A circle is a
shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie on ...
consisting of all
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
in a
plane that are at a given distance from a given point, the
centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is
constant. The distance between any point of the circle and the centre is called the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
. Usually, the radius is required to be a positive number. A circle with
(a single point) is a
degenerate case. This article is about circles in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, and, in particular, the
Euclidean plane, except where otherwise noted.
Specifically, a circle is a
simple closed
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
that divides the plane into two
regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a ''
disc''.
A circle may also be defined as a special kind of
ellipse in which the two
foci are coincident, the
eccentricity is 0, and the
semi-major and semi-minor axes are equal; or the two-dimensional shape enclosing the most area per unit perimeter squared, using
calculus of variations.
Euclid's definition
Topological definition
In the field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, 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
R3 upon itself (known as an
ambient isotopy).
Terminology
*
Annulus: a ring-shaped object, the region bounded by two
concentric
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center ...
circles.
*
Arc: any
connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle.
* Centre: the point equidistant from all points on the circle.
*
Chord
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ( ...
: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
*
Circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
: the length of one circuit along the circle, or the distance around the circle.
*
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
: 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.
*
Lens: the region common to (the intersection of) two overlapping discs.
* Passant: a
coplanar straight line that has no point in common with the circle.
* 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.
*
Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center 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 term ''segment'' is used only for regions not containing the center of the circle to which their arc belongs to.
*
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 center. 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 as ''open'', that is, not containing their boundaries, or as ''closed'', including their respective boundaries.
History
The word ''circle'' derives from the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
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
Circuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circu ...
'' are closely related.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the
wheel
A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
, which, with related inventions such as
gear
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic p ...
s, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.
Early
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
, particularly
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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.
Some highlights in the history of the circle are:
* 1700 BCE – The
Rhind papyrus gives a method to find the area of a circular field. The result corresponds to (3.16049...) as an approximate value of
.
* 300 BCE – Book 3 of
Euclid's ''Elements'' deals with the properties of circles.
* In
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
'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.
* 1880 CE –
Lindemann Lindemann is a German surname.
Persons
Notable people with the surname include:
Arts and entertainment
* Elisabeth Lindemann, German textile designer and weaver
* Jens Lindemann, trumpet player
* Julie Lindemann, American photographer
* Maggie ...
proves that is
transcendental
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
, effectively settling the millennia-old problem of squaring the circle.
Analytic results
Circumference
The ratio of a circle's circumference to its diameter is (pi), an
irrational constant approximately equal to 3.141592654. Thus the circumference ''C'' is related to the radius ''r'' and diameter ''d'' by:
:
Area enclosed
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 the circle's radius, which comes to multiplied by the radius squared:
:
Equivalently, denoting diameter by ''d'',
:
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 the calculus of variations, namely the
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
.
Equations
Cartesian coordinates
;Equation of a circle
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
:
This
equation, known as the ''equation of the circle'', follows from the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
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
:
;Parametric form
The equation can be written in
parametric form using the
trigonometric functions sine and cosine as
:
:
where ''t'' is a
parametric variable in the range 0 to 2, interpreted geometrically as the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
that the ray from (''a'', ''b'') to (''x'', ''y'') makes with the positive ''x'' axis.
An alternative parametrisation of the circle is
:
:
In this parameterisation, the ratio of ''t'' to ''r'' can be interpreted geometrically as the
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
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.
;3-point form
The equation of the circle determined by three points
not on a line is obtained by a conversion of the
''3-point form of a circle equation'':
:
;Homogeneous form
In
homogeneous coordinates, each
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
with the equation of a circle has the form
:
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 the
circular points at infinity.
Polar coordinates
In
polar coordinates, the equation of a circle is
:
where ''a'' 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., ''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 the circle). For a circle centred on the origin, i.e. , this reduces to . When , or when the origin lies on the circle, the equation becomes
:
In the general case, the equation can be solved for ''r'', giving
:
Note that without the ± sign, the equation would in some cases describe only half a circle.
Complex plane
In the
complex plane, a circle with a centre at ''c'' and radius ''r'' has the equation
:
In parametric form, this can be written as
:
The slightly generalised equation
:
for real ''p'', ''q'' and complex ''g'' is sometimes called a
generalised 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 a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
Tangent lines
The
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
through a point ''P'' on the circle is perpendicular to the diameter passing through ''P''. If 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 . Evaluating at (''x''
1, ''y''
1) determines the value of ''c'', and the result is that the equation of the tangent is
:
or
:
If , then the slope of this line is
:
This can also be found using
implicit differentiation.
When the centre of the circle is at the origin, then the equation of the tangent line becomes
:
and its slope is
:
Properties
* The circle is the shape with the largest area for a given length of perimeter (see
Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
).
* 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 centre for every angle. Its
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
is the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(2,''R''). The group of rotations alone is the
circle group T.
* All circles are
similar.
** A circle circumference and radius are
proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant
* Ratio, of one quantity to another, especially of a part compare ...
.
** The
area enclosed and the square of its radius are proportional.
** The constants of proportionality are 2 and respectively.
* The circle that is centred at the origin with radius 1 is called the
unit circle.
** Thought of as a
great circle of the
unit sphere, it becomes the
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or th ...
.
* 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. See
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every pol ...
.
Chord
* Chords are equidistant from the centre of a circle if and only if they are equal in length.
* The
perpendicular 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.
** The
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
through the centre bisecting a chord is
perpendicular to the chord.
* If a central angle and an
inscribed angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an i ...
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 are
supplementary
The term supplementary can refer to:
* Supplementary angles
* Supplementary Benefit, a former benefit payable in the United Kingdom
* Supplementary question, a type of question asked during a questioning time for prime minister
See also
* Sup ...
.
** 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 (see
Thales' 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 the
intersection of any two chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then .
* If the intersection of any two perpendicular chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then equals the square of the diameter.
* 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 8''r''
2 − 4''p''
2, where ''r'' is the circle radius, and ''p'' is the distance from the centre point to the point of intersection.
* 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.
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 at ''A'' and a tangent at ''B'' intersect at the exterior point ''P'', then denoting the centre as ''O'', the angles ∠''BOA'' and ∠''BPA'' are supplementary.
* If ''AD'' is tangent to the circle at ''A'' and if ''AQ'' is a chord of the circle, then .
Theorems
* The chord theorem states that if two chords, ''CD'' and ''EB'', intersect at ''A'', then .
* If two secants, ''AE'' and ''AD'', also cut the circle at ''B'' and ''C'' respectively, then (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 point ''A'' meets the circle at ''F'' and a secant from the external point ''A'' meets the circle at ''C'' and ''D'' respectively, then (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
°, then , where ''ℓ'' is the length of the chord, and ''r'' is the radius of the circle.
* 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 (
and
). That is,
, where ''O'' is the centre of the circle (secant–secant theorem).
Inscribed angles
An inscribed angle (examples are 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
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
(since the central angle is 180°).
Sagitta
The
sagitta (also known as the
versine) is a line segment drawn perpendicular to 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:
:
Another proof of this result, which relies only on two chord properties given above, 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 () 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 (. Solving for ''r'', we find the required result.
Compass and straightedge constructions
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
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
on the centre point, the movable leg on the point on the circle and rotate the compass.
Construction with given diameter
* Construct the
midpoint of the diameter.
* Construct the circle with centre passing through one of the endpoints of the diameter (it will also pass through the other endpoint).
Construction through three noncollinear points
* Name the points , and ,
* Construct the
perpendicular bisector of the segment .
* Construct the
perpendicular bisector of the segment .
* Label the point of intersection of these two perpendicular bisectors . (They meet because the points are not
collinear).
* Construct the circle with centre passing through one of the points , or (it will also pass through the other two points).
Circle of Apollonius
Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribut ...
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.
The proof is in two parts. First, one must prove that, given two foci ''A'' and ''B'' and a ratio of distances, any point ''P'' satisfying the ratio of distances must fall on a particular circle. Let ''C'' be another point, also satisfying the ratio and lying on segment ''AB''. By the
angle bisector theorem the line segment ''PC'' will bisect the
interior angle ''APB'', since the segments are similar:
:
Analogously, a line segment ''PD'' through some point ''D'' on ''AB'' extended bisects the corresponding exterior angle ''BPQ'' where ''Q'' is on ''AP'' extended. Since the interior and exterior angles sum to 180 degrees, the angle ''CPD'' is exactly 90 degrees; that is, a right angle. The set of points ''P'' such that angle ''CPD'' is a right angle forms a circle, of which ''CD'' is a diameter.
Second, see 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 the
cross-ratio of points in the complex plane. If ''A'', ''B'', and ''C'' are as above, then the circle of Apollonius for these three points is the collection of points ''P'' 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 is on the unit circle in the complex plane.
Generalised circles
If ''C'' is the midpoint of the segment ''AB'', then the collection of points ''P'' satisfying the Apollonius condition
:
is not a circle, but rather a line.
Thus, if ''A'', ''B'', and ''C'' are given distinct points in the plane, then the
locus of points ''P'' 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 every
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
a unique circle, called the
incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.
About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three
vertices.
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. 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.
A
hypocycloid 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 a
limiting case of each of various other figures:
* A
Cartesian oval is a set of points such that a
weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse 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.
* A
superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.
In the ...
has an equation of the form
for positive ''a'', ''b'', and ''n''. A supercircle has . A circle is the special case of a supercircle in which .
* A
Cassini 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.
* A
curve 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.
In other ''p''-norms
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
''p''-norm, distance is determined by
:
In Euclidean geometry, ''p'' = 2, giving the familiar
:
In
taxicab geometry, ''p'' = 1. Taxicab circles are
squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length
using a
Euclidean metric, where ''r'' is the circle's radius, its length in taxicab geometry is 2''r''. Thus, a circle's circumference is 8''r''. 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 and
:
in polar coordinates.
A circle of radius 1 (using this distance) is the
von Neumann neighborhood of its center.
A circle of radius ''r'' for the
Chebyshev distance (
L∞ metric) on a plane is also a square with side length 2''r'' 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 between L
1 and L
∞ metrics does not generalize to higher dimensions.
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 center is at the centroid of the given points.
A generalization for higher powers of distances is obtained if under
points the vertices of the regular polygon
are taken.
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, if
:
, where
=1,2,…,
-1;
whose center is the centroid of the
.
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.
Squaring the circle
Squaring the circle is the problem, proposed by
ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with
compass and straightedge.
In 1882, the task was proven to be impossible, as a consequence of the
Lindemann–Weierstrass theorem, which proves that pi () is a
transcendental number, rather than an
algebraic irrational number; that is, it is not the
root of any
polynomial with
rational coefficients. Despite the impossibility, this topic continues to be of interest for
pseudomath enthusiasts.
Significance in art and symbolism
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, the
Dharma wheel, a rainbow, mandalas, rose windows and so forth.
See also
*
Affine sphere
*
Apeirogon
*
Circle fitting
*
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem i ...
*
Inversion in a circle
*
Line–circle intersection
*
List of circle topics
*
Sphere
*
Three points determine a circle
In algebraic geometry, Cramer's theorem on algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-degenerate cases. This number is
:\frac 2,
...
*
Translation of axes
Specially named circles
*
Apollonian circles
In geometry, Apollonian circles are two families ( pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. T ...
*
Archimedean circle
*
Archimedes' twin circles
*
Bankoff circle
*
Carlyle circle
*
Chromatic circle
*
Circle of antisimilitude
*
Ford circle
*
Geodesic circle
*
Johnson circles
*
Schoch circles
*
Woo circles
Of a triangle
*
Apollonius circle of the excircles
*
Brocard circle
*
Excircle
*
Incircle
*
Lemoine circle
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the cor ...
*
Lester circle
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
The result is named after June Lester, who published it in 1997, and t ...
*
Malfatti circles
*
Mandart circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
*
Nine-point circle
*
Orthocentroidal circle
*
Parry circle
*
Polar circle (geometry)
*
Spieker circle
*
Van Lamoen circle
Of certain quadrilaterals
*
Eight-point circle of an orthodiagonal quadrilateral
Of a conic section
*
Director circle
*
Directrix circle
Of a torus
*
Villarceau circles
References
Further reading
*
"Circle" in The MacTutor History of Mathematics archive
External links
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{{Authority control
Elementary shapes
Conic sections
Pi