Misplaced Pages

#160839

113-604: The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 Tahrir al-Majisti (Commentary on the Almagest) as a solution for the latitudinal motion of the inferior planets and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy 's Almagest . The translation of the copy of Tusi's original description of his geometrical model alludes to at least one inversion of

226-454: A a 2 − x 2 = ± ( a 2 − x 2 ) ( 1 − e 2 ) . {\displaystyle y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.} The width and height parameters a , b {\displaystyle a,\;b} are called

339-528: A 2 x 1 b 2 ) {\displaystyle {\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}} is a tangent vector at point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} , which proves the vector equation. If ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( u , v ) {\displaystyle (u,v)} are two points of

452-621: A 2 + y 1 v b 2 ) + s 2 ( u 2 a 2 + v 2 b 2 ) = 0   . {\displaystyle {\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\ .} There are then cases: Using (1) one finds that ( − y 1

565-549: A 2 + y 2 b 2 = 1. {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.} Assuming a ≥ b {\displaystyle a\geq b} , the foci are ( ± c , 0 ) {\displaystyle (\pm c,0)} for c = a 2 − b 2 {\textstyle c={\sqrt {a^{2}-b^{2}}}} . The standard parametric equation is: ( x , y ) = (

678-466: A 2 b 2 . {\displaystyle {\begin{aligned}A&=a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta &B&=2\left(b^{2}-a^{2}\right)\sin \theta \cos \theta \\[1ex]C&=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta &D&=-2Ax_{\circ }-By_{\circ }\\[1ex]E&=-Bx_{\circ }-2Cy_{\circ }&F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}} These expressions can be derived from

791-542: A 2 cos 2 ⁡ θ + b 2 sin 2 ⁡ θ D = − 2 A x ∘ − B y ∘ E = − B x ∘ − 2 C y ∘ F = A x ∘ 2 + B x ∘ y ∘ + C y ∘ 2 −

904-462: A 2 − b 2 {\displaystyle c={\sqrt {a^{2}-b^{2}}}} . The eccentricity can be expressed as: e = c a = 1 − ( b a ) 2 , {\displaystyle e={\frac {c}{a}}={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}},} assuming a > b . {\displaystyle a>b.} An ellipse with equal axes (

1017-425: A ≥ b > 0   . {\displaystyle a\geq b>0\ .} In principle, the canonical ellipse equation x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} may have a < b {\displaystyle a<b} (and hence

1130-458: A + e x {\displaystyle a+ex} and a − e x {\displaystyle a-ex} . It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin. Throughout this article, the semi-major and semi-minor axes are denoted a {\displaystyle a} and b {\displaystyle b} , respectively, i.e.

1243-596: A = b {\displaystyle a=b} ) has zero eccentricity, and is a circle. The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum . One half of it is the semi-latus rectum ℓ {\displaystyle \ell } . A calculation shows: ℓ = b 2 a = a ( 1 − e 2 ) . {\displaystyle \ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).} The semi-latus rectum ℓ {\displaystyle \ell }

SECTION 10

#1732764985161

1356-429: A cos ⁡ ( t ) , b sin ⁡ ( t ) ) for 0 ≤ t ≤ 2 π . {\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .} Ellipses are the closed type of conic section : a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with

1469-418: A parabola ). An ellipse has a simple algebraic solution for its area, but for its perimeter (also known as circumference ), integration is required to obtain an exact solution. Analytically , the equation of a standard ellipse centered at the origin with width 2 a {\displaystyle 2a} and height 2 b {\displaystyle 2b} is: x 2

1582-534: A Hellenized form of Old Persian Pārsa ( 𐎱𐎠𐎼𐎿 ), which evolves into Fārs ( فارس ) in modern Persian. In the Bible , particularly in the books of Daniel , Esther , Ezra , and Nehemya , it is given as Pārās ( פָּרָס ). A Greek folk etymology connected the name to Perseus , a legendary character in Greek mythology . Herodotus recounts this story, devising a foreign son, Perses , from whom

1695-576: A center of interest, particularly during the annual international Shiraz Arts Festival and the 2,500th anniversary of the founding of the Persian Empire . The Pahlavi rulers modernized Iran, and ruled it until the 1979 Revolution . In modern Iran, the Persians make up the majority of the population. They are native speakers of the modern dialects of Persian , which serves as the country's official language. The Persian language belongs to

1808-590: A dialect version of Gujarati , and no longer speak in Persian. They do however continue to use Avestan as their liturgical language. The Parsis have adapted many practices and tendencies of the Indian groups that surrounded them, such as Indian dress norms, and the observance of many Indian festivals and ceremonies. From Persis and throughout the Median, Achaemenid, Parthian, and Sasanian empires of ancient Iran to

1921-410: A geometric system. The enclosure of this symmetrically arranged planting and irrigation by an infrastructure such as a palace created the impression of "paradise". The word paradise itself originates from Avestan pairidaēza ( Old Persian paridaida ; New Persian pardis , ferdows ), which literally translates to "walled-around". Characterized by its quadripartite ( čārbāq ) design,

2034-683: A major role in the downfall of the Neo-Assyrian Empire . The Medes , another group of ancient Iranian people, unified the region under an empire centered in Media , which would become the region's leading cultural and political power of the time by 612 BC. Meanwhile, under the dynasty of the Achaemenids , the Persians formed a vassal state to the central Median power. In 552 BC, the Achaemenid Persians revolted against

2147-435: A more available and detailed documentation than the earlier periods, and is especially more evident within the context of Zoroastrian musical rituals . Overall, Sasanian music was influential and was adopted in the subsequent eras. Iranian music, as a whole, utilizes a variety of musical instruments that are unique to the region, and has remarkably evolved since the ancient and medieval times. In traditional Sasanian music,

2260-725: A no-slip condition as its tangent point moves along the fixed outer circle. The term "Tusi couple" is a modern one, coined by Edward Stewart Kennedy in 1966. It is one of several late Islamic astronomical devices bearing a striking similarity to models in Nicolaus Copernicus 's De revolutionibus , including his Mercury model and his theory of trepidation . Historians suspect that Copernicus or another European author had access to an Arabic astronomical text, but an exact chain of transmission has not yet been identified, The 16th century scientist and traveler Guillaume Postel has been suggested as one possible facilitator. Since

2373-704: A period of artistic growth that left a legacy ranging from Cyrus the Great's solemn tomb at Pasargadae to the structures at Persepolis and Naqsh-e Rostam . The Bam Citadel , a massive structure at 1,940,000 square feet (180,000 m ) constructed on the Silk Road in Bam , is from around the 5th century BC. The quintessential feature of Achaemenid architecture was its eclectic nature, with elements from Median architecture, Assyrian architecture, and Asiatic Greek architecture all incorporated. The architectural heritage of

SECTION 20

#1732764985161

2486-522: A point on an ellipse and x → = ( x 1 y 1 ) + s ( u v ) {\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}} be the equation of any line g {\displaystyle g} containing ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} . Inserting

2599-507: A variety of Persian by the name of Aimaqi , which also belongs to the Dari dialect continuum. Persian-speaking communities native to modern Arab countries are generally designated as Ajam , including the Ajam of Bahrain , the Ajam of Iraq , and the Ajam of Kuwait . The Parsis are a Zoroastrian community of Persian descent who migrated to South Asia , to escape religious persecution after

2712-458: Is a constant. This constant ratio is the above-mentioned eccentricity: e = c a = 1 − b 2 a 2 . {\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.} Ellipses are common in physics , astronomy and engineering . For example, the orbit of each planet in the Solar System

2825-525: Is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics . The name, ἔλλειψις ( élleipsis , "omission"), was given by Apollonius of Perga in his Conics . An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane: The midpoint C {\displaystyle C} of

2938-562: Is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids . A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection . The ellipse

3051-719: Is attested as late as 465". Following the Arab conquest of the Sasanian Empire in the medieval times, the Arab caliphates established their rule over the region for the next several centuries, during which the long process of the Islamization of Iran took place. Confronting the cultural and linguistic dominance of the Persians, beginning by the Umayyad Caliphate , the Arab conquerors began to establish Arabic as

3164-447: Is equal to the radius of curvature at the vertices (see section curvature ). An arbitrary line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line , tangent and secant . Through any point of an ellipse there is a unique tangent. The tangent at a point ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} of

3277-688: Is essentially a reciprocal cultural exchange. Its legacy and impact on the kingdom of Macedon was also notably huge, even for centuries after the withdrawal of the Persians from Europe following the Greco-Persian Wars . During the Achaemenid era, Persian colonists settled in Asia Minor . In Lydia (the most important Achaemenid satrapy), near Sardis , there was the Hyrcanian plain , which, according to Strabo , got its name from

3390-650: Is from the Behistun Inscription , a multilingual inscription from the time of Achaemenid ruler Darius the Great carved on a cliff in western Iran. There are several ethnic groups and communities that are either ethnically or linguistically related to the Persian people, living predominantly in Iran, and also within Afghanistan, Tajikistan, Uzbekistan, the Caucasus, Turkey, Iraq, and the Arab states of

3503-473: Is included as a special type of ellipse. The equation | P F 2 | + | P F 1 | = 2 a {\displaystyle \left|PF_{2}\right|+\left|PF_{1}\right|=2a} can be viewed in a different way (see figure): c 2 {\displaystyle c_{2}} is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of

Tusi couple - Misplaced Pages Continue

3616-536: Is known to have one of the world's oldest and most influential literatures. Old Persian written works are attested on several inscriptions from between the 6th and the 4th centuries BC, and Middle Persian literature is attested on inscriptions from the Parthian and Sasanian eras and in Zoroastrian and Manichaean scriptures from between the 3rd to the 10th century AD. New Persian literature flourished after

3729-421: Is regarded as the world's oldest existing carpet, depicts elements of Assyrian and Achaemenid designs, including stylistic references to the stone slab designs found in Persian royal buildings. According to the accounts reported by Xenophon, a great number of singers were present at the Achaemenid court. However, little information is available from the music of that era. The music scene of the Sasanian Empire has

3842-625: Is the 2-argument arctangent function. Using trigonometric functions , a parametric representation of the standard ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} is: ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) ,   0 ≤ t < 2 π . {\displaystyle (x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \,.} The parameter t (called

3955-642: Is the immediate ancestor of Modern Persian and a variety of other Iranian dialects, became the official language of the empire and was greatly diffused among Iranians. The Parthians and the Sasanians would also extensively interact with the Romans culturally. The Roman–Persian wars and the Byzantine–Sasanian wars would shape the landscape of Western Asia , Europe , the Caucasus , North Africa, and

4068-427: Is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e {\displaystyle e} , a number ranging from e = 0 {\displaystyle e=0} (the limiting case of a circle) to e = 1 {\displaystyle e=1} (the limiting case of infinite elongation, no longer an ellipse but

4181-805: The eccentric anomaly in astronomy) is not the angle of ( x ( t ) , y ( t ) ) {\displaystyle (x(t),y(t))} with the x -axis, but has a geometric meaning due to Philippe de La Hire (see § Drawing ellipses below). With the substitution u = tan ⁡ ( t 2 ) {\textstyle u=\tan \left({\frac {t}{2}}\right)} and trigonometric formulae one obtains cos ⁡ t = 1 − u 2 1 + u 2   , sin ⁡ t = 2 u 1 + u 2 {\displaystyle \cos t={\frac {1-u^{2}}{1+u^{2}}}\ ,\quad \sin t={\frac {2u}{1+u^{2}}}} and

4294-638: The Arab conquest of Iran with its earliest records from the 9th century, and was developed as a court tradition in many eastern courts. The Shahnameh of Ferdowsi , the works of Rumi , the Rubaiyat of Omar Khayyam , the Panj Ganj of Nizami Ganjavi , the Divān of Hafez , The Conference of the Birds by Attar of Nishapur , and the miscellanea of Gulistan and Bustan by Saadi Shirazi are among

4407-772: The Greek colonies of the Tauric Chersonesos , and for a brief time the Roman province of Asia . After a long struggle with Rome in the Mithridatic Wars , Pontus was defeated; part of it was incorporated into the Roman Republic as the province of Bithynia and Pontus , and the eastern half survived as a client kingdom. Following the Macedonian conquests , the Persian colonists in Cappadocia and

4520-572: The Iranian calendar , which corresponds to around March 21 in the Gregorian calendar. An ancient tradition that has been preserved in Iran and several other countries that were under the influence of the ancient empires of Iran, Nowruz has been registered on UNESCO 's Intangible Cultural Heritage Lists . In Iran, the Nowruz celebrations (incl. Charshanbe Suri and Sizdebedar ) begin on the eve of

4633-579: The Old Assyrian form Parahše , designating a region belonging to the Sumerians . The name of this region was adopted by a nomadic ancient Iranian people who migrated to the region in the west and southwest of Lake Urmia , eventually becoming known as "the Persians". The ninth-century BC Neo-Assyrian inscription of the Black Obelisk of Shalmaneser III , found at Nimrud , gives it in

Tusi couple - Misplaced Pages Continue

4746-563: The Pahlavi dynasty issued a decree asking the international community to use the term Iran , the native name of the country, in formal correspondence. However, the term Persian is still historically used to designate the predominant population of the Iranian peoples living in the Iranian cultural continent . Persia is first attested in Assyrian sources from the third millennium BC in

4859-479: The Sasanian Empire , a national culture that was fully aware of being Iranian took shape, partially motivated by restoration and revival of the wisdom of "the old sages" ( dānāgān pēšēnīgān ). Other aspects of this national culture included the glorification of a great heroic past and an archaizing spirit. Throughout the period, Iranian identity reached its height in every aspect. Middle Persian , which

4972-470: The Western world , the names Persia and Persian came to refer to all of Iran and its subjects. Some medieval and early modern Islamic sources also used cognates of the term Persian to refer to various Iranian peoples and languages, including the speakers of Khwarazmian , Mazanderani , and Old Azeri . 10th-century Iraqi historian Al-Masudi refers to Pahlavi , Dari , and Azari as dialects of

5085-786: The degenerate cases from the non-degenerate case, let ∆ be the determinant Δ = | A 1 2 B 1 2 D 1 2 B C 1 2 E 1 2 D 1 2 E F | = A C F + 1 4 B D E − 1 4 ( A E 2 + C D 2 + F B 2 ) . {\displaystyle \Delta ={\begin{vmatrix}A&{\frac {1}{2}}B&{\frac {1}{2}}D\\{\frac {1}{2}}B&C&{\frac {1}{2}}E\\{\frac {1}{2}}D&{\frac {1}{2}}E&F\end{vmatrix}}=ACF+{\tfrac {1}{4}}BDE-{\tfrac {1}{4}}(AE^{2}+CD^{2}+FB^{2}).} Then

5198-410: The octave was divided into seventeen tones. By the end of the 13th century, Iranian music also maintained a twelve-interval octave, which resembled the western counterparts. The Iranian New Year 's Day, Nowruz , which translates to "new day", is celebrated by Persians and other peoples of Iran to mark the beginning of spring on the vernal equinox on the first day of Farvardin , the first month of

5311-491: The radicals by suitable squarings and using b 2 = a 2 − c 2 {\displaystyle b^{2}=a^{2}-c^{2}} (see diagram) produces the standard equation of the ellipse: x 2 a 2 + y 2 b 2 = 1 , {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,} or, solved for y : y = ± b

5424-557: The rational parametric equation of an ellipse { x ( u ) = a 1 − u 2 1 + u 2 y ( u ) = b 2 u 1 + u 2 − ∞ < u < ∞ {\displaystyle {\begin{cases}x(u)=a\,{\dfrac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)=b\,{\dfrac {2u}{1+u^{2}}}\\[10mu]-\infty <u<\infty \end{cases}}} which covers any point of

5537-423: The semi-major and semi-minor axes . The top and bottom points V 3 = ( 0 , b ) , V 4 = ( 0 , − b ) {\displaystyle V_{3}=(0,\,b),\;V_{4}=(0,\,-b)} are the co-vertices . The distances from a point ( x , y ) {\displaystyle (x,\,y)} on the ellipse to the left and right foci are

5650-774: The western group of the Iranian branch of the Indo-European language family . Modern Persian is classified as a continuation of Middle Persian , the official religious and literary language of the Sasanian Empire , itself a continuation of Old Persian , which was used by the time of the Achaemenid Empire . Old Persian is one of the oldest Indo-European languages attested in original text. Samples of Old Persian have been discovered in present-day Iran, Armenia , Egypt , Iraq , Romania ( Gherla ), and Turkey . The oldest attested text written in Old Persian

5763-653: The x - and y -axes. In analytic geometry , the ellipse is defined as a quadric : the set of points ( x , y ) {\displaystyle (x,\,y)} of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation A x 2 + B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0} provided B 2 − 4 A C < 0. {\displaystyle B^{2}-4AC<0.} To distinguish

SECTION 50

#1732764985161

5876-574: The Caucasus (primarily living in the Republic of Azerbaijan and the Russian republic of Dagestan ), speak a language ( Tat language ) that is closely related to Persian. The origin of the Tat people is traced to an Iranian-speaking population that was resettled in the Caucasus by the time of the Sasanian Empire. The Lurs , an ethnic Iranian people native to western Iran, are often associated with

5989-529: The Late Assyrian forms Parsua and Parsumaš as a region and a people located in the Zagros Mountains , the latter likely having migrated southward and transferred the name of the region with them to what would become Persis (Persia proper, i.e., modern-day Fars ), and that is considered to be the earliest attestation to the ancient Persian people. The ancient Persians played

6102-578: The Median monarchy, leading to the victory of Cyrus the Great over the throne in 550 BC. The Persians spread their influence to the rest of what is considered to be the Iranian Plateau , and assimilated with the non-Iranian indigenous groups of the region, including the Elamites and the Mannaeans . At its greatest extent, the Achaemenid Empire stretched from parts of Eastern Europe in

6215-686: The Mediterranean Basin for centuries. For a period of over 400 years, the Sasanians and the neighboring Byzantines were recognized as the two leading powers in the world. Cappadocia in Late Antiquity , now well into the Roman era, still retained a significant Iranian character; Stephen Mitchell notes in the Oxford Dictionary of Late Antiquity : "Many inhabitants of Cappadocia were of Persian descent and Iranian fire worship

6328-473: The Parthian era, Iranian identity had an ethnic, linguistic, and religious value. However, it did not yet have a political import. The Parthian language , which was used as an official language of the Parthian Empire, left influences on Persian, as well as on the neighboring Armenian language . The Parthian monarchy was succeeded by the Persian dynasty of the Sasanians in 224 AD. By the time of

6441-464: The Persian Gulf . The Tajiks are a people native to Tajikistan , Afghanistan , and Uzbekistan who speak Persian in a variety of dialects. The Tajiks of Tajikistan and Uzbekistan are native speakers of Tajik , which is the official language of Tajikistan, and those in Afghanistan speak Dari , one of the two official languages of Afghanistan. The Tat people , an Iranian people native to

6554-511: The Persian fire ceremony at the towns of Hyrocaesareia and Hypaepa . Mithridates III of Cius , a Persian nobleman and part of the Persian ruling elite of the town of Cius , founded the Kingdom of Pontus in his later life, in northern Asia Minor. At the peak of its power, under the infamous Mithridates VI the Great , the Kingdom of Pontus also controlled Colchis , Cappadocia , Bithynia ,

6667-538: The Persian garden was evolved and developed into various forms throughout history, and was also adopted in various other cultures in Eurasia. It was inscribed on UNESCO 's World Heritage List in June 2011. Carpet weaving is an essential part of the Persian culture, and Persian rugs are said to be one of the most detailed hand-made works of art. Achaemenid rug and carpet artistry is well recognized. Xenophon describes

6780-520: The Persian language are known as Tajiks , with the former two countries having their own dialects of Persian known as Dari and Tajiki , respectively; whereas those in the Caucasus (primarily in the present-day Republic of Azerbaijan and Dagestan , Russia ), albeit heavily assimilated, are known as Tats . Historically, however, the terms Tajik and Tat were used synonymously and interchangeably with Persian . Many influential Persian figures hailed from outside of Iran's present-day borders—to

6893-622: The Persian language. In 1333, medieval Moroccan traveler and scholar Ibn Battuta referred to the Afghans of Kabul as a specific sub-tribe of the Persians. Lady Mary (Leonora Woulfe) Sheil, in her observation of Iran during the Qajar era, states that the Kurds and the Leks would consider themselves as belonging to the race of the "old Persians". On 21 March 1935, the then-king of Iran Reza Shah of

SECTION 60

#1732764985161

7006-480: The Persian settlers that were moved from Hyrcania . Similarly near Sardis, there was the plain of Cyrus, which further signified the presence of numerous Persian settlements in the area. In all these centuries, Lydia and Pontus were reportedly the chief centers for the worship of the Persian gods in Asia Minor. According to Pausanias , as late as the second century AD, one could witness rituals which resembled

7119-459: The Persians is eclectic and has included contributions from both the east and the west. Due to the central location of Iran, Persian art has served as a fusion point between eastern and western traditions. Persians have contributed to various forms of art, including calligraphy , carpet weaving , glasswork , lacquerware , marquetry ( khatam ), metalwork , miniature illustration , mosaic , pottery , and textile design . The Persian language

7232-508: The Persians and the Kurds . They speak various dialects of the Luri language , which is considered to be a descendant of Middle Persian . The Hazaras , making up the third largest ethnic group in Afghanistan, speak a variety of Persian by the name of Hazaragi , which is more precisely a part of the Dari dialect continuum. The Aimaqs , a semi-nomadic people native to Afghanistan, speak

7345-564: The Persians took the name. Apparently, the Persians themselves knew the story, as Xerxes I tried to use it to suborn the Argives during his invasion of Greece, but ultimately failed to do so. Although Persis (Persia proper) was only one of the provinces of ancient Iran, varieties of this term (e.g., Persia ) were adopted through Greek sources and used as an exonym for all of the Persian Empire for many years. Thus, especially in

7458-1002: The Sasanian Empire includes, among others, castle fortifications such as the Fortifications of Derbent (located in North Caucasus , now part of Russia ), the Rudkhan Castle and the Shapur-Khwast Castle , palaces such as the Palace of Ardashir and the Sarvestan Palace , bridges such as the Shahrestan Bridge and the Shapuri Bridge , the Archway of Ctesiphon , and the reliefs at Taq-e Bostan . Architectural elements from

7571-432: The Tusi couple was developed within an astronomical context, later mathematicians and engineers developed similar versions of what came to be called hypocycloid straight-line mechanisms. The mathematician Gerolamo Cardano designed a system known as Cardan's movement (also known as a Cardan gear ). Nineteenth-century engineers James White, Matthew Murray , as well as later designers, developed practical applications of

7684-413: The Tusi-couple are still extant in Italy. Another possibility is that he encountered the manuscript of the "Straightening of the Curves" (Sefer Meyasher 'Aqov) while studying in Italy. There are other sources for this mathematical model for converting circular motions to reciprocating linear motion. It is found in Proclus's Commentary on the First Book of Euclid and the concept was known in Paris by

7797-577: The Tusi-couple was used by Copernicus in his reformulation of mathematical astronomy, there is a growing consensus that he became aware of this idea in some way. It has been suggested that the idea of the Tusi couple may have arrived in Europe leaving few manuscript traces, since it could have occurred without the translation of any Arabic text into Latin. One possible route of transmission may have been through Byzantine science ; Gregory Chioniades translated some of al-Tusi's works from Arabic into Byzantine Greek . Several Byzantine Greek manuscripts containing

7910-1060: The canonical equation X 2 a 2 + Y 2 b 2 = 1 {\displaystyle {\frac {X^{2}}{a^{2}}}+{\frac {Y^{2}}{b^{2}}}=1} by a Euclidean transformation of the coordinates ( X , Y ) {\displaystyle (X,\,Y)} : X = ( x − x ∘ ) cos ⁡ θ + ( y − y ∘ ) sin ⁡ θ , Y = − ( x − x ∘ ) sin ⁡ θ + ( y − y ∘ ) cos ⁡ θ . {\displaystyle {\begin{aligned}X&=\left(x-x_{\circ }\right)\cos \theta +\left(y-y_{\circ }\right)\sin \theta ,\\Y&=-\left(x-x_{\circ }\right)\sin \theta +\left(y-y_{\circ }\right)\cos \theta .\end{aligned}}} Conversely,

8023-1385: The canonical form parameters can be obtained from the general-form coefficients by the equations: a , b = − 2 ( A E 2 + C D 2 − B D E + ( B 2 − 4 A C ) F ) ( ( A + C ) ± ( A − C ) 2 + B 2 ) B 2 − 4 A C , x ∘ = 2 C D − B E B 2 − 4 A C , y ∘ = 2 A E − B D B 2 − 4 A C , θ = 1 2 atan2 ⁡ ( − B , C − A ) , {\displaystyle {\begin{aligned}a,b&={\frac {-{\sqrt {2{\big (}AE^{2}+CD^{2}-BDE+(B^{2}-4AC)F{\big )}{\big (}(A+C)\pm {\sqrt {(A-C)^{2}+B^{2}}}{\big )}}}}{B^{2}-4AC}},\\x_{\circ }&={\frac {2CD-BE}{B^{2}-4AC}},\\[5mu]y_{\circ }&={\frac {2AE-BD}{B^{2}-4AC}},\\[5mu]\theta &={\tfrac {1}{2}}\operatorname {atan2} (-B,\,C-A),\end{aligned}}} where atan2

8136-404: The carpet production in the city of Sardis, stating that the locals take pride in their carpet production. A special mention of Persian carpets is also made by Athenaeus of Naucratis in his Deipnosophistae , as he describes a "delightfully embroidered" Persian carpet with "preposterous shapes of griffins ". The Pazyryk carpet, a Scythian pile-carpet dating back to the 4th century BC that

8249-399: The center. The distance c {\displaystyle c} of the foci to the center is called the focal distance or linear eccentricity. The quotient e = c a {\displaystyle e={\tfrac {c}{a}}} is the eccentricity . The case F 1 = F 2 {\displaystyle F_{1}=F_{2}} yields a circle and

8362-456: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} except the left vertex ( − a , 0 ) {\displaystyle (-a,\,0)} . For u ∈ [ 0 , 1 ] , {\displaystyle u\in [0,\,1],} this formula represents

8475-479: The ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} has the coordinate equation: x 1 a 2 x + y 1 b 2 y = 1. {\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.} A vector parametric equation of

8588-620: The ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse. The general equation's coefficients can be obtained from known semi-major axis a {\displaystyle a} , semi-minor axis b {\displaystyle b} , center coordinates ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , and rotation angle θ {\displaystyle \theta } (the angle from

8701-427: The ellipse such that x 1 u a 2 + y 1 v b 2 = 0 {\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0} , then the points lie on two conjugate diameters (see below ). (If a = b {\displaystyle a=b} , the ellipse is a circle and "conjugate" means "orthogonal".) If

8814-418: The ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names x {\displaystyle x} and y {\displaystyle y} and the parameter names a {\displaystyle a} and b . {\displaystyle b.} This is the distance from the center to a focus: c =

8927-543: The ellipse, the x -axis is the major axis, and: For an arbitrary point ( x , y ) {\displaystyle (x,y)} the distance to the focus ( c , 0 ) {\displaystyle (c,0)} is ( x − c ) 2 + y 2 {\textstyle {\sqrt {(x-c)^{2}+y^{2}}}} and to the other focus ( x + c ) 2 + y 2 {\textstyle {\sqrt {(x+c)^{2}+y^{2}}}} . Hence

9040-469: The ellipse. This property should not be confused with the definition of an ellipse using a directrix line below. Using Dandelin spheres , one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone. The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of

9153-621: The expansion of the Persianate societies , particularly those of Turco-Persian and Indo-Persian blends. After over eight centuries of foreign rule within the region, the Iranian hegemony was reestablished by the emergence of the Safavid Empire in the 16th century. Under the Safavid Empire, focus on Persian language and identity was further revived, and the political evolution of the empire once again maintained Persian as

9266-571: The fall of the Sassanian Empire . They have had a significant role in the development of India , Pakistan and Sri Lanka , and also played a role in the development of Iranian nationalism during the late Qajar years and Pahlavi dynasty. They are primarily located in the western regions of India principally the states of Gujarat and Maharashtra , with smaller communities in other parts of India and in South and Southeast Asia. They speak

9379-669: The fall of the Achaemenid Persian Empire, records only traces of Persians in western Asia Minor; however, he considered Cappadocia "almost a living part of Persia". The Iranian dominance collapsed in 330 BC following the conquest of the Achaemenid Empire by Alexander the Great , but reemerged shortly after through the establishment of the Parthian Empire in 247 BC, which was founded by a group of ancient Iranian people rising from Parthia . Until

9492-489: The famous works of medieval Persian literature. A thriving contemporary Persian literature has also been formed by the works of writers such as Ahmad Shamlou , Forough Farrokhzad , Mehdi Akhavan-Sales , Parvin E'tesami , Sadegh Hedayat , and Simin Daneshvar , among others. Not all Persian literature is written in Persian, as works written by Persians in other languages—such as Arabic and Greek—might also be included. At

9605-710: The first time after two centuries of no attestation of the language, now having received the Arabic script and a large Arabic vocabulary. Persian language and culture continued to prevail after the invasions and conquests by the Mongols and the Turks (including the Ilkhanate , Ghaznavids , Seljuks , Khwarazmians , and Timurids ), who were themselves significantly Persianized , further developing in Asia Minor , Central Asia , and South Asia , where Persian culture flourished by

9718-428: The good and beautiful things that the soil produce." The Persian garden , the earliest examples of which were found throughout the Achaemenid Empire, has an integral position in Persian architecture. Gardens assumed an important place for the Achaemenid monarchs, and utilized the advanced Achaemenid knowledge of water technologies, including aqueducts , earliest recorded gravity-fed water rills, and basins arranged in

9831-495: The hypocycloid straight-line mechanism. A practical and mechanically simple version of the Tusi couple, which avoids the use of an external rim gear, was developed in 2021 by John Goodman in order to provide linear motion. It uses 3 standard spur gears. A rotating (blue) arm is mounted on a central shaft, to which a fixed (yellow) gear is mounted. A (red) idler gear on the arm meshes with the fixed gear. A third (green) gear meshes with

9944-406: The idler. The third gear has half the number of teeth of the fixed gear. An (orange) arm is fixed to the third gear. If the length of the arm equals the distance between the fixed and outer gears = d, the arm will describe a straight line of throw = 2d. An advantage of this design is that, if standard modulus gears that do not provide the required throw, the idler gear does not have to be colinear with

10057-410: The last Wednesday of the preceding year in the Iranian calendar and last on the 13th day of the new year. Islamic festivals are also widely celebrated by Muslim Persians. Ellipse In mathematics , an ellipse is a plane curve surrounding two focal points , such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle , which

10170-420: The line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis , and the line perpendicular to it through the center is the minor axis . The major axis intersects the ellipse at two vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance a {\displaystyle a} to

10283-558: The line's equation into the ellipse equation and respecting x 1 2 a 2 + y 1 2 b 2 = 1 {\textstyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1} yields: ( x 1 + s u ) 2 a 2 + ( y 1 + s v ) 2 b 2 = 1   ⟹ 2 s ( x 1 u

10396-617: The main language of the country. During the times of the Safavids and subsequent modern Iranian dynasties such as the Qajars , architectural and iconographic elements from the time of the Sasanian Persian Empire were reincorporated, linking the modern country with its ancient past. Contemporary embracement of the legacy of Iran's ancient empires, with an emphasis on the Achaemenid Persian Empire, developed particularly under

10509-527: The majority of the population of Iran . They share a common cultural system and are native speakers of the Persian language as well as of the languages that are closely related to Persian . The ancient Persians were originally an ancient Iranian people who had migrated to the region of Persis (corresponding to the modern-day Iranian province of Fars ) by the 9th century BCE. Together with their compatriot allies, they established and ruled some of

10622-473: The middle of the 14th Century. In his questiones on the Sphere (written before 1362), Nicole Oresme described how to combine circular motions to produce a reciprocating linear motion of a planet along the radius of its epicycle. Oresme's description is unclear and it is not certain whether this represents an independent invention or an attempt to come to grips with a poorly understood Arabic text. Although

10735-405: The model to be seen in the diagrams: The description is not coherent and appears to arbitrarily combine features of several both possible and impossible inversions of the geometric model. Algebraically, the model can be expressed with complex numbers as Other commentators have observed that the Tusi couple can be interpreted as a rolling curve where the rotation of the inner circle satisfies

10848-441: The neighboring Greek city states and the kingdom of Macedon , and later throughout the medieval Islamic world , all the way to modern Iran and others parts of Eurasia , Persian culture has been extended, celebrated, and incorporated. This is due mainly to its geopolitical conditions, and its intricate relationship with the ever-changing political arena once as dominant as the Achaemenid Empire. The artistic heritage of

10961-568: The northeast in Afghanistan and Central Asia , and to a lesser extent within the Caucasus proper to the northwest. In historical contexts, especially in English , "Persian" may be defined more loosely (often as a national identity) to cover all subjects of the ancient Persian polities, regardless of their ethnic background. The term Persian , meaning "from Persia", derives from Latin Persia , itself deriving from Greek Persís ( Περσίς ),

11074-412: The other two forms of conic sections, parabolas and hyperbolas , both of which are open and unbounded . An angled cross section of a right circular cylinder is also an ellipse. An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix : for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix

11187-436: The other two gears A property of the Tusi couple is that points on the inner circle that are not on the circumference trace ellipses . These ellipses, and the straight line traced by the classic Tusi couple, are special cases of hypotrochoids . Persian people The Persians ( / ˈ p ɜːr ʒ ən z / PUR -zhənz or / ˈ p ɜːr ʃ ən z / PUR -shənz ) are an Iranian ethnic group who comprise

11300-737: The parameter [ u : v ] {\displaystyle [u:v]} is considered to be a point on the real projective line P ( R ) {\textstyle \mathbf {P} (\mathbf {R} )} , then the corresponding rational parametrization is [ u : v ] ↦ ( a v 2 − u 2 v 2 + u 2 , b 2 u v v 2 + u 2 ) . {\displaystyle [u:v]\mapsto \left(a{\frac {v^{2}-u^{2}}{v^{2}+u^{2}}},b{\frac {2uv}{v^{2}+u^{2}}}\right).} Then [ 1 : 0 ] ↦ ( −

11413-399: The point ( x , y ) {\displaystyle (x,\,y)} is on the ellipse whenever: ( x − c ) 2 + y 2 + ( x + c ) 2 + y 2 = 2 a   . {\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a\ .} Removing

11526-429: The positive horizontal axis to the ellipse's major axis) using the formulae: A = a 2 sin 2 ⁡ θ + b 2 cos 2 ⁡ θ B = 2 ( b 2 − a 2 ) sin ⁡ θ cos ⁡ θ C =

11639-421: The primary language of the subject peoples throughout their empire, sometimes by force, further confirming the new political reality over the region. The Arabic term ʿAjam , denoting "people unable to speak properly", was adopted as a designation for non-Arabs (or non-Arabic speakers), especially the Persians. Although the term had developed a derogatory meaning and implied cultural and ethnic inferiority, it

11752-489: The reign of the Pahlavi dynasty , providing the motive of a modern nationalistic pride. Iran's modern architecture was then inspired by that of the country's classical eras, particularly with the adoption of details from the ancient monuments in the Achaemenid capitals Persepolis and Pasargadae and the Sasanian capital Ctesiphon . Fars, corresponding to the ancient province of Persia, with its modern capital Shiraz , became

11865-606: The rest of Asia Minor were cut off from their co-religionists in Iran proper, but they continued to practice the Iranian faith of their forefathers. Strabo, who observed them in the Cappadocian Kingdom in the first century BC, records (XV.3.15) that these "fire kindlers" possessed many "holy places of the Persian Gods", as well as fire temples. Strabo, who wrote during the time of Augustus ( r.  27 BC – AD 14 ), almost three hundred years after

11978-432: The right upper quarter of the ellipse moving counter-clockwise with increasing u . {\displaystyle u.} The left vertex is the limit lim u → ± ∞ ( x ( u ) , y ( u ) ) = ( − a , 0 ) . {\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.} Alternately, if

12091-426: The same time, not all literature written in Persian is written by ethnic Persians or Iranians, as Turkic, Caucasian, and Indic authors have also used Persian literature in the environment of Persianate cultures . The most notable examples of ancient Persian architecture are the works of the Achaemenids hailing from Persis . Achaemenid architecture , dating from the expansion of the empire around 550 BC, flourished in

12204-618: The standard ellipse is shifted to have center ( x ∘ , y ∘ ) {\displaystyle \left(x_{\circ },\,y_{\circ }\right)} , its equation is ( x − x ∘ ) 2 a 2 + ( y − y ∘ ) 2 b 2 = 1   . {\displaystyle {\frac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\frac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1\ .} The axes are still parallel to

12317-606: The tangent is: x → = ( x 1 y 1 ) + s ( − y 1 a 2 x 1 b 2 ) , s ∈ R . {\displaystyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s\left({\begin{array}{r}-y_{1}a^{2}\\x_{1}b^{2}\end{array}}\right),\quad s\in \mathbb {R} .} Proof: Let ( x 1 , y 1 ) {\displaystyle (x_{1},\,y_{1})} be

12430-554: The time of Iran's ancient Persian empires have been adopted and incorporated in later period. They were used especially during the modernization of Iran under the reign of the Pahlavi dynasty to contribute to the characterization of the modern country with its ancient history. Xenophon , in his Oeconomicus , states: "The Great King [Cyrus II]...in all the districts he resides in and visits, takes care that there are parádeisos ("paradise") as they [Persians] call them, full of

12543-529: The west to the Indus Valley in the east, making it the largest empire the world had yet seen. The Achaemenids developed the infrastructure to support their growing influence, including the establishment of the cities of Pasargadae and Persepolis . The empire extended as far as the limits of the Greek city states in modern-day mainland Greece , where the Persians and Athenians influenced each other in what

12656-479: The world's most powerful empires that are well-recognized for their massive cultural, political, and social influence, which covered much of the territory and population of the ancient world . Throughout history, the Persian people have contributed greatly to art and science . Persian literature is one of the world's most prominent literary traditions. In contemporary terminology, people from Afghanistan , Tajikistan , and Uzbekistan who natively speak

12769-544: Was gradually accepted as a synonym for "Persian" and still remains today as a designation for the Persian-speaking communities native to the modern Arab states of the Middle East. A series of Muslim Iranian kingdoms were later established on the fringes of the declining Abbasid Caliphate , including that of the ninth-century Samanids , under the reign of whom the Persian language was used officially for

#160839