In geometry, the cardioid is an epicycloid which has one and only one cusp. That is, a cardioid is a curve that can be produced as a locus — by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed but which has the same radius as the rolling circle. Geometry (Greek Γεωμετρια, geo = earth, metria = measure (check accuracy of this)) arose as the field of knowledge dealing with spatial relationships. ... In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around without slipping around a fixed circle. ... A principle, especially in software development, that values avoiding the hazards of redundancy. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, a locus (plural loci) is a collection of points which share a common property. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ... RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...

The cardioid is also a special type of limaçon: it is the limaçon with one cusp. (The cusp is formed when the ratio of a to b in the equation is equal to one.) In mathematics, limaçons, also known as limaçons of Pascal (pronounced with a soft c), are heart-shaped mathematical curves. ... In algebra, a ratio is the relationship between two quantities. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

The name comes from the heart shape of the curve (Greek kardioeides = kardia:heart + eidos:shape). Compared to the ♥ symbol, though, a cardiod does not come to a sharp point. It is rather shaped more like the outline of the cross section of a plum. The heart and lungs (from an older edition of Grays Anatomy) The heart (Latin cor) is a hollow, muscular organ that pumps blood through the blood vessels by repeated, rhythmic contractions. ... In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ... Species See text A plum is a stone fruit tree in the genus Prunus, subgenus Prunus. ...

The cardioid is an inverse transform of a parabola. In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... A parabola A parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone and a plane tangent to the cone or parallel to some plane tangent to the cone. ...

The large, central, black figure in a Mandelbrot set is a cardioid. This cardioid is surrounded by a fractal arrangement of circles. A rendering of the Mandelbrot set: black points represent the stable points under the iterative map In mathematics, the Mandelbrot set is a fractal that is defined as the set of points c in the complex plane for which the iteratively defined sequence: does not tend to infinity. ... The Mandelbrot set, named after its discoverer, is a famous example of a fractal. ...

Equations

Since the cardioid is an epicycloid with one cusp, its parametric equations are In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around without slipping around a fixed circle. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

The same shape can be defined in polar coordinates by the equation This article describes some of the common coordinate systems that appear in elementary mathematics. ...

    For a proof, see cardioid proofs.

// Theorem The parametric equations are The same shape can be defined in polar coordinates by the equation Proof Start from . ...

Graphs

Image File history File links Download high resolution version (651x686, 32 KB)A set of four cardioids, oriented in the four cardinal directions, with their respective equations. ...

Four graphs of cardioids oriented in the four cardinal directions, with their respective polar equations.

Cardinal directions or cardinal points are the four principal directions or points of the compass, north, east, south and west. ...

Area

The area of a cardioid which is congruent to

ρ(θ) = a(1 − cosθ)

is

.

See proof. // Theorem The parametric equations are The same shape can be defined in polar coordinates by the equation Proof Start from . ...

External link

• Alexander Bogomolny, Hearty Munching on Cardioids at Cut-the-Knot, (1998)
• Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).