Cardioid
The curve given by the polar equation
 |
(1)
|
sometimes also written
 |
(2)
|
where 
.
The cardioid has Cartesian equation
 |
(3)
|
and the parametric equations
The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with 
) 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 
.
The cardioid may also be generated as follows. Draw a circle 
and fix a point 
on it. Now draw
a set of circles centered on the circumference
of 
and passing through 
. The envelope
of these circles is then a cardioid (Pedoe 1995). Let
the circle 
be centered at
the origin and have radius 1, and let the fixed point
be 
. Then the radius
of a circle centered at an angle 
from (1, 0) is
If the fixed point 
is not on the circle, then the resulting
envelope is a limaçon
instead of a cardioid.
The arc length, curvature,
and tangential angle are
The perimeter and area of
the curve are
SEE ALSO: Cardioid Coordinates,
Circle,
Cissoid,
Coin
Paradox,
Conchoid,
Heart
Curve,
Lemniscate,
Limaçon,
Logarithmic Spiral,
Mandelbrot
Set,
Nephroid
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. http://mathworld.wolfram.com/Cardioid.html
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