonies. They were key participants in a rich folklore, today largely forgotten.

Cuisine
Basque cuisine is at the heart of Basque culture, influenced by the neighboring communities and produce from the sea and the land. A 20th-century feature of Basque culture is the phenomenon of gastronomical societies (called txoko in Basque), food clubs where men gather to cook and enjoy their own food. Until recently, women were allowed entry only one day in the year. Cider houses (Sagardotegiak) are popular restaurants in Gipuzkoa open for a few months while the cider is in season.

Cultural production

At the end of the 20th century, despite ETA violence (ended in 2010) and the crisis of heavy industries, the Basque economic condition recovered remarkably. They emerged from the Franco regime with a revitalized language and culture. The Basque language expanded geographically led by large increases in the major urban centers of Pamplona, Bilbao, and Bayonne, where only a few decades ago the Basque language had all but disappeared. Nowadays, the number of Basque speakers is maintaining its level or increasing slightly.

Religion
Traditionally Basques have been mostly Catholics. In the 19th century and well into the 20th, Basques as a group remained notably devout and churchgoing. In recent years church attendance has fallen off, as in most of Western Europe. The region has been a source of missionaries like Francis Xavier and Michel Garicoïts. Ignatius Loyola, founder of the Society of Jesus, was a Basque. California Franciscan Fermín Lasuén was born in Vitoria. Lasuén was the successor to Franciscan Padre Junípero Serra and founded 9 of the 21 extant California Missions along the coast.

A sprout of Protestantism in the continental Basque Country produced the first translation of the new Testament into Basque by Joanes Leizarraga. Queen Jeanne III of Navarre, a devout Huguenot, commissioned the translation of the New Testament into Basque and Béarnese for the benefit of her subjects. By the time Henry III of Navarre converted to Catholicism in order to become king of France, Protestantism virtually disappeared from the Basque community.

Bayonne held a Jewish community composed mainly of Sephardi Jews fleeing from the Spanish and Portuguese Inquisitions. There were also important Jewish and Muslim communities in Navarre before the Castilian invasion of 1512–21.

Nowadays, according to one single opinion poll, only slightly more than 50% of Basques profess some kind of belief in God, while the rest are either agnostic or atheist. The number of religious skeptics increases noticeably for the younger generations, while the older ones are more religious. Catholicism is, by far, the largest religion in Basque Country. In 2019, the proportion of Basques that identify themselves as Roman Catholic was 60%, while it is one of the most secularized communities of Spain: 24.6% were non-religious and 12.3% of Basques were atheist.

Pre-Christian religion and mythology

The Christianisation of the Basque Country has been the topic of some discussion. There are, broadly speaking, two views. According to one, Christianity arrived in the Basque Country during the 4th and 5th centuries but according to the other, it did not take place until the 12th and 13th centuries. The Encyclopaedia Britannica says that the Basques were not Christianized until the tenth century, however, and that their earlier animism survives in their folklore. The main issue lies in the different interpretations of what is considered Christianisation. Early traces of Christianity can be found in the major urban areas from the 4th century onwards, a bishopric from 589 in Pamplona and three hermit cave concentrations (two in Álava, one in Navarre) that were in use from the 6th century onwards. In this sense, Christianity arrived "early".

Pre-Christian belief seems to have focused on a goddess called Mari. A number of place-names contain her name, which would suggest these places were related to worship of her such as Anbotoko Mari who appears to have been related to the weather. According to one tradition, she travelled every seven years between a cave on Mount Anboto and one on another mountain (the stories vary); the weather would be wet when she was in Anboto, dry when she was in Aloña, or Supelegor, or Gorbea. One of her names, Mari Urraca possibly ties her to an historical Navarrese princess of the 11th and 12th century, with other legends giving her a brother or cousin who was a Roman Catholic priest. So far the discussions about whether the name Mari is original and just happened to coincide closely with the Christian name María or if Mari is an early Basque attempt to give a Christian veneer to pagan worship have remained speculative. At any rate, Mari (Andramari) is one of the oldest worshipped Christian icons in Basque territories.

Mari's consort is Sugaar. This chthonic couple seems to bear the superior ethical power and the power of creation and destruction. It's said that when they gathered in the high caves of the sacred peaks, they engendered the storms. These meetings typically happened on Friday nights, the day of historical akelarre or coven. Mari was said to reside in Mount Anboto; periodically she crossed the skies as a bright light to reach her other home at Mount Txindoki.

Legends also speak of many and abundant genies, like jentilak (equivalent to giants), lamiak (equivalent to nymphs), mairuak (builders of the cromlechs or stone circles, literally Moors), iratxoak (imps), sorginak (witches, priestess of Mari), and so on. Basajaun is a Basque version of the Woodwose. There is a trickster named San Martin Txiki ("St Martin the Lesser").

It is unclear whether Neolithic stone structures called dolmens have a religious significance or were built to house animals or resting shepherds. Some of the dolmens and cromlechs are burial sites serving also as border markers. The jentilak ('Giants'), on the other hand, are a legendary people which explains the disappearance of a people of Stone Age culture that used to live in the high lands and with no knowledge of iron. Many legends about them tell that they were bigger and taller, with a great force, but were displaced by the ferrons, or workers of ironworks foundries, until their total fade-out. They were pagans, but one of them, Olentzero, accepted Christianity and became a sort of Basque Santa Claus. They gave name to several toponyms, as Jentilbaratza.

Society

Historically, Basque society can be described as being somewhat at odds with Roman and later European societal norms. Strabo's account of the north of Spain in his Geographica (written between approximately 20 BC and 20 AD) makes a mention of "a sort of woman-rule—not at all a mark of civilization" (Hadington 1992), a first mention of the—for the period—unusual position of women: "Women could inherit and control property as well as officiate in churches." The evidence for this assertion is rather sparse however.

This preference for female dominance existed well into the 20th century: "... matrilineal inheritance laws, and agricultural work performed by women continued in Basque country until the early twentieth century. For more than a century, scholars have widely discussed the high status of Basque women in law codes, as well as their positions as judges, inheritors, and arbitrators through ante-Roman, medieval, and modern times. The system of laws governing succession in the French Basque region reflected total equality between the sexes. Up until the eve of the French Revolution, the Basque woman was truly 'the mistress of the house', hereditary guardian, and head of the lineage."

While women continued to have a higher position in Basque than other western European societies, it is highly unlikely that any point the society was 'matriarchal', as is often falsely claimed about pre-Indo-European peoples in general. The 'Basque matriarchy' argument is typically tied to 20th century nationalism and is at odds with earlier accounts of the society.

Although the Kingdom of Navarre did adopt feudalism, most Basques also possessed unusual social institutions different from those of the rest of feudal Europe. Some aspects of this include the elizate tradition where local house-owners met in front of the church to elect a representative to send to the juntas and Juntas Generales (such as the Juntas Generales de Vizcaya or Guipúzcoa) which administered much larger areas. Another example was that in the medieval period most land was owned by the farmers, not the Church or a king.

Pelota
The great family of ball games has its unique offspring among Basque ball games, known generically as pilota (Spanish: pelota). Some variants have been exported to the United States and Macau under the name of Jai Alai.

Rural sports

There are several sports derived by Basques from everyday chores. Heavy workers were challenged and bets placed upon them. Examples are:
 * estropadak rowing regattas: from fishermen activities.
 * sokatira: tug-of-war.
 * harri-jasotzea: stone-lifting, from quarry works.
 * aizkolaritza and trontzalaritza: wood-chopping and log sawing.
 * sega jokoa: cutting grass with a scythe.
 * Giza-abere probak: stone block pulling, from construction works:
 * idi probak with teams of oxen.
 * asto probak with donkeys.
 * zaldi probak with horses.
 * gizon probak with human teams.
 * txinga eramatea: carrying of weights, one in each hand, representing milk canisters.
 * ahari topaketa: ram fights.
 * harri zulaketa competitions: drilling stone blocks with a metal bar, only in the former mining areas of West Biscay.
 * Basque sheepdog trials competitions.

Bull runs and bullock games
The encierro (bull run) in Pamplona's fiestas Sanfermines started as a transport of bulls to the ring. These encierros, as well as other bull and bullock related activities are not exclusive to Pamplona but are traditional in many towns and villages of the Basque country.

Football
There are several clubs within the Basque Country, such as Athletic Bilbao, Real Sociedad, Deportivo Alavés, SD Eibar and, as Navarre club, the CA Osasuna (the only club in La Liga that has a Basque name—osasuna means "health"). In the 2016–17 season these five clubs played together in La Liga, the first instance where five Basque clubs have reached that level at the same time. Athletic's recruitment policy has meant the club refuses to sign any non-Basque players, with "Basque" currently defined to include either ethnic Basques or players of any ethnicity trained by a Basque club. Real Sociedad also previously employed such a policy.

Basketball
The Basque Country also features several professional basketball teams, the most notable of which is Saski Baskonia from Vitoria-Gasteiz, one of the 11 clubs that own stakes in Euroleague Basketball, the company that operates the continent-wide EuroLeague and EuroCup. They are currently joined in the Spanish top flight, Liga ACB, by Bilbao Basket, with the two clubs involved in a longstanding rivalry. Another club from the Basque Country, Gipuzkoa Basket from Donostia, currently plays in the second-level LEB Oro.

Rugby union
Rugby union is a popular sport among French Basques, with major clubs Biarritz Olympique and Aviron Bayonnais traditional powerhouses in the premier division of French Rugby (the Top 14). Biarritz regularly play Champions Cup matches, especially knockout matches, at Estadio Anoeta in San Sebastian. Games between the Basque clubs and Catalan club USA Perpignan are always hard fought.

Professional cycling
Cycling is popular and the professional cycling team, partly sponsored by the Basque Government participated in the UCI World Tour division until 2014. Known for their orange tops and hill-climbing ability, their fans were famous for lining the famous Pyrenean climbs in the Tour de France, in support of their compatriots.

Each April the week-long Tour of the Basque Country showcases the beautiful rolling Basque countryside. Miguel Indurain, born in Villava is one of the most celebrated cyclists in the world having won 5 consecutive Tours de France.

Politics

While there is no independent Basque state, Spain's autonomous community of the Basque Country, made up of the provinces of Álava (Araba), Biscay (Bizkaia) and Gipuzkoa, is primarily a historical consequence and an answer to the wide autonomy claim of its population.

Navarre has a separate statute of autonomy, a contentious arrangement designed during Spanish transition to democracy (the Amejoramiento, an 'upgrade' of its previous status during dictatorship). It refers back to the kingdom status of Navarre (up to 1841) and their traditional institutional and legal framework (charters). Basque, the original and main language of Navarre up to the late 18th century, has kept family transmission especially in the northern part of Navarre and central areas to a lesser extent, designated as Basque speaking or mixed area in Navarrese law. Questions of political, linguistic and cultural allegiance and identity are highly complex in Navarre. Politically some Basque nationalists would like to integrate with the Basque Autonomous Community.

The French Basque Country today does not exist as a formal political entity and is officially simply part of the French department of Pyrénées Atlantiques, centered in Béarn. In recent years the number of mayors of the region supporting the creation of a separate Basque department has grown to 63.87%. So far, their attempts have been unsuccessful.

Language
Both the Spanish and French governments have, at times, suppressed Basque linguistic and cultural identity. The French Republics, the epitome of the nation-state, have a long history of attempting the complete cultural absorption of cultural minority groups. Spain has, at most points in its history, granted some degree of linguistic, cultural, and even political autonomy to its Basques, but under the regime of Francisco Franco, the Spanish government reversed the advances of Basque nationalism, as it had fought in the opposite side of the Spanish Civil War: cultural activity in Basque was limited to folkloric issues and the Catholic Church.

Today, the Southern Basque Country within Spain enjoys an extensive cultural and political autonomy. The majority of schools under the jurisdiction of the Basque education system use Basque as the primary medium of teaching. However, the situation is more delicate in the Northern Basque Country within France, where Basque is not officially recognized, and where lack of autonomy and monolingual public schooling in French exert great pressure on the Basque language.

In Navarre, Basque has been declared an endangered language, since the anti-Basque and conservative government of Navarrese People's Union opposes the symbols of Basque culture, highlighting a Spanish identity for Navarre.

Basque is also spoken by immigrants in the major cities of Spain and France, in Australia, in many parts of Latin America, and in the United States, especially in Nevada, Idaho, and California.

Political status and violence

Since its articulation by Sabino Arana in the late 19th century, the more radical currents of Basque nationalism have demanded the right of self-determination and even independence. Within the Basque country, this element of Basque politics is often in balance with the conception of the Basque Country as just another part of the Spanish state, a view more commonly espoused on the right of the political spectrum. In contrast, the desire for greater autonomy or independence is particularly common among leftist Basque nationalists. The right of self-determination was asserted by the Basque Parliament in 2002 and 2006. Since self-determination is not recognized in the Spanish Constitution of 1978, a wide majority of Basques abstained (55%) and some even voted against it (23.5%) in the ratification referendum of December 6 of that year. However, it was approved by clear majority overall in Spain (87%). The autonomous regime for the Basque Country was approved in a 1979 referendum but the autonomy of Navarre (Amejoramiento del Fuero: "improvement of the charter") was never subject to a referendum but only approved by the Navarrese Cortes (parliament).

Classification
As with their language, the Basques are clearly a distinct cultural group in their region. They regard themselves as culturally and especially linguistically distinct from their surrounding neighbours. Some Basques identify themselves as Basques only whereas others identify themselves both as Basque and Spanish. Many Basques regard the designation as a "cultural minority" as incomplete, favouring instead the definition as a nation, the commonly accepted designation for the Basque people up to the rise of the nation-states and the definition imposed by the 1812 Spanish Constitution.

In modern times, as a European people living in a highly industrialized area, cultural differences from the rest of Europe are inevitably blurred, although a conscious cultural identity as a people or nation remains very strong, as does an identification with their homeland, even among many Basques who have emigrated to other parts of Spain or France, or to other parts of the world.

The strongest distinction between the Basques and their traditional neighbours is linguistic. Surrounded by Romance-language speakers, the Basques traditionally spoke (and many still speak) a language that was not only non-Romance but non-Indo-European. The prevailing belief amongst Basques, and forming part of their national identity, is that their language has continuity with the people who were in this region since not only pre-Roman and pre-Celtic times, but since the Stone Age.

Notable Basques
Among the most notable Basque people are Juan Sebastián Elcano (who led the first successful expedition to circumnavigate the globe after Ferdinand Magellan died mid-journey); Sancho III of Navarre; and Ignatius of Loyola and Francis Xavier, founders of the Society of Jesus.

Don Diego María de Gardoqui y Arriquibar (1735–1798) was also a Basque who became Spain's first Ambassador to the United States, and Miguel de Unamuno was a noted novelist and philosopher of the late 19th and the 20th century, was also a Basque.

Another well-known Basque was Father Alberto Hurtado, S.J. (1901–1952), a Jesuit priest who founded the charitable housing system Hogar de Cristo, meaning hearth, or home, of Christ, in Chile. El Hogar provided a home-like milieu for the homeless. Hurtado also founded the Chilean Trade Union Association to promote a union movement based on the social teachings of the Catholic Church. He was a friend and savior to all the poor and homeless, and was beatified by Pope John Paul II on October 16, 1994. He was canonized by Pope Benedict XVI on October 23, 2005.

title: Basques


text:
Blot

Blot may refer to:

Surname

 * Guillaume Blot (born 1985), French racing cyclist
 * Harold W. Blot (born 1938), served as United States Deputy Chief of Staff for Aviation
 * Jean-François Joseph Blot (1781–1857), French soldier and politician
 * Yvan Blot (1948–2018), French conservative political figure, founder and president of the Club de l'Horloge
 * Jean Blot (1923–2019), French writer, translator, and senior civil servant of Russian origin
 * Laetitia Blot (born 1983), French judoka

Religion

 * Blót, a sacrifice to the gods or other beings in Germanic paganism and modern Germanic paganism
 * Blot (album), a 2003 album by Einherjer
 * "Hefja Blot", a song by Danheim for his album Friðr
 * "Blotjarl", a song by Danheim and Heldom for his album Skapanir
 * "Vetrnátta Blot", a song by Danheim and Heldom for his album Skapanir

Other

 * Blot (biology), method of transferring proteins, DNA, RNA or a protein onto a carrier
 * The Blot, a 1921 silent film
 * Another name of a trick-taking card game Belot
 * Blot (Transformers), a character from the Transformers franchise
 * Ink blots, as used in the Rorschach test
 * Blot (1994 film), a 1994 film

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Blot


text:
Bookkeeping

Bookkeeping is the recording of financial transactions, and is part of the process of accounting in business and other organizations. It involves preparing source documents for all transactions, operations, and other events of a business. Transactions include purchases, sales, receipts and payments by an individual person or an organization/corporation. There are several standard methods of bookkeeping, including the single-entry and double-entry bookkeeping systems. While these may be viewed as "real" bookkeeping, any process for recording financial transactions is a bookkeeping process.

The person in an organisation who is employed to perform bookkeeping functions is usually called the bookkeeper (or book-keeper). They usually write the daybooks (which contain records of sales, purchases, receipts, and payments), and document each financial transaction, whether cash or credit, into the correct daybook—that is, petty cash book, suppliers ledger, customer ledger, etc.—and the general ledger. Thereafter, an accountant can create financial reports from the information recorded by the bookkeeper. The bookkeeper brings the books to the trial balance stage, from which an accountant may prepare financial reports for the organisation, such as the income statement and balance sheet.

History
The origin of book-keeping is lost in obscurity, but recent research indicates that methods of keeping accounts have existed from the remotest times of human life in cities. Babylonian records written with styli on small slabs of clay have been found dating to 2600 BC. Mesopotamian bookkeepers kept records on clay tablets that may date back as far as 7,000 years. Use of the modern double entry bookkeeping system was described by Luca Pacioli in 1494.

The term "waste book" was used in colonial America, referring to the documenting of daily transactions of receipts and expenditures. Records were made in chronological order, and for temporary use only. Daily records were then transferred to a daybook or account ledger to balance the accounts and to create a permanent journal; then the waste book could be discarded, hence the name.

Process
The primary purpose of bookkeeping is to record the financial effects of transactions. An important difference between a manual and an electronic accounting system is the former's latency between the recording of a financial transaction and its posting in the relevant account. This delay, which is absent in electronic accounting systems due to nearly instantaneous posting to relevant accounts, is characteristic of manual systems, and gave rise to the primary books of accounts—cash book, purchase book, sales book, etc.—for immediately documenting a financial transaction.

In the normal course of business, a document is produced each time a transaction occurs. Sales and purchases usually have invoices or receipts. Historically, deposit slips were produced when lodgements (deposits) were made to a bank account; and checks (spelled "cheques" in the UK and several other countries) were written to pay money out of the account. Nowadays such transactions are mostly made electronically. Bookkeeping first involves recording the details of all of these source documents into multi-column journals (also known as books of first entry or daybooks). For example, all credit sales are recorded in the sales journal; all cash payments are recorded in the cash payments journal. Each column in a journal normally corresponds to an account. In the single entry system, each transaction is recorded only once. Most individuals who balance their check-book each month are using such a system, and most personal-finance software follows this approach.

After a certain period, typically a month, each column in each journal is totalled to give a summary for that period. Using the rules of double-entry, these journal summaries are then transferred to their respective accounts in the ledger, or account book. For example, the entries in the Sales Journal are taken and a debit entry is made in each customer's account (showing that the customer now owes us money), and a credit entry might be made in the account for "Sale of class 2 widgets" (showing that this activity has generated revenue for us). This process of transferring summaries or individual transactions to the ledger is called posting. Once the posting process is complete, accounts kept using the "T" format (debits on the left side of the "T" and credits on the right side) undergo balancing, which is simply a process to arrive at the balance of the account.

As a partial check that the posting process was done correctly, a working document called an unadjusted trial balance is created. In its simplest form, this is a three-column list. Column One contains the names of those accounts in the ledger which have a non-zero balance. If an account has a debit balance, the balance amount is copied into Column Two (the debit column); if an account has a credit balance, the amount is copied into Column Three (the credit column). The debit column is then totalled, and then the credit column is totalled. The two totals must agree—which is not by chance—because under the double-entry rules, whenever there is a posting, the debits of the posting equal the credits of the posting. If the two totals do not agree, an error has been made, either in the journals or during the posting process. The error must be located and rectified, and the totals of the debit column and the credit column recalculated to check for agreement before any further processing can take place.

Once the accounts balance, the accountant makes a number of adjustments and changes the balance amounts of some of the accounts. These adjustments must still obey the double-entry rule: for example, the inventory account and asset account might be changed to bring them into line with the actual numbers counted during a stocktake. At the same time, the expense account associated with use of inventory is adjusted by an equal and opposite amount. Other adjustments such as posting depreciation and prepayments are also done at this time. This results in a listing called the adjusted trial balance. It is the accounts in this list, and their corresponding debit or credit balances, that are used to prepare the financial statements.

Finally financial statements are drawn from the trial balance, which may include:
 * the income statement, also known as the statement of financial results, profit and loss account, or P&L
 * the balance sheet, also known as the statement of financial position
 * the cash flow statement
 * the statement of changes in equity, also known as the statement of total recognised gains and losses

Single-entry system
The primary bookkeeping record in single-entry bookkeeping is the cash book, which is similar to a checking account register (in UK: cheque account, current account), except all entries are allocated among several categories of income and expense accounts. Separate account records are maintained for petty cash, accounts payable and accounts receivable, and other relevant transactions such as inventory and travel expenses. To save time and avoid the errors of manual calculations, single-entry bookkeeping can be done today with do-it-yourself bookkeeping software.

Double-entry system
A double-entry bookkeeping system is a set of rules for recording financial information in a financial accounting system in which every transaction or event changes at least two different ledger accounts.

Daybooks
A daybook is a descriptive and chronological (diary-like) record of day-to-day financial transactions; it is also called a book of original entry. The daybook's details must be transcribed formally into journals to enable posting to ledgers. Daybooks include:

 * Sales daybook, for recording sales invoices.
 * Sales credits daybook, for recording sales credit notes.
 * Purchases daybook, for recording purchase invoices.
 * Purchases debits daybook, for recording purchase debit notes.
 * Cash daybook, usually known as the cash book, for recording all monies received and all monies paid out. It may be split into two daybooks: a receipts daybook documenting every money-amount received, and a payments daybook recording every payment made.
 * General Journal daybook, for recording journal entries.

Petty cash book
A petty cash book is a record of small-value purchases before they are later transferred to the ledger and final accounts; it is maintained by a petty or junior cashier. This type of cash book usually uses the imprest system: a certain amount of money is provided to the petty cashier by the senior cashier. This money is to cater for minor expenditures (hospitality, minor stationery, casual postage, and so on) and is reimbursed periodically on satisfactory explanation of how it was spent. The balance of petty cash book is Asset.

Journals
Journals are recorded in the general journal daybook. A journal is a formal and chronological record of financial transactions before their values are accounted for in the general ledger as debits and credits. A company can maintain one journal for all transactions, or keep several journals based on similar activity (e.g., sales, cash receipts, revenue, etc.), making transactions easier to summarize and reference later. For every debit journal entry recorded, there must be an equivalent credit journal entry to maintain a balanced accounting equation.

Ledgers
A ledger is a record of accounts. The ledger is a permanent summary of all amounts entered in supporting Journals which list individual transactions by date. These accounts are recorded separately, showing their beginning/ending balance. A journal lists financial transactions in chronological order, without showing their balance but showing how much is going to be entered in each account. A ledger takes each financial transaction from the journal and records it into the corresponding account for every transaction listed. The ledger also determines the balance of every account, which is transferred into the balance sheet and/or the income statement. There are three different kinds of ledgers that deal with book-keeping:
 * Sales ledger, which deals mostly with the accounts receivable account. This ledger consists of the records of the financial transactions made by customers to the business.
 * Purchase ledger is the record of the company's purchasing transactions; it goes hand in hand with the Accounts Payable account.

Abbreviations used in bookkeeping
• A/c or Acc – Account

• A/R – Accounts receivable

• A/P – Accounts payable

• B/S – Balance sheet

• c/d – Carried down

• b/d – Brought down

• c/f – Carried forward

• b/f – Brought forward

• Dr – Debit side of a ledger. "Dr" stands for "Debit register"

• Cr – Credit side of a ledger. "Cr" stands for "Credit register"

• G/L – General ledger; (or N/L – nominal ledger)

• PL – Profit and loss; (or I/S – income statement)

• P/L – Purchase Ledger (Accounts payable)

• P/R – Payroll

• PP&E – Property, plant and equipment

• S/L - Sales Ledger (Accounts receivable)

• TB – Trial Balance

• GST – Goods and services tax

• SGST – State goods & service tax

• CGST – Central goods & service tax

• IGST- integrated goods & service tax

• VAT – Value added tax

• CST – Central sale tax

• TDS – Tax deducted at source

• AMT – Alternate minimum tax

• EBT – Earnings before tax

• EAT – Earnings after tax

• PAT – Profit after tax

• PBT – Profit before tax

• Dep or Depr – Depreciation

• CPO – Cash paid out

• CP - Cash Payment

• w.e.f. - with effect from

• @ - at the rate of

• L/F – ledger folio

• J/F – Journal Folio

• M/s- Messrs Account

• Co- Company

• V/N or V.no. – voucher number

• In no -invoice Number

Chart of accounts
A chart of accounts is a list of the accounts codes that can be identified with numeric, alphabetical, or alphanumeric codes allowing the account to be located in the general ledger. The equity section of the chart of accounts is based on the fact that the legal structure of the entity is of a particular legal type. Possibilities include sole trader, partnership, trust, and company.

Computerized bookkeeping
Computerized bookkeeping removes many of the paper "books" that are used to record the financial transactions of a business entity; instead, relational databases are used today, but typically, these still enforce the norms of bookkeeping including the single-entry and double-entry bookkeeping systems. Certified Public Accountants (CPAs) supervise the internal controls for computerized bookkeeping systems, which serve to minimize errors in documenting the numerous activities a business entity may initiate or complete over an accounting period.

title: Bookkeeping


text:
Bézier curve

A Bézier curve is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. The Bézier triangle is a special case of the latter.

In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify.

Bézier curves are also used in the time domain, particularly in animation, user interface design and smoothing cursor trajectory in eye gaze controlled interfaces. For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.

Invention
The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën. De Casteljau's method was patented in France but not published until the 1980s while the Bézier polynomials were widely publicised in the 1960s by the French engineer Pierre Bézier, who discovered them independently and used them to design automobile bodies at Renault.

Specific cases
A Bézier curve is defined by a set of control points P0 through Pn, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve. The sums in the following sections are to be understood as affine combinations – that is, the coefficients sum to 1.

Linear Bézier curves
Given distinct points P0 and P1, a linear Bézier curve is simply a line between those two points. The curve is given by

 * $$\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1,\ 0 \le t \le 1$$

This is the simplest and is equivalent to linear interpolation. The quantity $$\mathbf{P}_1-\mathbf{P}_0$$ represents the displacement vector from the start point to the end point.

Quadratic Bézier curves
A quadratic Bézier curve is the path traced by the function B(t), given points P0, P1, and P2,

 * $$\mathbf{B}(t) = (1 - t)[(1 - t) \mathbf P_0 + t \mathbf P_1] + t [(1 - t) \mathbf P_1 + t \mathbf P_2],\ 0 \le t \le 1$$,

which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P0 to P1 and from P1 to P2 respectively. Rearranging the preceding equation yields:

 * $$\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2,\ 0 \le t \le 1.$$

This can be written in a way that highlights the symmetry with respect to P1:

 * $$\mathbf{B}(t) = \mathbf{P}_1+(1 - t)^{2}(\mathbf{P}_0 - \mathbf{P}_1) + t^{2}(\mathbf{P}_2-\mathbf{P}_1),\ 0 \le t \le 1.$$

Which immediately gives the derivative of the Bézier curve with respect to t:

 * $$\mathbf{B}'(t) = 2 (1 - t) (\mathbf{P}_1 - \mathbf{P}_0) + 2 t (\mathbf{P}_2 - \mathbf{P}_1),$$

from which it can be concluded that the tangents to the curve at P0 and P2 intersect at P1. As t increases from 0 to 1, the curve departs from P0 in the direction of P1, then bends to arrive at P2 from the direction of P1.

The second derivative of the Bézier curve with respect to t is

 * $$\mathbf{B}''(t) = 2(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0).$$

Cubic Bézier curves
Four points P0, P1, P2 and P3 in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at P0 going toward P1 and arrives at P3 coming from the direction of P2. Usually, it will not pass through P1 or P2; these points are only there to provide directional information. The distance between P1 and P2 determines "how far" and "how fast" the curve moves towards P1 before turning towards P2.

Writing BP i,Pj,Pk (t) for the quadratic Bézier curve defined by points Pi, Pj, and Pk, the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves:

 * $$\mathbf{B}(t) = (1-t)\mathbf{B}_{\mathbf P_0,\mathbf P_1,\mathbf P_2}(t) + t \mathbf{B}_{\mathbf P_1,\mathbf P_2,\mathbf P_3}(t),\ 0 \le t \le 1.$$

The explicit form of the curve is:

 * $$\mathbf{B}(t) = (1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3,\ 0 \le t \le 1.$$

For some choices of P1 and P2 the curve may intersect itself, or contain a cusp.

Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order. Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to t = 1/3 and t = 2/3, the control points for the original Bézier curve can be recovered.

The derivative of the cubic Bézier curve with respect to t is
 * $$\mathbf{B}'(t) = 3(1-t)^2(\mathbf{P}_1 - \mathbf{P}_0) + 6(1-t)t(\mathbf{P}_2 - \mathbf{P}_1) + 3t^2(\mathbf{P}_3 - \mathbf{P}_2) \,.$$

The second derivative of the Bézier curve with respect to t is
 * $$\mathbf{B}''(t) = 6(1-t)(\mathbf{P}_2 - 2 \mathbf{P}_1 + \mathbf{P}_0) + 6t(\mathbf{P}_3 - 2 \mathbf{P}_2 + \mathbf{P}_1) \,.$$

General definition
Bézier curves can be defined for any degree n.

Recursive definition
A recursive definition for the Bézier curve of degree n expresses it as a point-to-point linear combination (linear interpolation) of a pair of corresponding points in two Bézier curves of degree n − 1.

Let $$\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_k}$$ denote the Bézier curve determined by any selection of points P0, P1, ..., Pk. Then to start,

 * $$\mathbf{B}_{\mathbf{P}_0}(t) = \mathbf{P}_0 \text{, and}$$

 * $$\mathbf{B}(t) = \mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_n}(t) = (1-t)\mathbf{B}_{\mathbf{P}_0\mathbf{P}_1\ldots\mathbf{P}_{n-1}}(t) + t\mathbf{B}_{\mathbf{P}_1\mathbf{P}_2\ldots\mathbf{P}_n}(t)$$

This recursion is elucidated in the animations below.

Explicit definition
The formula can be expressed explicitly as follows (where t0 and (1-t)0 are extended continuously to be 1 throughout [0,1]):

 * $$\begin{align}

\mathbf{B}(t) &= \sum_{i=0}^n {n\choose i}(1 - t)^{n - i}t^i\mathbf{P}_i \\ &=(1 - t)^n\mathbf{P}_0 + {n\choose 1}(1 - t)^{n - 1}t\mathbf{P}_1 + \cdots + {n\choose n - 1}(1 - t)t^{n - 1}\mathbf{P}_{n - 1} + t^n\mathbf{P}_n, && 0 \leqslant t \leqslant 1 \end{align}$$

where $$\scriptstyle {n \choose i}$$ are the binomial coefficients.

For example, when n = 5:

 * $$\begin{align}

\mathbf{B}(t) &= (1 - t)^5\mathbf{P}_0 + 5t(1 - t)^4\mathbf{P}_1 + 10t^2(1 - t)^3 \mathbf{P}_2 + 10t^3 (1-t)^2 \mathbf{P}_3 + 5t^4(1-t) \mathbf{P}_4 + t^5 \mathbf{P}_5, && 0 \leqslant t \leqslant 1. \end{align}$$

Terminology
Some terminology is associated with these parametric curves. We have

 * $$\mathbf{B}(t) = \sum_{i=0}^n b_{i, n}(t)\mathbf{P}_i,\ \ \ 0 \le t \le 1$$

where the polynomials

 * $$b_{i,n}(t) = {n \choose i} t^i (1 - t)^{n - i},\ \ \ i = 0, \ldots, n$$

are known as Bernstein basis polynomials of degree n.

t0 = 1, (1 − t)0 = 1, and the binomial coefficient, $$\scriptstyle {n \choose i}$$, is:

 * $${n \choose i} = \frac{n!}{i!(n - i)!}.$$

The points Pi are called control points for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P0 and finishing with Pn, is called the Bézier polygon (or control polygon). The convex hull of the Bézier polygon contains the Bézier curve.

Polynomial form
Sometimes it is desirable to express the Bézier curve as a polynomial instead of a sum of less straightforward Bernstein polynomials. Application of the binomial theorem to the definition of the curve followed by some rearrangement will yield

 * $$\mathbf{B}(t) = \sum_{j = 0}^n {t^j \mathbf{C}_j}$$

where
 * $$\mathbf{C}_j = \frac{n!}{(n - j)!} \sum_{i = 0}^j \frac{(-1)^{i + j} \mathbf{P}_i}{i! (j - i)!} =

\prod_{m = 0}^{j - 1} (n - m) \sum_{i = 0}^j \frac{(-1)^{i + j} \mathbf{P}_i}{i! (j - i)!}.$$

This could be practical if $$\mathbf{C}_j$$ can be computed prior to many evaluations of $$\mathbf{B}(t)$$; however one should use caution as high order curves may lack numeric stability (de Casteljau's algorithm should be used if this occurs). Note that the empty product is 1.

Properties

 * The curve begins at $$\mathbf P_0$$ and ends at $$\mathbf P_n$$; this is the so-called endpoint interpolation property.
 * The curve is a line if and only if all the control points are collinear.
 * The start and end of the curve is tangent to the first and last section of the Bézier polygon, respectively.
 * A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
 * Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve; though a four-piece cubic Bézier curve can approximate a circle (see composite Bézier curve), with a maximum radial error of less than one part in a thousand, when each inner control point (or offline point) is the distance $$\textstyle\frac{4\left(\sqrt {2}-1\right)}{3}$$ horizontally or vertically from an outer control point on a unit circle. More generally, an n-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance $$\textstyle\frac{4}{3}\tan(t/4)$$ from an outer control point on a unit circle, where $t = 2\pi/n$  (i.e. $$t=360^\circ/n$$), and $$n>2$$.
 * Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points $$\mathbf{P}_0,\,\dots,\,\mathbf{P}_n$$ is equivalent (including the parametrization) to the degree n + 1 curve with control points $$\mathbf{P}'_0,\,\dots,\,\mathbf {P}'_{n+1}$$, where $$\mathbf P'_k=\tfrac{k}{n+1}\mathbf P_{k-1}+\left(1-\tfrac{k}{n+1}\right)\mathbf P_k$$, $$\forall k = 0,\,1,\,\dots,\,n,\,n+1$$ and define $$\mathbf{P}_{n+1} := \mathbf{P}_{0}$$, $$\mathbf{P}_{-1} := \mathbf{P}_{n}$$.
 * Bézier curves have the variation diminishing property. What this means in intuitive terms is that a Bézier curve does not "undulate" more than the polygon of its control points, and may actually "undulate" less than that.
 * There is no local control in degree n Bézier curves—meaning that any change to a control point requires recalculation of and thus affects the aspect of the entire curve, "although the further that one is from the control point that was changed, the smaller is the change in the curve".
 * A Bézier curve of order higher than two may intersect itself or have a cusp for certain choices of the control points.

Second-order curve is a parabolic segment

A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows:

 * 1) Tangents to the parabola at the endpoints of the curve (A and B) intersect at its control point (C).
 * 2) If D is the midpoint of AB, the tangent to the curve which is perpendicular to CD (dashed cyan line) defines its vertex (V). Its axis of symmetry (dash-dot cyan) passes through V and is perpendicular to the tangent.
 * 3) E is either point on the curve with a tangent at 45° to CD (dashed green). If G is the intersection of this tangent and the axis, the line passing through G and perpendicular to CD is the directrix (solid green).
 * 4) The focus (F) is at the intersection of the axis and a line passing through E and perpendicular to CD (dotted yellow). The latus rectum is the line segment within the curve (solid yellow).

Derivative
The derivative for a curve of order n is

 * $$\mathbf{B}'(t) = n \sum_{i=0}^{n-1} b_{i,n-1}(t) (\mathbf{P}_{i+1} - \mathbf{P}_i).$$

Linear curves
Let t denote the fraction of progress (from 0 to 1) the point B(t) has made along its traversal from P0 to P1. For example, when t=0.25, B(t) is one quarter of the way from point P0 to P1. As t varies from 0 to 1, B(t) draws a line from P0 to P1.

Quadratic curves
For quadratic Bézier curves one can construct intermediate points Q0 and Q1 such that as t varies from 0 to 1:
 * Point Q0(t) varies from P0 to P1 and describes a linear Bézier curve.
 * Point Q1(t) varies from P1 to P2 and describes a linear Bézier curve.
 * Point B(t) is interpolated linearly between Q0(t) to Q1(t) and describes a quadratic Bézier curve.

Higher-order curves
For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points Q0, Q1, and Q2 that describe linear Bézier curves, and points R0 and R1 that describe quadratic Bézier curves:

For fourth-order curves one can construct intermediate points Q0, Q1, Q2 and Q3 that describe linear Bézier curves, points R0, R1 and R2 that describe quadratic Bézier curves, and points S0 and S1 that describe cubic Bézier curves:

For fifth-order curves, one can construct similar intermediate points.

These representations rest on the process used in De Casteljau's algorithm to calculate Bézier curves.

Offsets (or stroking) of Bézier curves
The curve at a fixed offset from a given Bézier curve, called an offset or parallel curve in mathematics (lying "parallel" to the original curve, like the offset between rails in a railroad track), cannot be exactly formed by a Bézier curve (except in some trivial cases). In general, the two-sided offset curve of a cubic Bézier is a 10th-order algebraic curve and more generally for a Bézier of degree n the two-sided offset curve is an algebraic curve of degree 4n − 2. However, there are heuristic methods that usually give an adequate approximation for practical purposes.

In the field of vector graphics, painting two symmetrically distanced offset curves is called stroking (the Bézier curve or in general a path of several Bézier segments). The conversion from offset curves to filled Bézier contours is of practical importance in converting fonts defined in Metafont, which require stroking of Bézier curves, to the more widely used PostScript type 1 fonts, which only require (for efficiency purposes) the mathematically simpler operation of filling a contour defined by (non-self-intersecting) Bézier curves.

Degree elevation
A Bézier curve of degree n can be converted into a Bézier curve of degree n + 1 with the same shape. This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation.

To do degree elevation, we use the equality $$\mathbf{B}(t) = (1-t)\mathbf{B}(t) + t\mathbf{B}(t).$$ Each component $$\mathbf{b}_{i,n}(t)\mathbf{P}_i$$ is multiplied by (1 − t) and t, thus increasing a degree by one, without changing the value. Here is the example of increasing degree from 2 to 3.

 * $$\begin{align}

(1 - t)^2 \mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^2 \mathbf{P}_2 &= (1 - t)^3 \mathbf{P}_0 + 2(1 - t)^2 t\mathbf{P}_1 + (1 - t)t^2 \mathbf{P}_2 + (1 - t)^{2}t\mathbf{P}_0 + 2(1 - t)t^2 \mathbf{P}_1 + t^3 \mathbf{P}_2 \\ &= (1 - t)^3 \mathbf{P}_0 + (1 - t)^2 t \left( \mathbf{P}_0 + 2\mathbf{P}_1\right) + (1 - t) t^2 \left(2\mathbf{P}_1 + \mathbf{P}_2\right) + t^{3}\mathbf{P}_2 \\ &= (1 - t)^3 \mathbf{P}_0 + 3(1 - t)^2 t \tfrac{1}{3} \left( \mathbf{P}_0 + 2\mathbf{P}_1 \right) + 3(1 - t) t^2 \tfrac{1}{3} \left( 2\mathbf{P}_1 + \mathbf{P}_2 \right) + t^{3}\mathbf{P}_2 \end{align}$$

In other words, the original start and end points are unchanged. The new control points are $$\mathbf{P'}_1 = \tfrac{1}{3} \left( \mathbf{P}_0 + 2\mathbf{P}_1 \right)$$ and $$\mathbf{P'}_2 = \tfrac{1}{3} \left( 2\mathbf{P}_1 + \mathbf{P}_2 \right)$$.

For arbitrary n we use equalities

 * $$\begin{cases} {n + 1 \choose i}(1 - t)\mathbf{b}_{i, n} = {n \choose i} \mathbf{b}_{i, n + 1} \\ {n + 1 \choose i + 1} t\mathbf{b}_{i, n} = {n \choose i} \mathbf{b}_{i + 1, n + 1} \end{cases} \quad \implies \quad \begin{cases} (1 - t)\mathbf{b}_{i, n} = \frac{n + 1 - i}{n + 1} \mathbf{b}_{i, n + 1} \\ t\mathbf{b}_{i, n} = \frac{i + 1}{n + 1} \mathbf{b}_{i + 1, n + 1} \end{cases} $$

Therefore:

 * $$\begin{align}

\mathbf{B}(t) &= (1 - t)\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i + t\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i \\ &= \sum_{i=0}^n \frac{n + 1 - i}{n + 1}\mathbf{b}_{i, n + 1}(t)\mathbf{P}_i + \sum_{i=0}^n \frac{i + 1}{n + 1}\mathbf{b}_{i + 1, n + 1}(t)\mathbf{P}_i \\ &= \sum_{i=0}^{n + 1} \left(\frac{i}{n + 1}\mathbf{P}_{i - 1} + \frac{n + 1 - i}{n + 1}\mathbf{P}_i\right) \mathbf{b}_{i, n + 1}(t) \\ &= \sum_{i=0}^{n + 1} \mathbf{b}_{i, n + 1}(t)\mathbf{P'}_i \end{align}$$

introducing arbitrary $$\mathbf{P}_{-1}$$ and $$\mathbf{P}_{n + 1}$$.

Therefore, new control points are

 * $$\mathbf{P'}_i = \frac{i}{n + 1}\mathbf{P}_{i - 1} + \frac{n + 1 - i}{n + 1}\mathbf{P}_i,\quad i=0, \ldots, n + 1.$$

Repeated degree elevation
The concept of degree elevation can be repeated on a control polygon R to get a sequence of control polygons R, R1, R2, and so on. After r degree elevations, the polygon Rr has the vertices P0,r, P1,r, P2,r, ..., Pn+r,r given by

 * $$\mathbf{P}_{i,r} = \sum_{j=0}^n \mathbf{P}_j \tbinom nj \frac{\tbinom{r}{i-j}}{\tbinom{n+r}{i}}.$$

It can also be shown that for the underlying Bézier curve B,

 * $$\mathbf{\lim_{r \to \infty}R_r} = \mathbf{B}.$$

Degree reduction
Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree. A number of approximation algorithms have been proposed and used in practice.

Rational Bézier curves
The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of Bernstein polynomials. Rational Bézier curves can, among other uses, be used to represent segments of conic sections exactly, including circular arcs.

Given n + 1 control points P0, ..., Pn, the rational Bézier curve can be described by
 * $$\mathbf{B}(t) = \frac{

\sum_{i=0}^n b_{i,n}(t) \mathbf{P}_{i}w_i } { \sum_{i=0}^n b_{i,n}(t) w_i },$$ or simply
 * $$\mathbf{B}(t) = \frac{

\sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}\mathbf{P}_{i}w_i } { \sum_{i=0}^n {n \choose i} t^i (1-t)^{n-i}w_i }.$$ The expression can be extended by using number systems besides reals for the weights. In the complex plane the points {1}, {-1}, and {1} with weights {$$i$$}, {1}, and {$$-i$$} generate a full circle with radius one. For curves with points and weights on a circle, the weights can be scaled without changing the curve's shape. Scaling the central weight of the above curve by 1.35508 gives a more uniform parameterization.

Computer graphics
Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation and rotation can be applied on the curve by applying the respective transform on the control points of the curve.

Quadratic and cubic Bézier curves are most common. Higher degree curves are more computationally expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a composite Bézier curve. A composite Bézier curve is commonly referred to as a "path" in vector graphics languages (like PostScript), vector graphics standards (like SVG) and vector graphics programs (like Artline, Timeworks Publisher, Adobe Illustrator, CorelDraw, Inkscape, and Allegro). In order to join Bézier curves into a composite Bézier curve without kinks, a property called G1 continuity suffices to force the control point at which two constituent Bézier curves meet to lie on the line defined by the two control points on either side.

The simplest method for scan converting (rasterizing) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0.5 and 0.5 ≤ t ≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation.

Analytical methods where a Bézier is intersected with each scan line involve finding roots of cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.

The rasterisation algorithm used in Metafont is based on discretising the curve, so that it is approximated by a sequence of "rook moves" that are purely vertical or purely horizontal, along the pixel boundaries. To that end, the plane is first split into eight 45° sectors (by the coordinate axes and the two lines $$ y = \pm x $$), then the curve is decomposed into smaller segments such that the direction of a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the $$t$$ values where it is parallel to one of these lines can be done by solving quadratic equations. Within each segment, either horizontal or vertical movement dominates, and the total number of steps in either direction can be read off from the endpoint coordinates; in for example the 0–45° sector horizontal movement to the right dominates, so it only remains to decide between which steps to the right the curve should make a step up.

There is also a modified curve form of Bresenham's line drawing algorithm by Zingl that performs this rasterization by subdividing the curve into rational pieces and calculating the error at each pixel location such that it either travels at a 45° angle or straight depending on compounding error as it iterates through the curve. This reduces the next step calculation to a series of integer additions and subtractions.

Animation
In animation applications, such as Adobe Flash and Synfig, Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path.

In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation. Bézier curves are now very frequently used to control the animation easing in CSS, JavaScript, JavaFx and Flutter SDK.

Fonts
TrueType fonts use composite Bézier curves composed of quadratic Bézier curves. Other languages and imaging tools (such as PostScript, Asymptote, Metafont, and SVG) use composite Béziers composed of cubic Bézier curves for drawing curved shapes. OpenType fonts can use either kind of curve, depending on which font technology underlies the OpenType wrapper.

Font engines, like FreeType, draw the font's curves (and lines) on a pixellated surface using a process known as font rasterization. Typically font engines and vector graphics engines render Bézier curves by splitting them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. The "smooth curve" feature of charts in Microsoft Excel also uses this algorithm.

Because arcs of circles and ellipses cannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also preserves the curve definition under all linear or perspective 2D and 3D transforms and projections.

Robotics
Because the control polygon allows to tell whether or not the path collides with any obstacles, Bézier curves are used in producing trajectories of the end effectors. Furthermore, joint space trajectories can be accurately differentiated using Bézier curves. Consequently, the derivatives of joint space trajectories are used in the calculation of the dynamics and control effort (torque profiles) of the robotic manipulator.

title: Bézier curve


text:
Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra $$A$$ over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy $$\|x \, y\| \ \leq \|x\| \, \|y\| \quad \text{ for all } x, y \in A.$$

This ensures that the multiplication operation is continuous.

A Banach algebra is called unital if it has an identity element for the multiplication whose norm is $$1,$$ and commutative if its multiplication is commutative. Any Banach algebra $$A$$ (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra $$A_e$$ so as to form a closed ideal of $$A_e$$. Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering $$A_e$$ and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

Banach algebras can also be defined over fields of $p$-adic numbers. This is part of $p$-adic analysis.

Examples
The prototypical example of a Banach algebra is $$C_0(X)$$, the space of (complex-valued) continuous functions, defined on a locally compact Hausdorff space $$X$$, that vanish at infinity. $$C_0(X)$$ is unital if and only if $$X$$ is compact. The complex conjugation being an involution, $$C_0(X)$$ is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra by definition.

 * The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value.
 * The set of all real or complex $$n$$-by-$$n$$ matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
 * Take the Banach space $$\R^n$$ (or $$\Complex^n$$) with norm $$\|x\| = \max_{} |x_i|$$ and define multiplication componentwise: $$\left(x_1, \ldots, x_n\right) \left(y_1, \ldots, y_n\right) = \left(x_1 y_1, \ldots, x_n y_n\right).$$
 * The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
 * The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra.
 * The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.
 * The algebra of all continuous linear operators on a Banach space $$E$$ (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on $$E$$ is a Banach algebra and closed ideal. It is without identity if $$\dim E = \infty.$$
 * If $$G$$ is a locally compact Hausdorff topological group and $$\mu$$ is its Haar measure, then the Banach space $$L^1(G)$$ of all $$\mu$$-integrable functions on $$G$$ becomes a Banach algebra under the convolution $$x y(g) = \int x(h) y\left(h^{-1} g\right) d \mu(h)$$ for $$x, y \in L^1(G).$$
 * Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra $$C(X)$$ with the supremum norm and that contains the constants and separates the points of $$X$$ (which must be a compact Hausdorff space).
 * Natural Banach function algebra: A uniform algebra all of whose characters are evaluations at points of $$X.$$
 * C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.
 * Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution of measures.
 * The algebra of the quaternions $$\H$$ is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.
 * An affinoid algebra is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in rigid analytic geometry.

Properties
Several elementary functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a topological group under multiplication.

If a Banach algebra has unit $$\mathbf{1},$$ then $$\mathbf{1}$$ cannot be a commutator; that is, $$xy - yx \neq \mathbf{1}$$  for any $$x, y \in A.$$ This is because $$x y$$ and $$y x$$ have the same spectrum except possibly $$0.$$

The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:

 * Every real Banach algebra that is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the Gelfand–Mazur theorem.)
 * Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.
 * Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
 * Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
 * Permanently singular elements in Banach algebras are topological divisors of zero, that is, considering extensions $$B$$ of Banach algebras $$A$$ some elements that are singular in the given algebra $$A$$ have a multiplicative inverse element in a Banach algebra extension $$B.$$ Topological divisors of zero in $$A$$ are permanently singular in any Banach extension $$B$$ of $$A.$$

Spectral theory
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element $$x \in A,$$ denoted by $$\sigma(x)$$, consists of all those complex scalars $$\lambda$$ such that $$x - \lambda \mathbf{1}$$ is not invertible in $$A.$$ The spectrum of any element $$x$$ is a closed subset of the closed disc in $$\Complex$$ with radius $$\|x\|$$ and center $$0,$$ and thus is  compact. Moreover, the spectrum $$\sigma(x)$$ of an element $$x$$ is non-empty and satisfies the spectral radius formula: $$\sup \{|\lambda| : \lambda \in \sigma(x)\} = \lim_{n \to \infty} \|x^n\|^{1/n}.$$

Given $$x \in A,$$ the holomorphic functional calculus allows to define $$f(x) \in A$$ for any function $$f$$ holomorphic in a neighborhood of $$\sigma(x).$$ Furthermore, the spectral mapping theorem holds: $$\sigma(f(x)) = f(\sigma(x)).$$

When the Banach algebra $$A$$ is the algebra $$L(X)$$ of bounded linear operators on a complex Banach space $$X$$ (for example, the algebra of square matrices), the notion of the spectrum in $$A$$ coincides with the usual one in operator theory. For $$f \in C(X)$$ (with a compact Hausdorff space $$X$$), one sees that: $$\sigma(f) = \{f(t) : t \in X\}.$$

The norm of a normal element $$x$$ of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.

Let $$A$$ be a complex unital Banach algebra in which every non-zero element $$x$$ is invertible (a division algebra). For every $$a \in A,$$ there is $$\lambda \in \Complex$$ such that $$a - \lambda \mathbf{1}$$ is not invertible (because the spectrum of $$a$$ is not empty) hence $$a = \lambda \mathbf{1}:$$ this algebra $$A$$ is naturally isomorphic to $$\Complex$$ (the complex case of the Gelfand–Mazur theorem).

Ideals and characters
Let $$A$$ be a unital commutative Banach algebra over $$\Complex.$$ Since $$A$$ is then a commutative ring with unit, every non-invertible element of $$A$$ belongs to some maximal ideal of $$A.$$ Since a maximal ideal $$\mathfrak m$$ in $$A$$ is closed, $$A / \mathfrak m$$ is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of $$A$$ and the set $$\Delta(A)$$ of all nonzero homomorphisms from $$A$$ to $$\Complex.$$ The set $$\Delta(A)$$ is called the "structure space" or "character space" of $$A,$$ and its members "characters".

A character $$\chi$$ is a linear functional on $$A$$ that is at the same time multiplicative, $$\chi(a b) = \chi(a) \chi(b),$$ and satisfies $$\chi(\mathbf{1}) = 1.$$ Every character is automatically continuous from $$A$$ to $$\Complex,$$ since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on $$A$$ (that is, the topology induced by the weak-* topology of $$A^*$$), the character space, $$\Delta(A),$$ is a Hausdorff compact space.

For any $$x \in A,$$ $$\sigma(x) = \sigma(\hat x)$$ where $$\hat x$$ is the Gelfand representation of $$x$$ defined as follows: $$\hat x$$ is the continuous function from $$\Delta(A)$$ to $$\Complex$$ given by $$\hat x(\chi) = \chi(x).$$ The spectrum of $$\hat x,$$ in the formula above, is the spectrum as element of the algebra $$C(\Delta(A))$$ of complex continuous functions on the compact space $$\Delta(A).$$  Explicitly, $$\sigma(\hat x) = \{\chi(x) : \chi \in \Delta(A)\}.$$

As an algebra, a unital commutative Banach algebra is semisimple (that is, its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when $$A$$ is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between $$A$$ and $$C(\Delta(A)).$$

Banach *-algebras
A Banach *-algebra $$A$$ is a Banach algebra over the field of complex numbers, together with a map $${}^* : A \to A$$ that has the following properties:
 * 1) $$\left(x^*\right)^* = x$$ for all $$x \in A$$ (so the map is an involution).
 * 2) $$(x + y)^* = x^* + y^*$$ for all $$x, y \in A.$$
 * 3) $$(\lambda x)^* = \bar{\lambda}x^*$$ for every $$\lambda \in \Complex$$ and every $$x \in A;$$ here, $$\bar{\lambda}$$ denotes the complex conjugate of $$\lambda.$$
 * 4) $$(x y)^* = y^* x^*$$ for all $$x, y \in A.$$

In other words, a Banach *-algebra is a Banach algebra over $$\Complex$$ that is also a *-algebra.

In most natural examples, one also has that the involution is isometric, that is, $$\|x^*\| = \|x\| \quad \text{ for all } x \in A.$$ Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying $$\|x^* x\| = \|x^*\| \|x\|$$ is a C*-algebra.

title: Banach algebra