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83-428: 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 the diameter . A circle bounds a region of the plane called a disc . The circle has been known since before

166-412: 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

249-404: 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 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

332-570: 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

415-450: A " p " have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if

498-565: 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 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,

581-423: 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 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

664-557: 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,

747-423: 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 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

830-410: A coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has

913-459: 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

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996-564: A glorification of the Buddha’s preaching symbolised by the crowning wheel, but also through the cosmological implications of 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

1079-476: 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, 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,

1162-428: 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 a certain point within it to the bounding line, are equal. The bounding line

1245-405: A mirror is the same shape as the original, and not a distinct shape. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles , squares , and pentagons . Other shapes may be bounded by curves such as the circle or

1328-599: A outline and boundary so you can see it and is not just regular dots on a regular paper. The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis . In particular, Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis ). All similar triangles have

1411-421: A reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have

1494-428: A set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation , and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes: In this paper ‘shape’

1577-435: A shape defined by n − 2 complex numbers S ( z j , z j + 1 , z j + 2 ) ,   j = 1 , . . . , n − 2. {\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.} The polygon bounds a convex set when all these shape components have imaginary components of the same sign. Human vision relies on

1660-472: A translation of age-old Indian and Asiatic cosmology into artistic terms of essentially foreign origin and dedicated, like all Asoka’s monuments, to 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

1743-440: A triangle. The shape of a quadrilateral is associated with two complex numbers p , q . If the quadrilateral has vertices u , v , w , x , then p = S( u , v , w ) and q = S( v , w , x ) . Artzy proves these propositions about quadrilateral shapes: A polygon ( z 1 , z 2 , . . . z n ) {\displaystyle (z_{1},z_{2},...z_{n})} has

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1826-476: A wide range of shape representations. Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons . Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability , compactness and spikiness . When comparing shape similarity, however, at least 22 independent dimensions are needed to account for

1909-538: Is a symbol in the sramana religion of Budhha Dhamma. Wheel symbolism was also used in Indian temples in places that underwent 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

1992-466: 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 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

2075-414: 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

2158-403: Is by homeomorphisms . Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of

2241-417: 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 , was connected to the divine for most medieval scholars , and many believed that there

2324-431: Is constrained to lie on a plane , in contrast to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure ) may lie on a more general curved surface (a two-dimensional space ). Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc. Each of these

2407-611: Is divided into smaller categories; triangles can be equilateral , isosceles , obtuse , acute , scalene , etc. while quadrilaterals can be rectangles , rhombi , trapezoids , squares , etc. Other common shapes are points , lines , planes , and conic sections such as ellipses , circles , and parabolas . Among the most common 3-dimensional shapes are polyhedra , which are shapes with flat faces; ellipsoids , which are egg-shaped or sphere-shaped objects; cylinders ; and cones . If an object falls into one of these categories exactly or even approximately, we can use it to describe

2490-611: Is its rim (or felly) because it terminates it. The remaining ten links [of Dependent Origination] are its spokes [i.e. saṅkhāra up to 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

2573-587: Is present in the modern flag of India , representing the pan-Indian concept of Dharma . The modern State Emblem of India 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 ,

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2656-480: Is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size. A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. One way of modeling non-rigid movements

2739-450: 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, 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

2822-735: 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:

2905-477: Is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size. Objects that have the same shape or mirror image shapes are called geometrically similar , whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity

2988-406: Is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.’ Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on

3071-1307: 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

3154-522: 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 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

3237-667: 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 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

3320-655: 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 the Nebra sky disc and jade discs called Bi . The Egyptian Rhind papyrus , dated to 1700 BCE, gives

3403-555: 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,

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3486-493: 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

3569-1596: The complex plane , z ↦ a z + b , a ≠ 0 , {\displaystyle z\mapsto az+b,\quad a\neq 0,}   a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry . The shape p = S( u , v , w ) depends on the order of the arguments of function S, but permutations lead to related values. For instance, 1 − p = 1 − u − w u − v = w − v u − v = v − w v − u = S ( v , u , w ) . {\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).} Also p − 1 = S ( u , w , v ) . {\displaystyle p^{-1}=S(u,w,v).} Combining these permutations gives S ( v , w , u ) = ( 1 − p ) − 1 . {\displaystyle S(v,w,u)=(1-p)^{-1}.} Furthermore, p ( 1 − p ) − 1 = S ( u , v , w ) S ( v , w , u ) = u − w v − w = S ( w , v , u ) . {\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).} These relations are "conversion rules" for shape of

3652-415: The ellipse . Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons . Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and

3735-620: The shape of triangle ( u , v , w ) . Then the shape of the equilateral triangle is 0 − 1 + i 3 2 0 − 1 = 1 + i 3 2 = cos ⁡ ( 60 ∘ ) + i sin ⁡ ( 60 ∘ ) = e i π / 3 . {\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.} For any affine transformation of

3818-404: The sphere . A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape. There are multiple ways to compare the shapes of two objects: Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters " b " and " d " are

3901-401: 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

3984-549: 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 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

4067-427: 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

4150-424: 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 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

4233-461: 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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4316-1774: 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

4399-478: 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 the equation of the circle , follows from the Pythagorean theorem applied to any point on

4482-520: 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 ) ±

4565-595: 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

4648-413: 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. Shape A plane shape or plane figure

4731-618: The dharmachakra was often used as a decoration in East Asian statues and inscriptions , beginning with the earliest period of East Asian culture to 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

4814-470: 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 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,

4897-495: The different number of spokes may represent different aspects of the Buddha's Dharma (teaching). In 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

4980-514: 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

5063-461: 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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5146-406: 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

5229-422: 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 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 ,

5312-476: 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 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

5395-474: The many realms of rebirth in Buddhist cosmology . The spokes of a wheel are also often used as symbols of 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)

5478-467: 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 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

5561-454: The naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations , rotations (together also called rigid transformations ), and uniform scalings . In other words, the shape of

5644-448: The object is scaled non-uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object. Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent . An object

5727-407: The physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry , or as fractals . Some common shapes include: Circle , Square , Triangle , Rectangle , Oval , Star (polygon) , Rhombus , Semicircle . Regular polygons starting at pentagon follow

5810-407: 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

5893-406: 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

5976-422: The same shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic geometric objects such as a line , a curve , a plane , a plane figure (e.g. square or circle ), or a solid figure (e.g. cube or sphere ). However, most shapes occurring in

6059-482: The same shape. These shapes can be classified using complex numbers u , v , w for the vertices, in a method advanced by J.A. Lester and Rafael Artzy . For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i√3)/2 representing its vertices. Lester and Artzy call the ratio S ( u , v , w ) = u − w u − v {\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}

6142-454: The shape of the object. Thus, we say that the shape of a manhole cover is a disk , because it is approximately the same geometric object as an actual geometric disk. A geometric shape consists of the geometric information which remains when location , scale , orientation and reflection are removed from the description of a geometric object . That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in

6225-403: The size and placement in space of the object. For instance, a " d " and a " p " have the same shape, as they can be perfectly superimposed if the " d " is translated to the right by a given distance, rotated upside down and magnified by a given factor (see Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a " b " and

6308-498: 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 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

6391-709: 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 – 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

6474-483: The universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, 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

6557-457: 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 −

6640-579: The way natural shapes vary. There is also clear evidence that shapes guide human attention . Dharmachakra The dharmachakra ( Sanskrit : धर्मचक्र, Pali : dhammacakka ) or wheel of dharma 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,

6723-527: 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 ", 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 ,

6806-480: 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 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

6889-496: 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 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

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