Acircle is ashape consisting of allpoints in aplane that are at a given distance from a given point, thecentre. The distance between any point of the circle and the centre is called theradius. The length of a line segment connecting two points on the circle and passing through the centre is called thediameter. A circle bounds a region of the plane called adisc.
The circle has been known since before the beginning of recorded history. Natural circles are common, such as thefull moon or a slice of round fruit. The circle is the basis for thewheel, which, with related inventions such asgears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry,astronomy andcalculus.
Terminology
Annulus: a ring-shaped object, the region bounded by twoconcentric circles.
Arc: anyconnected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
Centre: the point equidistant from all points on the circle.
Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
Circumference: thelength of one circuit along the circle, or the distance around the circle.
Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
Disc: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc (or disk), while in everyday use the term "circle" may also refer to a disc.
Lens: the region common to (the intersection of) two overlapping discs.
Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted and required to be a positive number. A circle with is adegenerate case consisting of a single point.
Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the termsegment is used only for regions not containing the centre of the circle to which their arc belongs.
Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").
All of the specified regions may be considered asopen, that is, not containing their boundaries, or asclosed, including their respective boundaries.
Chord, secant, tangent, radius, and diameter
Arc, sector, and segment
Etymology
The wordcircle derives from theGreek κίρκος/κύκλος (kirkos/kuklos), itself ametathesis of theHomeric Greek κρίκος (krikos), meaning "hoop" or "ring".[1] The origins of the wordscircus andcircuit are closely related.
The EgyptianRhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to256/81 (3.16049...) as an approximate value ofπ.[3]
Book 3 ofEuclid'sElements 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 is called its circumference and the point, its centre.
InPlato'sSeventh 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. Earlyscience, particularlygeometry andastrology and astronomy, was connected to the divine for mostmedieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[5][6]
With the advent ofabstract 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.[8][9]
Symbolism and religious use
Thecompass in this 13th-century manuscript is a symbol of God's act ofCreation. Notice also the circular shape of thehalo.
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 – 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, 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, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, theDharma wheel, a rainbow, mandalas, rose windows and so forth.[10]Magic circles are part of some traditions ofWestern esotericism.
The ratio of a circle's circumference to its diameter isπ (pi), anirrationalconstant approximately equal to 3.141592654. The ratio of a circle's circumference to its radius is2π.[a] Thus the circumferenceC is related to the radiusr and diameterd by:
Area enclosed
Area enclosed by a circle =π × area of the shaded square
As proved byArchimedes, in hisMeasurement of a Circle, thearea 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,[11] which comes toπ multiplied by the radius squared:
Equivalently, denoting diameter byd,that is, approximately 79% of thecircumscribing square (whose side is of lengthd).
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 theisoperimetric inequality.
If a circle of radiusr is centred at thevertex of anangle, and that angle intercepts anarc of the circle with anarc length ofs, then theradian measure 𝜃 of the angle is the ratio of the arc length to the radius:
The circular arc is said tosubtend the angle, known as thecentral angle, at the centre of the circle. One radian is the measure of the central angle subtended by a circular arc whose length is equal to its radius. The angle subtended by a complete circle at its centre is acomplete angle, which measures2π radians, 360degrees, or oneturn.
Using radians, the formula for the arc lengths of a circular arc of radiusr and subtending a central angle of measure 𝜃 is
and the formula for the areaA of acircular sector of radiusr and with central angle of measure 𝜃 is
In the special case𝜃 = 2π, these formulae yield the circumference of a complete circle and area of a complete disc, respectively.
Equations
Cartesian coordinates
Circle of radiusr = 1, centre (a, b) = (1.2, −0.5)
Thisequation, known as theequation of the circle, follows from thePythagorean 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
One coordinate as a function of the other
Upper semicircle with radius1 and center(0, 0) and its derivative.
The circle of radius with center at in the– plane can be broken into two semicircles each of which is thegraph of a function, and, respectively:for values of ranging from to.
Parametric form
The equation can be written inparametric form using thetrigonometric functions sine and cosine aswheret is aparametric variable in the range 0 to 2π, interpreted geometrically as theangle that the ray from (a, b) to (x, y) makes with the positivex axis.
An alternative parametrisation of the circle is
In this parameterisation, the ratio oft tor can be interpreted geometrically as thestereographic projection of the line passing through the centre parallel to thex axis (seeTangent half-angle substitution). However, this parameterisation works only ift 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.
3-point form
The equation of the circle determined by three points not on a line is obtained by a conversion of the3-point form of a circle equation:
It can be proven that a conic section is a circle exactly when it contains (when extended to thecomplex projective plane) the pointsI(1:i: 0) andJ(1: −i: 0). These points are called thecircular points at infinity.
wherea is the radius of the circle, are the polar coordinates of a generic point on the circle, and are the polar coordinates of the centre of the circle (i.e.,r0 is the distance from the origin to the centre of the circle, andφ is the anticlockwise angle from the positivex axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e.r0 = 0, this reduces tor =a. Whenr0 =a, or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved forr, givingWithout the ± sign, the equation would in some cases describe only half a circle.
Complex plane
In thecomplex plane, a circle with a centre atc and radiusr has the equation
In parametric form, this can be written as
The slightly generalised equation
for realp,q and complexg is sometimes called ageneralised circle. This becomes the above equation for a circle with, since. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or aline.
Thetangent line through a pointP on the circle is perpendicular to the diameter passing throughP. IfP = (x1,y1) and the circle has centre (a,b) and radiusr, then the tangent line is perpendicular to the line from (a,b) to (x1,y1), so it has the form(x1 −a)x + (y1 –b)y =c. Evaluating at (x1,y1) determines the value ofc, and the result is that the equation of the tangent isor
Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. Seecircumcircle.
Chord
Chords are equidistant from the centre of a circle if and only if they are equal in length.
Theperpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
A perpendicular line from the centre of a circle bisects the chord.
If a central angle and aninscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
If two angles are inscribed on the same chord and on opposite sides of the chord, then they aresupplementary.
An inscribed angle subtended by a diameter is a right angle (seeThales' theorem).
The diameter is the longest chord of the circle.
Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
If theintersection of any two chords divides one chord into lengthsa andb and divides the other chord into lengthsc andd, thenab =cd.
If the intersection of any two perpendicular chords divides one chord into lengthsa andb and divides the other chord into lengthsc andd, thena2 +b2 +c2 +d2 equals the square of the diameter.[13]
The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8r2 − 4p2, wherer is the circle radius, andp is the distance from the centre point to the point of intersection.[14]
The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.[15]: p.71
Tangent
A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
If a tangent atA and a tangent atB intersect at the exterior pointP, then denoting the centre asO, the angles ∠BOA and ∠BPA are supplementary.
IfAD is tangent to the circle atA and ifAQ is a chord of the circle, then∠DAQ =1/2arc(AQ).
The chord theorem states that if two chords,CD andEB, intersect atA, thenAC ×AD =AB ×AE.
If two secants,AE andAD, also cut the circle atB andC respectively, thenAC ×AD =AB ×AE (corollary of the chord theorem).
A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external pointA meets the circle atF and a secant from the external pointA meets the circle atC andD respectively, thenAF2 =AC ×AD (tangent–secant theorem).
The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).
If the angle subtended by the chord at the centre is 90°, thenℓ =r √2, whereℓ is the length of the chord, andr is the radius of the circle.
If two secants are inscribed in the circle as shown at right, then the measurement of angleA is equal to one half the difference of the measurements of the enclosed arcs ( and). That is,, whereO is the centre of the circle (secant–secant theorem).
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the correspondingcentral 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 aright angle (since the central angle is 180°).
Sagitta
The sagitta is the vertical segment.
Thesagitta (also known as theversine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
Given the lengthy of a chord and the lengthx of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines:
Another proof of this result, which relies only on two chord properties given above, is as follows. Given a chord of lengthy and with sagitta of lengthx, 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 (2r −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 the first chord, we find that (2r −x)x = (y / 2)2. Solving forr, we find the required result.
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 thecompass on the centre point, the movable leg on the point on the circle and rotate the compass.
Construct the circle with centreM passing through one of the endpoints of the diameter (it will also pass through the other endpoint).
Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre.
Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constantratio (other than 1) of distances to two fixed foci,A andB.[16][17] (The set of points where the distances are equal is the perpendicular bisector of segmentAB, a line.) That circle is sometimes said to be drawnabout two points.
The proof is in two parts. First, one must prove that, given two fociA andB and a ratio of distances, any pointP satisfying the ratio of distances must fall on a particular circle. LetC be another point, also satisfying the ratio and lying on segmentAB. By theangle bisector theorem the line segmentPC will bisect theinterior angleAPB, since the segments are similar:
Analogously, a line segmentPD through some pointD onAB extended bisects the corresponding exterior angleBPQ whereQ is onAP extended. Since the interior and exterior angles sum to 180 degrees, the angleCPD is exactly 90 degrees; that is, a right angle. The set of pointsP such that angleCPD is a right angle forms a circle, of whichCD is a diameter.
Second, see[18]: 15 for a proof that every point on the indicated circle satisfies the given ratio.
Cross-ratios
A closely related property of circles involves the geometry of thecross-ratio of points in the complex plane. IfA,B, andC are as above, then the circle of Apollonius for these three points is the collection of pointsP for which the absolute value of the cross-ratio is equal to one:
Stated another way,P is a point on the circle of Apollonius if and only if the cross-ratio[A,B;C,P] is on the unit circle in the complex plane.
IfC is the midpoint of the segmentAB, then the collection of pointsP satisfying the Apollonius conditionis not a circle, but rather a line.
Thus, ifA,B, andC are given distinct points in the plane, then thelocus of pointsP satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.
Inscription in or circumscription about other figures
In everytriangle a unique circle, called theincircle, can be inscribed such that it is tangent to each of the three sides of the triangle.[19]
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's threevertices.[20]
Acyclic polygon is any convex polygon about which acircle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called abicentric polygon.
Ahypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.
Limiting case of other figures
The circle can be viewed as alimiting case of various other figures:
ACartesian oval is a set of points such that aweighted sum of the distances from any of its points to two fixed points (foci) is a constant. Anellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
Asuperellipse has an equation of the form for positivea,b, andn. A supercircle hasb =a. A circle is the special case of a supercircle in whichn = 2.
ACassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
Acurve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.
Locus of constant sum
Consider a finite set of points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points.[22]A generalisation for higher powers of distances is obtained if, instead of points, the vertices of the regular polygon are taken.[23] The locus of points such that the sum of the-th power of distances to the vertices of a given regular polygon with circumradius is constant is a circle, ifwhose centre is the centroid of the.
In the case of theequilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For theregular pentagon the constant sum of the eighth powers of the distances will be added and so forth.
Squaring the circle is the problem, proposed byancientgeometers, of constructing a square with the same area as a given circle by using only a finite number of steps withcompass and straightedge.
Illustrations of unit circles (see alsosuperellipse) in differentp-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the correspondingp).
Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. Inp-norm, distance is determined byIn Euclidean geometry,p = 2, giving the familiar
Intaxicab geometry,p = 1. Taxicab circles aresquares with sides oriented at a 45° angle to the coordinate axes. While each side would have length using aEuclidean metric, wherer is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to is 4 in this geometry. The formula for the unit circle in taxicab geometry is in Cartesian coordinates andin polar coordinates.
A circle of radiusr for theChebyshev distance (L∞ metric) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence betweenL1 andL∞ metrics does not generalise to higher dimensions.
Intopology, a circle is not limited to the geometric concept, but to all of itshomeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation ofR3 upon itself (known as anambient isotopy).[24]
Apeirogon – Polygon with an infinite number of sides
Circle fitting – Process of constructing a curve that has the best fit to a series of data pointsPages displaying short descriptions of redirect targets
^Abdullahi, Yahya (29 October 2019). "The Circle from East to West". In Charnier, Jean-François (ed.).The Louvre Abu Dhabi: A World Vision of Art. Rizzoli International Publications, Incorporated.ISBN9782370741004.
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