166 relations: Affine sphere, Algebraic number, Angle, Angle bisector theorem, Annulus (mathematics), Apeirogon, Apollonian circles, Apollonius of Perga, Arc (geometry), Archimedean circle, Archimedes, Area, Area of a circle, Arthur Koestler, Astrology and astronomy, Astronomy, Bankoff circle, Bicentric polygon, Bisection, Brocard circle, C. Stanley Ogilvy, Calculus of variations, Carlyle circle, Cartesian coordinate system, Cartesian oval, Cassini oval, Central angle, Centre (geometry), Chord (geometry), Chromatic circle, Circle group, Circle of antisimilitude, Circular points at infinity, Circular sector, Circular segment, Circumference, Circumscribed circle, Circus, Classical antiquity, Collinearity, Compass (drawing tool), Compass-and-straightedge construction, Complex plane, Complex projective plane, Concentric objects, Conic section, Connected space, Constant (mathematics), Convex polygon, Coordinate system, ..., Coplanarity, Cramer's theorem (algebraic curves), Creation myth, Cross-ratio, Curve, Curve fitting, Curve of constant width, Cut-the-Knot, Cyclic quadrilateral, Degree (angle), Diameter, Director circle, Disk (mathematics), Eccentricity (mathematics), Ellipse, Equation, Euclid, Euclid's Elements, Euclidean geometry, Ferdinand von Lindemann, Focus (geometry), Ford circle, Gear, Generalised circle, Geometry, Great circle, Greek language, Halo (religious iconography), History of science, Homeric Greek, Homogeneous coordinates, Hypocycloid, Implicit function, Incircle and excircles of a triangle, Inscribed angle, Interior (topology), Internal and external angles, Intersecting chords theorem, Irrational number, Isoperimetric inequality, Johnson circles, Lens (geometry), Lester's theorem, Limiting case (mathematics), Lindemann–Weierstrass theorem, Line (geometry), Line segment, List of circle topics, List of geometers, Malfatti circles, Mathematical constant, Measurement of a Circle, Metathesis (linguistics), Midpoint, Nine-point circle, Orthocentroidal circle, Orthodiagonal quadrilateral, Orthogonal group, Parametric equation, Parry point (triangle), Perpendicular, Pi, Plane (geometry), Plato, Point (geometry), Polar circle (geometry), Polar coordinate system, Polynomial, Proclus, Proportionality (mathematics), Pythagorean theorem, Radius, Rational number, Reflection symmetry, Regular polygon, Rhind Mathematical Papyrus, Riemannian circle, Right angle, Rotational symmetry, Sagitta (geometry), Schoch circles, Science, Secant line, Semicircle, Seventh Letter, Shape, Similarity (geometry), Sphere, Spieker circle, Squaring the circle, Stereographic projection, Superellipse, Symmedian, Symmetry group, Tangent, Tangent half-angle substitution, Tangential polygon, Tangential quadrilateral, Thales's theorem, The College Mathematics Journal, The Sleepwalkers (Koestler book), Transcendental number, Translation of axes, Triangle, Trigonometric functions, Twin circles, Unit circle, Unit sphere, Van Lamoen circle, Versine, Vertex (geometry), Villarceau circles, Weight function, Wheel, Woo circles, Zero of a function. Expand index (116 more) » « Shrink index
In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point.
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
In plane 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.
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle.
In mathematics, an annulus (the Latin word for "little ring" is anulus/annulus, with plural anuli/annuli) is a ring-shaped object, a region bounded by two concentric circles.
In geometry, an apeirogon (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") is a generalized polygon with a countably infinite number of sides.
Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa.
Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.
In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve.
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles.
Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
In geometry, the area enclosed by a circle of radius is.
Arthur Koestler, (Kösztler Artúr; 5 September 1905 – 1 March 1983) was a Hungarian-British author and journalist.
Astrology and astronomy were archaically treated together (astrologia), and were only gradually separated in Western 17th century philosophy (the "Age of Reason") with the rejection of astrology.
Astronomy (from ἀστρονομία) is a natural science that studies celestial objects and phenomena.
In geometry, the Bankoff circle or Bankoff triplet circle is a certain Archimedean circle that can be constructed from an arbelos; an Archimedean circle is any circle with area equal to each of Archimedes' twin circles.
In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon.
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector.
In geometry, the Brocard circle (or seven-point circle) for a triangle is a circle defined from a given triangle.
Charles Stanley Ogilvy (1913–2000) was an American mathematician, sailor, and author.
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
In geometry, a Cartesian oval, named after René Descartes, is a plane curve, the set of points that have the same linear combination of distances from two fixed points.
A Cassini oval is a quartic plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant.
Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians).
In geometry, a centre (or center) (from Greek κέντρον) of an object is a point in some sense in the middle of the object.
A chord of a circle is a straight line segment whose endpoints both lie on the circle.
The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, α and β, is a reference circle for which α and β are inverses of each other.
In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.
A circular sector or circle sector (symbol: ⌔), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.
In geometry, a circular segment (symbol: ⌓) is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord.
In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.
A circus is a company of performers who put on diverse entertainment shows that include clowns, acrobats, trained animals, trapeze acts, musicians, dancers, hoopers, tightrope walkers, jugglers, magicians, unicyclists, as well as other object manipulation and stunt-oriented artists.
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th or 6th century AD centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world.
In geometry, collinearity of a set of points is the property of their lying on a single line.
A pair of compasses, also known simply as a bow compass, is a technical drawing instrument that can be used for inscribing circles or arcs.
Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space.
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis.
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
In mathematics, the adjective constant means non-varying.
A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all.
In mathematics, 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.
A creation myth (or cosmogonic myth) is a symbolic narrative of how the world began and how people first came to inhabit it.
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.
In geometry, a curve of constant width is a convex planar shape whose width (defined as the perpendicular distance between two distinct parallel lines each having at least one point in common with the shape's boundary but none with the shape's interior) is the same regardless of the orientation of the curve.
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.
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.
In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
In geometry, a disk (also spelled disc).
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section.
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
In mathematics, an equation is a statement of an equality containing one or more variables.
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that pi (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.
In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed.
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers.
A gear or cogwheel is a rotating machine part having cut like teeth, or cogs, which mesh with another toothed part to transmit torque.
A generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere.
Greek (Modern Greek: ελληνικά, elliniká, "Greek", ελληνική γλώσσα, ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean and the Black Sea.
A halo (from Greek ἅλως, halōs; also known as a nimbus, aureole, glory, or gloriole) is a crown of light rays, circle or disk of light that surrounds a person in art.
The history of science is the study of the development of science and scientific knowledge, including both the natural and social sciences.
Homeric Greek is the form of the Greek language that was used by Homer in the Iliad and Odyssey and in the Homeric Hymns.
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle.
In mathematics, an implicit equation is a relation of the form R(x_1,\ldots, x_n).
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides.
In geometry, an inscribed angle is the angle formed in the interior of a circle when two secant lines (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle.
In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint.
The intersecting chords theorem or just chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle.
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.
In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.
In geometry, a set of Johnson circles comprises three circles of equal radius r sharing one common point of intersection H. In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point H that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection).
In 2-dimensional geometry, a lens is a convex set bounded by two circular arcs joined to each other at their endpoints.
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.
In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values.
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers.
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
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 endpoints.
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.
A geometer is a mathematician whose area of study is geometry.
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle.
A mathematical constant is a special number that is "significantly interesting in some way".
Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis) is a treatise that consists of three propositions by Archimedes, ca.
Metathesis (from Greek, from "I put in a different order"; Latin: trānspositiō) is the transposition of sounds or syllables in a word or of words in a sentence.
In geometry, the midpoint is the middle point of a line segment.
In geometry, the nine-point circle is a circle that can be constructed for any given triangle.
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and its centroid at opposite ends of a diameter.
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles.
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
In geometry, the Parry point is a special point associated with a plane triangle.
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).
The number is a mathematical constant.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.
In modern mathematics, a point refers usually to an element of some set called a space.
In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is r^2 &.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Proclus Lycaeus (8 February 412 – 17 April 485 AD), called the Successor (Greek Πρόκλος ὁ Διάδοχος, Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major classical philosophers (see Damascius).
In mathematics, two variables are proportional if there is always a constant ratio between them.
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
In classical geometry, a radius 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.
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of Egyptian mathematics.
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance.
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.
In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the center of the arc to the center of its base.
In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch.
R. P. Feynman, The Feynman Lectures on Physics, Vol.1, Chaps.1,2,&3.
In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points.
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle.
The Seventh Letter of Plato is an epistle that tradition has ascribed to Plato.
A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker.
Squaring the circle is a problem proposed by ancient geometers.
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.
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 geometry, symmedians are three particular geometrical lines associated with every triangle.
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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.
In integral calculus, the tangent half-angle substitution is a substitution used for finding antiderivatives, and hence definite integrals, of rational functions of trigonometric functions.
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an incircle).
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral.
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, then the angle ∠ABC is a right angle.
The College Mathematics Journal, published by the Mathematical Association of America, is an expository journal aimed at teachers of college mathematics, particular those teaching the first two years.
The Sleepwalkers: A History of Man's Changing Vision of the Universe is a 1959 book by Arthur Koestler.
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.
In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away.
A triangle is a polygon with three edges and three vertices.
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
In geometry, specifically in the study of the arbelos, the twin circles are two special circles associated with it.
In mathematics, a unit circle is a circle with a radius of one.
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.
In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle T. It contains the circumcenters of the six triangles that are defined inside T by its three medians.
The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables.
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.
In geometry, Villarceau circles are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set.
A wheel is a circular component that is intended to rotate on an axle bearing.
In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles.
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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