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Cardioid

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Cardioid
CardioidFrames

The curve given by the polar equation

r=a(1-costheta),
(1)

sometimes also written

r=2b(1-costheta),
(2)

where b=a/2.

The cardioid has Cartesian equation

(x^2+y^2+ax)^2=a^2(x^2+y^2),
(3)

and the parametric equations

x=acost(1-cost)
(4)
y=asint(1-cost).
(5)

The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with r=r) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.

The cardioid has a cusp at the origin.

The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is 2a.

CardioidEnvelope
Cardioid envelope

The cardioid may also be generated as follows. Draw a circle C and fix a point A on it. Now draw a set of circles centered on the circumference of C and passing through A. The envelope of these circles is then a cardioid (Pedoe 1995). Let the circle C be centered at the origin and have radius 1, and let the fixed point be A=(1,0). Then the radius of a circle centered at an angle theta from (1, 0) is

r^2=(0-costheta)^2+(1-sintheta)^2
(6)
=cos^2theta+1-2sintheta+sin^2theta
(7)
=2(1-sintheta).
(8)

If the fixed point A is not on the circle, then the resulting envelope is a limaçon instead of a cardioid.

The arc length, curvature, and tangential angle are

s=8asin^2(1/4t)
(9)
kappa=(3csc(1/2t))/(4a)
(10)
phi=3/2t.
(11)

The perimeter and area of the curve are

L=8a
(12)
A=3/2pia^2.
(13)

See also

Cardioid Coordinates, Circle, Cissoid, Coin Paradox, Conchoid, Heart Curve, Lemniscate, Limaçon, Logarithmic Spiral, Mandelbrot Set, Nephroid

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References

Archibald, R. C. "The Cardioide and Some of Its Related Curves." Inaugural dissertation der Mathematischen und Naturwissenschaftlichen Facultät der Kaiser-Wilhelms-Universität, Strassburg zur Erlangung der Doctorwürde. Strassburg, France: J. Singer, 1900.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 214, 1987.Gray, A. "Cardioids." §3.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 54-55, 1997.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 123, 2002.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 118-121, 1972.Lockwood, E. H. "The Cardioid." Ch. 4 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 34-43, 1967.MacTutor History of Mathematics Archive. "Cardioid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cardioid.html.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 326, 1958.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 24-25, 1991.Yates, R. C. "The Cardioid." Math. Teacher 52, 10-14, 1959.Yates, R. C. "Cardioid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 4-7, 1952.

Cite this as:

Weisstein, Eric W. "Cardioid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cardioid.html

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