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A ______ 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 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 the development of geometry, astronomy and calculus.
The word 2______ derives from the Greek / (3_____________), itself a metathesis of the Homeric Greek (4______), meaning "hoop" or "ring".[1] The origins of the words 5______ and 6_______ are closely related.
Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings.[2] Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi.
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.
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 was something intrinsically "divine" or "perfect" that could be found in circles.[5][6]
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, the Dharma wheel, a rainbow, mandalas, rose windows and so forth.[8] Magic circles are part of some traditions of Western esotericism.
In this parameterisation, the ratio of 7_ to 8_ can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the 9_ axis (see Tangent half-angle substitution). However, this parameterisation works only if 10_ 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.
where 11_ is the radius of the circle, ( r , ) {\displaystyle (r,\theta )} are the polar coordinates of a generic point on the circle, and ( r 0 , ) {\displaystyle (r_{0},\phi )} are the polar coordinates of the centre of the circle (i.e., 12_0 is the distance from the origin to the centre of the circle, and 13 is the anticlockwise angle from the positive 14_ axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. 15_0 = 0, this reduces to 16_ = 17_. When 18_0 = 19_, or when the origin lies on the circle, the equation becomes
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 the compass.
Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant 20_____ (other than 1) of distances to two fixed foci, 21_ and 22_.[13][14] (The set of points where the distances are equal is the perpendicular bisector of segment 23__, a line.) That circle is sometimes said to be drawn 24_____ two points.
The proof is in two parts. First, one must prove that, given two foci 25_ and 26_ and a ratio of distances, any point 27_ satisfying the ratio of distances must fall on a particular circle. Let 28_ be another point, also satisfying the ratio and lying on segment 29__. By the angle bisector theorem the line segment 30__ will bisect the interior angle 31___, since the segments are similar:
Analogously, a line segment 32__ through some point 33_ on 34__ extended bisects the corresponding exterior angle 35___ where 36_ is on 37__ extended. Since the interior and exterior angles sum to 180 degrees, the angle 38___ is exactly 90 degrees; that is, a right angle. The set of points 39_ such that angle 40___ is a right angle forms a circle, of which 41__ is a diameter.
A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If 42_, 43_, and 44_ are as above, then the circle of Apollonius for these three points is the collection of points 45_ for which the absolute value of the cross-ratio is equal to one:
Thus, if 46_, 47_, and 48_ are given distinct points in the plane, then the locus of points 49_ 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.
A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.[18] Every regular polygon and every triangle is a tangential polygon.
A cyclic polygon is any convex polygon about which a circle 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 a bicentric polygon.
Consider a finite set of n {\displaystyle n} 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.[19]A generalization for higher powers of distances is obtained if under n {\displaystyle n} points the vertices of the regular polygon P n {\displaystyle P_{n}} are taken.[20] The locus of points such that the sum of the 2 m {\displaystyle 2m} -th power of distances d i {\displaystyle d_{i}} to the vertices of a given regular polygon with circumradius R {\displaystyle R} is constant is a circle, if
In the case of the equilateral 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 the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.
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. In 50_-norm, distance is determined by
In topology, a circle is not limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if one can be transformed into the other via a deformation of 1_3 upon itself (known as an ambient isotopy).[21] 5376163bf9