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In geometry, the cardioid is an epicycloid with one cusp. That is, a cardioid is a curve that can be produced as the path (or locus) of a point on the circumference of a circle as that circle rolls around another fixed circle with the same radius.

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.

The name comes from the heart shape of the curve (Greek kardioeides = kardia:heart + eidos:shape). Compared to the heart symbol (♥), though, a cardioid only has one sharp point (or cusp). It is rather shaped more like the outline of the cross section of a plum.

The cardioid is an inverse transform of a parabola.

The large central figure in the Mandelbrot set is a cardioid.

Caustics can take the shape of cardioids. The caustic seen at the bottom of a coffee cup, for instance, may be a cardioid. The specific curve depends on the angle the light source makes relative to the bottom of the cup. The shape can be a nephroid, which looks quite similar.

Equations

Since the cardioid is an epicycloid with one cusp, in cartesian coordinates it has parametric equations





where r is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The cusp is at (r,0).

The polar equation



yields a cardioid with the same shape. It is the same curve as the cardioid given above, shifted to the left r units, so the cusp is at the origin.

For a proof, see cardioid proofs.

Graphs

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

Area

The area of a cardioid with polar equation
is
.


See proof.

See also

References

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by:

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Cusp may refer to:
  • Cusp (singularity), a singular point of a curve
  • Cusp form in modular form theory
  • Cuspidal representation, a generalization of cusp forms in the theory of automorphic representations

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In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle.
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The word locus (plural loci) is Latin for "place":
  • Locus (psychology), in industrial psychology, having an internal or external locus of control.
  • Locus (genetics), the position of a gene (or other significant sequence) on a chromosome.

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circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
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In mathematics, limaçons (pronounced with a soft c), also known as limaçons of Pascal, are heart-shaped mathematical curves. The cardioid is a special case, with a cusp.
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This article or section is in need of attention from an expert on the subject.
Please help recruit one or [ improve this article] yourself. See the talk page for details.
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equation is a mathematical statement, in symbols, that two things are the same (or equivalent). Equations are written with an equal sign, as in
.


The equation above is an example of an equality: a proposition which states that two constants are equal.
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heart is a muscular organ responsible for pumping blood through the blood vessels by repeated, rhythmic contractions, or a similar structure in the annelids, mollusks, and arthropods.
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Shape (OE. sceap Eng. created thing), refers to the external two-dimensional outline, appearance or configuration of some thing — in contrast to the matter or content or substance of which it is composed.
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Cusp may refer to:
  • Cusp (singularity), a singular point of a curve
  • Cusp form in modular form theory
  • Cuspidal representation, a generalization of cusp forms in the theory of automorphic representations

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Prunus

Species
See text.
A plum or gage is a stone fruit tree in the genus Prunus, subgenus Prunus. The subgenus is distinguished from other subgenera (peaches, cherries, bird cherries, etc) in the shoots having a terminal bud and
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In geometry, an inversion is a particular type of transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius).
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parabola (from the Greek: παραβολή) (IPA pronunciation: /pəˈrab(ə)lə/
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Mandelbrot set is a set of points in the complex plane that forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial x2 + c
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In optics, a caustic is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. Caustic
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The nephroid is a plane curve whose name means kidney-shaped (compare nephrology.) Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Proctor in 1878.
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epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by:

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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables.
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polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of
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Theorem

The curve defined by the parametric equations





has the same shape as the curve defined in polar coordinates by the equation

Proof


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cardinal directions, they are north, east, south and west. Though the names may seem arbitrary, there are simple specific means to establish each direction, which should work anywhere on Earth where there is a view of the sky.
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Wittgenstein's rod is a thought experiment attributed to Ludwig Wittgenstein.

A rod passes through a sleeve, in which it can smoothly slide. The sleeve is fixed to a wall, forming a pivot. One end of the rod is moved in a circle whose plane is perpendicular to the wall.
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microphone, sometimes referred to as a mike or mic (both IPA pronunciation: [maɪk]), is an acoustic to electric transducer or sensor that converts sound into an electrical signal.
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cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. The site has won more than 20 awards from scientific and educational publications [1]
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