Circle

This article is about the shape and mathematical concept of circle. For list of topics related to circles, see List of circle topics. This list of circle topics includes things related to the geometric shape either abstractly as in idealizations studied by geometers or concretely in physical space For all other uses, see Circle (disambiguation).

Circle illustration showing a radius, a diameter, the center and the circumference.

Tycho crater, one of many examples of circles that arise in nature. Tycho is a prominent lunar Impact crater located in the southern lunar highlands named after the Danish Astronomer Tycho Brahe. NASA photo. The National Aeronautics and Space Administration ( NASA, ˈnæsə is an agency of the United States government, responsible for the nation's public space program

Circles are simple shapes of Euclidean geometry. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. A circle consists of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume Distance is a numerical description of how far apart objects are Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers A circle with center A is sometimes denoted by the symbol A.

A chord of a circle is a line segment whose both endpoints lie on the circle. A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve A diameter is a chord passing through the center. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the The length of a diameter is twice the radius. A diameter is the largest chord in a circle.

Circles are simple closed curves which divide the plane into an interior and an exterior. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. An arc is any connected part of a circle. In Geometry, an arc is a closed segment of a Differentiable Curve in the two-dimensional plane; for example a circular In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of

A circle is a special ellipse in which the two foci are coincident. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed

A circle of infinite radius is considered to be a straight line.

1 Analytic results
2 Properties
3 Apollonius circle
- 3.1 Cross-ratios
- 3.2 Generalized circles
4 References
5 See also
- 5.1 External links

Analytic results

Circle of radius r=1, center (a, b)=(1. 2, -0. 5).

Chord, secant, tangent, and diameter.

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

$\left( x - a \right)^2 + \left( y - b \right)^2=r^2.$

The equation of the circle follows from the Pythagorean theorem applied to any point on the circle. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry If the circle is centred at the origin (0, 0), then this formula can be simplified to

$x^2 + y^2 = r^2. \!\$

When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as

$x = a+r\,\cos t,\,\!$

$y = b+r\,\sin t\,\!$

where t is a parametric variable, understood as many the angle the ray to (x, y) makes with the x-axis. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information In Mathematics, parametric equations are a method of defining a curve In Mathematics, parametric equations are a method of defining a curve Alternatively, in stereographic coordinates, the circle has a parametrization

$x = a + r \frac{2t}{1+t^2}$

$y = b + r \frac{1-t^2}{1+t^2}$

In homogeneous coordinates each conic section with equation of a circle is

$\ ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.$

It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface These points are called the circular points at infinity. In Projective geometry, the circular points at infinity in the Complex projective plane (also called cyclic points or isotropic points) are

In polar coordinates the equation of a circle is

$r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2.\,$

In the complex plane, a circle with a center at c and radius (r) has the equation $| z - c | 2 = r 2$ . In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$ , the slightly generalised equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalised circle. A Circle &Gamma is the set of points p that lie at Radius r from a center point &gamma. Not all generalised circles are actually circles: a generalized circle is either a (true) circle or a line.

Tangent lines

The tangent line through a point P on a circle is perpendicular to the diameter passing through P. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. The equation of the tangent line to a circle of radius r centered at the origin at the point (x₁, y₁) is

$xx_1+yy_1=r^2 \!\$

Hence, the slope of a circle at (x₁, y₁) is given by:

$\frac{dy}{dx} = - \frac{x_1}{y_1}.$

More generally, the slope at a point (x, y) on the circle $(x - a) 2 + (y - b) 2 = r 2$ , i. Slope is used to describe the steepness incline gradient or grade of a straight line. Slope is used to describe the steepness incline gradient or grade of a straight line. e. , the circle centered at (a, b) with radius r units, is given by

$\frac{dy}{dx} = \frac{a-x}{y-b},$

provided that $y \neq b$ .

Pi ( $π$ )

For more details on this topic, see Pi. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems

Pi or π is the ratio of a circle's circumference to its diameter. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the

The numeric value of $π$ never changes.

In modern English, it is pronounced /ˈpaɪ/ (as in apple pie).

Area enclosed

Area of the circle = π × area of the shaded square

Main article: Area of a disk

The area enclosed by a circle is the radius squared, multiplied by $π$ . The area of a disk (the region inside a Circle) is &pi r 2 when the circle has Radius r. The area of a disk (the region inside a Circle) is &pi r 2 when the circle has Radius r.

$Area = r^2 \cdot \pi$

Using a square with side lengths equal to the diameter of the circle, then dividing the square into four squares with side lengths equal to the radius of the circle, take the area of the smaller square and multiply by $π$ .

$A = \frac{d^2\cdot\pi}{4} \approx 0{.}7854 \cdot d^2,$ that is, approximately 79% of the circumscribing square. The circle is the plane curve enclosing the maximum area for a given arclength. This relates the circle to a problem in the calculus of variations. Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions.

Properties

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetry)
The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the center for every angle. The isoperimetric inequality is a geometric Inequality involving the square of the Circumference of a Closed curve in the plane and the Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. Its symmetry group is the orthogonal group O(2,R). The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n The group of rotations alone is the circle group T. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex
All circles are similar.
- A circle's circumference and radius are proportional,
- The area enclosed and the square of its radius are proportional. This article is about proportionality the mathematical relation This article is about proportionality the mathematical relation
- The constants of proportionality are 2π and π, respectively. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems
The circle centered at the origin with radius 1 is called the unit circle. In Mathematics, a unit circle is
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 center of the circle and the radius in terms of the coordinates of the three given points. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane See circumcircle. In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon

Chord properties

Chords equidistant from the center of a circle are equal (length).
Equal (length) chords are equidistant from the center.
The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
- A perpendicular line from the center of a circle bisects the chord.
- The line segment (Circular segment) through the center bisecting a chord is perpendicular to the chord. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points In Geometry, a circular segment (also circle segment) is an area of a Circle informally defined as an area which is "cut off" from the rest of
If a central angle and an inscribed 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. In Geometry, an inscribed angle is formed when two Secant lines of a Circle (or in a Degenerate case, when one Secant line and
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 are supplemental.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
An inscribed angle subtended by a diameter is a right angle.
The diameter is longest chord of the circle.

Sagitta properties

The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference 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 which will fit around the two lines:

$r=\frac{y^2}{8x}+ \frac{x}{2}.$

Another proof of this result which relies only on 2 chord properties given above is as follows. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is 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 the first chord, we find that (2*r-x)(x)=(y/2)^2. Solving for r, we find:

$r=\frac{y^2}{8x}+ \frac{x}{2}.$

as required.

Tangent properties

The line drawn perpendicular to the end point of a radius is a tangent to the circle.
A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
Tangents drawn from a point outside the circle are equal in length.
Two tangents can always be drawn from a point outside of the circle.

Theorems

Secant-secant theorem

See also: Power of a point

The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA. In Geometry, the power of a point is a Real number h that reflects the relative distance of a given point from a given circle (Chord theorem)
If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC² = DG×DE. (tangent-secant theorem)
If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG=DF×DE. Secant is a term in mathematics It comes from the Latin secare (to cut (Corollary of the tangent-secant theorem)
The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
If the angle subtended by the chord at the center is 90 degrees then l = √2 × r, where l is the length of the chord and r is the radius of the circle. This article describes the unit of angle For other meanings see Degree.
If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle $ψ$ is exactly half of the corresponding central angle $θ$ (see Figure). In Geometry, an inscribed angle is formed when two Secant lines of a Circle (or in a Degenerate case, when one Secant line and central angle is an Angle whose Line is the center of a Circle, and whose sides pass through a pair of points on the circle thereby Subtending Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles $ψ$ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle

Apollonius circle

$\frac{d_1}{d_2}=\textrm{constant}$ Apollonius' definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points in plane having a constant ratio of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points^[1].

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

$\frac{AP}{BP} = \frac{AC}{BC}.$

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ}$ , the angle CPD is exactly $90^{\circ}$ , i. e. , a right angle. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

Cross-ratios

Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Astrology and astronomy are historically one and the same discipline ( Latin: astrologia) and were only gradually recognized as separate in western In the Middle Ages, Science progressed dramatically from the time of antiquity in areas as diverse as Astronomy, Medicine, and Mathematics Notice, even, the circular shape of the halo. HaLo ( Ayako Hirakata) is a Japanese J-Pop Musician. Hirakata can be heard on Lori Carson 's The Finest Thing. The compass in this 13th century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3} In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis If A, B, and C are as above, then the Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

| [A, B; C, P] | = 1.

Stated another way, P is a point on the Apollonius circle if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane. In Mathematics, a unit circle is

Generalized circles

See also: Generalized circle

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

$\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}$ (1)

is not a circle, but rather a line. A Circle &Gamma is the set of points p that lie at Radius r from a center point &gamma. The midpoint (also known as class mark in relation to Histogram) is the middle point of a Line segment.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying (1) is called a generalized circle. It may either be a true circle or a line.

References

^ Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.

^ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co. , p. 30.

Dictionary

-noun

(geometry): A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.
A two-dimensional geometric figure, a disk, consisting of the set of all those points of a plane at a distance less than or equal to a fixed distance from another point.
Any thin three-dimensional equivalent of the geometric figures.
A curve that more or less forms part or all of a circle.
Orbit.
A specific group of persons.
(cricket) A line comprising two semicircles of 30 yds radius centred on the wickets joined by straight lines parallel to the pitch used to enforce field restrictions in a one-day match.

-verb

(transitive) To travel around along a curved path.
(transitive) To surround.
(transitive) To place or mark a circle around.
(intransitive) To travel in circles.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents