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A cardoid generated by a rolling circle.
Four graphs of cardioids oriented in the four cardinal directions, with their respective polar equations.

In geometry, a cardioid is the curve traced by a point on the edge of a circular wheel that is rolling around a fixed wheel of the same size. The resulting curve is roughly heart-shaped[1], with a cusp at the place where the point touches the fixed wheel.

The cardioid is a roulette, and can be viewed as either an epicycloid with one cusp or as a member of the family of limaçons. It is also a type of sinusoidal spiral, and is the inverse curve of a parabola[2] with the focus as the center of inversion[3].

Contents

Name

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.

Equations

Based on the rolling circle description, the cardioid is given by the following parametric equations:

 x(t) = 2r \left( \cos t - {1 \over 2} \cos 2 t \right), \,
 y(t) = 2r \left( \sin t - {1 \over 2} \sin 2 t \right). \,

Here 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).

Shifting the curve to the left r units, so the cusp is at the origin, produces a new formula for x(t):

 x(t) = 2r \left( \cos t - {1 \over 2} \cos 2 t - {1 \over 2}\right). \,

By applying various trigonometric identities, this can be rewritten as alternative parametric equations for a cardioid are written as follows:

 x(t) = 2r \cos t \, (1 - \cos t),\,
 y(t) = 2r \sin t \, (1 - \cos t).\,

In this form it is apparent that the equation for this cardioid may be written in polar coordinates as

 \rho(\theta) = 2r(1 - \cos \theta)\,

where θ replaces the parameter t.

In Cartesian coordinates, the equation for this cardioid is

 \left(x^2+y^2+2rx\right)^2 \,=\, 4r^2\left(x^2 + y^2\right).\,

Area

The area enclosed by a cardioid with polar equation

 \rho (\theta) = 2r(1 - \cos \theta) \,

can be computed using the formula (see Polar coordinate system#Integral calculus)

A = \frac12\int_0^{2\pi}\ \rho(\theta)^2\ d\theta = 2r^2\int_0^{2\pi}\ (1 - \cos \theta)^2\ d\theta

which when evaluated becomes

A = 6πr2.

Arc length

The arc length of a cardioid can be computed exactly. For the cardioid with polar equation

 \rho (\theta) = 2r(1 - \cos \theta) \,

the total length is

 L = 4 \pi r.\,.

Inverse curve

The green cardioid is obtained by inverting the red parabola across the dashed circle.

The cardioid is one possible inverse curve for a parabola. Specifically, if a parabola is inverted across any circle whose center lies at the focus of the parabola, the result is a cardioid. The cusp of the resulting cardioid will lie at the center of the circle, and corresponds to the vanishing point of the parabola.

In terms of stereographic projection, this says that a parabola in the Euclidean plane is the projection of a cardioid drawn on the sphere whose cusp is at the north pole.

Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of the parabola, then the result is a cissoid of Diocles.

The picture to the right shows the parabola with polar equation

\rho(\theta) \,=\, \frac{1}{1 - \cos \theta}.\,

In Cartesian coordinates, this is the parabola y2 = 2x + 1. When this parabola is inverted across the unit circle, the result is a cardioid with the reciprocal equation

\rho(\theta) \,=\, 1 - \cos \theta.\,
The central bulb of the Mandelbrot set is a cardioid.

Cardioids in complex analysis

In complex analysis, the image of any circle through the origin under the map z\to z^2 is a cardioid. One application of this result is that the boundary of the central bulb of the Mandelbrot set is a cardioid given by the equation

 c \,=\, \frac{1 - \left(e^{it}-1\right)^2}{4}.\,

The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid.

The caustic appearing on the surface of this cup of coffee is a cardioid.

Caustics

Certain caustics can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone.[4] The shape at a the curve at the bottom of a cylindrical cup takes the form of a nephroid, which looks quite similar.

See also

Bibliography

  • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 24–25. ISBN 0-14-011813-6.  

References

  1. ^ Weisstein, Eric W. "Heart Curve." From MathWorld--A Wolfram Web Resource.
  2. ^ Weisstein, Eric W. "Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCurve.html
  3. ^ Weisstein, Eric W. "Parabola Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParabolaInverseCurve.html
  4. ^ "Surface Caustique" at Encyclopédie des Formes Mathématiques Remarquables

External links

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1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

CARDIOID, a curve so named by G. F. M. M. Castillon (1708-1791), on account of its heart-like form (Gr. KapSia, heart). It was mathematically treated by Louis Carre in 1705 and Koersma in 1741. It is a particular form of the limagon (q.v.) and is generated in the same way. It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. The polar equation to the cardioid is r=a(1-}-cos 0). There is symmetry about the initial line and a cusp at the origin. The area is -ira 2, i.e. 12 times the area of the generating circle; the length of the curve is 8a. (For a figure see LIMAgoN.)


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