57-457: Kreis is the German word for circle . Kreis may also refer to: Circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre . The distance between any point of the circle and the centre is called the radius . The length of a line segment connecting two points on the circle and passing through the centre is called
114-412: A 2 − r 0 2 sin 2 ( θ − ϕ ) . {\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.} Without the ± sign, the equation would in some cases describe only half a circle. In the complex plane , a circle with a centre at c and radius r has
171-570: A ) ( x − a ) + ( y 1 − b ) ( y − b ) = r 2 . {\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.} If y 1 ≠ b , then the slope of this line is d y d x = − x 1 − a y 1 − b . {\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.} This can also be found using implicit differentiation . When
228-407: A certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. In Plato 's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early science , particularly geometry and astrology and astronomy ,
285-557: A chord, between the midpoint of that chord and the arc of the circle. Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: r = y 2 8 x + x 2 . {\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.} Another proof of this result, which relies only on two chord properties given above,
342-484: A compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel , a rainbow, mandalas, rose windows and so forth. Magic circles are part of some traditions of Western esotericism . The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. The ratio of a circle's circumference to its radius
399-574: A complex and multivalent term which refers to the eternal cosmic law, universal moral order and in Buddhism, the very teaching and path expounded by the Buddha. In the Buddhist Art at early sites such as Bharhut and Sanchi , the dharmachakra was often used as a symbol of Gautama Buddha himself. The symbol is often paired with the triratna (triple jewel) or trishula (trident) symbolizing
456-459: A generalised circle is either a (true) circle or a line . The tangent line through a point P on the circle is perpendicular to the diameter passing through P . If P = ( x 1 , y 1 ) and the circle has centre ( a , b ) and radius r , then the tangent line is perpendicular to the line from ( a , b ) to ( x 1 , y 1 ), so it has the form ( x 1 − a ) x + ( y 1 – b ) y = c . Evaluating at ( x 1 , y 1 ) determines
513-481: A religious transformation from Buddhism, such as Jagannath temple, whose deity is believed by some scholars to have a Buddhist origin . It also finds use in other ancient temples of Odisha, the most famous of which is the Konark Sun Temple . The 24 spoke Ashoka dharmachakra is present in the modern flag of India , representing the pan-Indian concept of Dharma . The modern State Emblem of India
570-591: A symbol of both faiths. It is one of the oldest known Indian symbols found in Indian art , appearing with the first surviving post- Indus Valley Civilisation Indian iconography in the time of the Buddhist king Ashoka . The Buddha is said to have set the "wheel of dharma" in motion when he delivered his first sermon, which is described in the Dhammacakkappavattana Sutta . This "turning of
627-409: Is 2 π . Thus the circumference C is related to the radius r and diameter d by: C = 2 π r = π d . {\displaystyle C=2\pi r=\pi d.} As proved by Archimedes , in his Measurement of a Circle , the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals
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#1732764874649684-518: Is a symbol used in the Dharmic religions . It has a widespread use in Buddhism. In Hinduism , the symbol is particularly used in places that underwent religious transformation. The symbol also finds its usage in modern India. Historically, the dharmachakra was often used as a decoration in East Asian statues and inscriptions , beginning with the earliest period of East Asian culture to
741-725: Is a depiction of the Lion Capital of Ashoka (Sanchi), which includes the dharmachakra. An integral part of the emblem is the motto inscribed in Devanagari script: Satyameva Jayate (English: Truth Alone Triumphs ). This is a quote from the Mundaka Upanishad , the concluding part of the Vedas . Sarvepalli Radhakrishnan , the first Vice President of India, stated that the Ashoka Chakra of India represents
798-414: Is as follows. Given a chord of length y and with sagitta of length x , since the sagitta intersects the midpoint of the chord, we know that it is a part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is ( 2 r − x ) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting
855-504: Is said to be a chariot of one wheel (cakra). Mitra , a form of Surya, is described as "the eye of the world", and thus the sun is conceived of as an eye (cakṣu) which illuminates and perceives the world. Such a wheel is also the main attribute of Vishnu . Thus, a wheel symbol might also be associated with light and knowledge. In Buddhism, the Dharma Chakra is widely used to represent the Buddha's Dharma ( Buddha 's teaching and
912-735: Is sometimes called a generalised circle . This becomes the above equation for a circle with p = 1 , g = − c ¯ , q = r 2 − | c | 2 {\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}} , since | z − c | 2 = z z ¯ − c ¯ z − c z ¯ + c c ¯ {\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}} . Not all generalised circles are actually circles:
969-1307: The x {\displaystyle x} – y {\displaystyle y} plane can be broken into two semicircles each of which is the graph of a function , y + ( x ) {\displaystyle y_{+}(x)} and y − ( x ) {\displaystyle y_{-}(x)} , respectively: y + ( x ) = y 0 + r 2 − ( x − x 0 ) 2 , y − ( x ) = y 0 − r 2 − ( x − x 0 ) 2 , {\displaystyle {\begin{aligned}y_{+}(x)=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}},\\[5mu]y_{-}(x)=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}},\end{aligned}}} for values of x {\displaystyle x} ranging from x 0 − r {\displaystyle x_{0}-r} to x 0 + r {\displaystyle x_{0}+r} . The equation can be written in parametric form using
1026-474: The Digha Nikaya describes this wheel as having a nave (nābhi), a thousand spokes (sahassārāni) and a felly (nemi), all of which are perfect in every respect. Siddhartha Gautama was said to have been a "mahapurisa" (great man) who could have chosen to become a wheel turning king, but instead became the spiritual counterpart to such a king, a wheel turning sage, that is, a Buddha . In his explanation of
1083-509: The Nebra sky disc and jade discs called Bi . The Egyptian Rhind papyrus , dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π . Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from
1140-555: The angle that the ray from ( a , b ) to ( x , y ) makes with the positive x axis. An alternative parametrisation of the circle is x = a + r 1 − t 2 1 + t 2 , y = b + r 2 t 1 + t 2 . {\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}} In this parameterisation,
1197-493: The circular points at infinity . In polar coordinates , the equation of a circle is r 2 − 2 r r 0 cos ( θ − ϕ ) + r 0 2 = a 2 , {\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},} where a is the radius of the circle, ( r , θ ) {\displaystyle (r,\theta )} are
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#17327648746491254-401: The circumscribing square (whose side is of length d ). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality . If a circle of radius r is centred at the vertex of an angle , and that angle intercepts an arc of the circle with an arc length of s , then
1311-414: The diameter . A circle bounds a region of the plane called a disc . The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel , which, with related inventions such as gears , makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire
1368-747: The equation of the circle , follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length | x − a | and | y − b |. If the circle is centred at the origin (0, 0), then the equation simplifies to x 2 + y 2 = r 2 . {\displaystyle x^{2}+y^{2}=r^{2}.} The circle of radius r {\displaystyle r} with center at ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} in
1425-451: The radian measure 𝜃 of the angle is the ratio of the arc length to the radius: θ = s r . {\displaystyle \theta ={\frac {s}{r}}.} The circular arc is said to subtend the angle, known as the central angle , at the centre of the circle. The angle subtended by a complete circle at its centre is a complete angle , which measures 2 π radians, 360 degrees , or one turn . Using radians,
1482-401: The trigonometric functions sine and cosine as x = a + r cos t , y = b + r sin t , {\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}} where t is a parametric variable in the range 0 to 2 π , interpreted geometrically as
1539-400: The triple gem , umbrellas ( chatra ), symbols of sovereignty and royal power, gems and garlands. It is also sometimes depicted alongside animals such as lions, or deer. There are different designs of the Buddhist dharmachakra with 8, 12, 24 or more spokes . In different Buddhist traditions, the different number of spokes may represent different aspects of the Buddha's Dharma (teaching). In
1596-498: The Buddhist doctrine of dependent origination . According to the Theravada scholar Buddhaghosa : “It is the beginningless round of rebirths that is called the ’Wheel of the round of rebirths’ (saṃsāracakka). Ignorance (avijjā) is its hub (or nave) because it is its root. Ageing-and-death (jarā-maraṇa) is its rim (or felly) because it terminates it. The remaining ten links [of Dependent Origination] are its spokes [i.e. saṅkhāra up to
1653-573: The Indo-Tibetan Buddhist tradition for example, the 8 spoked wheel represents the noble eightfold path , and the hub, rim and spokes are also said to represent the three trainings ( sila , prajña and samadhi ). In Buddhism, the cyclical movement of a wheel is also used to symbolize the cyclical nature of life in the world (also referred to as the "wheel of samsara ", samsara-chakra or the "wheel of becoming" , bhava-cakra ). This wheel of suffering can be reversed or "turned" through
1710-427: The blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180°). The sagitta (also known as the versine ) is a line segment drawn perpendicular to
1767-461: The centre of the circle is at the origin, then the equation of the tangent line becomes x 1 x + y 1 y = r 2 , {\displaystyle x_{1}x+y_{1}y=r^{2},} and its slope is d y d x = − x 1 y 1 . {\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.} An inscribed angle (examples are
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1824-1774: The circle determined by three points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})} not on a line is obtained by a conversion of the 3-point form of a circle equation : ( x − x 1 ) ( x − x 2 ) + ( y − y 1 ) ( y − y 2 ) ( y − y 1 ) ( x − x 2 ) − ( y − y 2 ) ( x − x 1 ) = ( x 3 − x 1 ) ( x 3 − x 2 ) + ( y 3 − y 1 ) ( y 3 − y 2 ) ( y 3 − y 1 ) ( x 3 − x 2 ) − ( y 3 − y 2 ) ( x 3 − x 1 ) . {\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.} In homogeneous coordinates , each conic section with
1881-408: The circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines,
1938-419: The circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits. The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example,
1995-496: The circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2 . {\displaystyle \mathrm {Area} =\pi r^{2}.} Equivalently, denoting diameter by d , A r e a = π d 2 4 ≈ 0.7854 d 2 , {\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},} that is, approximately 79% of
2052-520: The circle). For a circle centred on the origin, i.e. r 0 = 0 , this reduces to r = a . When r 0 = a , or when the origin lies on the circle, the equation becomes r = 2 a cos ( θ − ϕ ) . {\displaystyle r=2a\cos(\theta -\phi ).} In the general case, the equation can be solved for r , giving r = r 0 cos ( θ − ϕ ) ±
2109-477: The compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B . (The set of points where the distances are equal is the perpendicular bisector of segment AB , a line.) That circle is sometimes said to be drawn about two points. Dharmachakra The dharmachakra ( Sanskrit : धर्मचक्र, Pali : dhammacakka ) or wheel of dharma
2166-747: The development of geometry, astronomy and calculus . All of the specified regions may be considered as open , that is, not containing their boundaries, or as closed , including their respective boundaries. The word circle derives from the Greek κίρκος/κύκλος ( kirkos/kuklos ), itself a metathesis of the Homeric Greek κρίκος ( krikos ), meaning "hoop" or "ring". The origins of the words circus and circuit are closely related. Prehistoric people made stone circles and timber circles , and circular elements are common in petroglyphs and cave paintings . Disc-shaped prehistoric artifacts include
2223-712: The early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions. From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity –
2280-514: The equation | z − c | = r . {\displaystyle |z-c|=r.} In parametric form, this can be written as z = r e i t + c . {\displaystyle z=re^{it}+c.} The slightly generalised equation p z z ¯ + g z + g z ¯ = q {\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q} for real p , q and complex g
2337-461: The equation of a circle has the form x 2 + y 2 − 2 a x z − 2 b y z + c z 2 = 0. {\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.} It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane ) the points I (1: i : 0) and J (1: − i : 0). These points are called
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2394-407: The first chord, we find that ( 2 r − x ) x = ( y / 2) . Solving for r , we find the required result. There are many compass-and-straightedge constructions resulting in circles. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate
2451-446: The formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is s = θ r , {\displaystyle s=\theta r,} and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.} In
2508-508: The glory of Buddhism and the royal house.” According to Harrison, the symbolism of "the wheel of the law" and the order of Nature is also visible in the Tibetan prayer wheels . The moving wheels symbolize the movement of cosmic order ( ṛta ). The dharmachakra is a symbol in the sramana religion of Budhha Dhamma. Wheel symbolism was also used in Indian temples in places that underwent
2565-546: The most ancient in all Indian history. Madhavan and Parpola note that a wheel symbol appears frequently in Indus Valley civilization artifacts, particularly on several seals . Notably, it is present in a sequence of ten signs on the Dholavira Signboard . Some historians associate the ancient chakra symbols with solar symbolism . In the Vedas , the god Surya is associated with the solar disc, which
2622-407: The polar coordinates of a generic point on the circle, and ( r 0 , ϕ ) {\displaystyle (r_{0},\phi )} are the polar coordinates of the centre of the circle (i.e., r 0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of
2679-469: The practice of the Buddhist path. The Buddhist terms for "suffering" ( dukkha ) and happiness ( sukha ) may also originally be related to the proper or improper fitting of wheels on a chariot's axle . The Indo-Tibetan tradition has developed elaborate depictions called Bhavacakras which depict the many realms of rebirth in Buddhist cosmology . The spokes of a wheel are also often used as symbols of
2736-523: The present. It remains a major symbol of the Buddhist religion today. The Sanskrit noun dharma ( धर्म ) is a derivation from the root dhṛ 'to hold, maintain, keep', and means 'what is established or firm'. The word derives from the Vedic Sanskrit n -stem dharman- with the meaning "bearer, supporter". The historical Vedic religion apparently conceived of dharma as an aspect of Ṛta . Similar chakra (spoked-wheel) symbols are one of
2793-655: The process of becoming, bhava].” The earliest Indian monument featuring dharmachakras are the Ashokan Pillars , such as the lion pillar at Sanchi, built at the behest of the Mauryan emperor Ashoka . According to Benjamin Rowland: ”The Sārnāth column may be interpreted, therefore, not only as a glorification of the Buddha’s preaching symbolised by the crowning wheel, but also through the cosmological implications of
2850-406: The ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution ). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted. The equation of
2907-506: The special case 𝜃 = 2 π , these formulae yield the circumference of a complete circle and area of a complete disc, respectively. In an x – y Cartesian coordinate system , the circle with centre coordinates ( a , b ) and radius r is the set of all points ( x , y ) such that ( x − a ) 2 + ( y − b ) 2 = r 2 . {\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.} This equation , known as
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#17327648746492964-478: The term "turning the wheel of Dharma", the Theravada exegete Buddhaghosa explains that this "wheel" which the Buddha turned is primarily to be understood as wisdom, knowledge, and insight ( ñāṇa ). This wisdom has two aspects, paṭivedha-ñāṇa, the wisdom of self-realisation of the Truth and desanā-ñāṇa, the wisdom of proclamation of the Truth. The dharmachakra symbol also points to the central Indian idea of " Dharma ",
3021-612: The universal moral order), Gautama Buddha himself and the walking of the path to enlightenment , since the time of Early Buddhism . The symbol is also sometimes connected to the Four Noble Truths , the Noble Eightfold Path and Dependent Origination. The pre-Buddhist dharmachakra ( Pali : dhammacakka ) is considered one of the ashtamangala (auspicious signs) in Hinduism and Buddhism and often used as
3078-457: The value of c , and the result is that the equation of the tangent is ( x 1 − a ) x + ( y 1 − b ) y = ( x 1 − a ) x 1 + ( y 1 − b ) y 1 , {\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},} or ( x 1 −
3135-411: The wheel" signifies a great and revolutionary change with universal consequences, brought about by an exceptional human being. Buddhism adopted the wheel as a symbol from the Indian mythical idea of the ideal king, called a chakravartin ("wheel-turner", or "universal monarch"), who was said to possess several mythical objects, including the ratana cakka (the ideal wheel). The Mahā Sudassana Sutta of
3192-535: The whole pillar as a symbol of the universal extension of the power of the Buddha’s Law as typified by the sun that dominates all space and all time, and simultaneously an emblem of the universal extension of Mauryan imperialism through the Dharma. The whole structure is then a translation of age-old Indian and Asiatic cosmology into artistic terms of essentially foreign origin and dedicated, like all Asoka’s monuments, to
3249-402: Was connected to the divine for most medieval scholars , and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles. In 1880 CE, Ferdinand von Lindemann proved that π is transcendental , proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass. With the advent of abstract art in
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