A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.
The name was coined by de Castillon in 1741 but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk.
A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred on the microphone which is the "stalk" of the apple.
Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius a, the cardioid is given by the following parametric equations:
In the complex plane this becomes
Here a is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The point generating the curve touches the fixed circle at (a, 0), the cusp. The parameter t can be eliminated giving
or, in rectangular coordinates,
These equations can be simplified somewhat by shifting the fixed circle to the right a units and choosing the point on the rolling circle so that it touches the fixed circle at the origin; this changes the orientation of the curve so that the cusp is on the left. The parametric equations are then:
or, in the complex plane,
With the substitution u=tan t/2,
giving a rational parameterization:
The parametrization can also be written
and in this form it is apparent that the equation for this cardioid may be written in polar coordinates as
where θ replaces the parameter t. This can also be written
which implies that the curve is a member of the family of sinusoidal spirals.
In Cartesian coordinates, the equation for this cardioid is
The area enclosed by a cardioid can be computed from the polar equation:
This is 6 times the area of the circles used in the construction with rolling circles, or 1.5 times the area of the circle used in the construction with circle and tangent lines.
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.
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
Cardioids in complex analysis
In complex analysis, the image of any circle through the origin under the map 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
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.
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. The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar.
- Wittgenstein's rod
- Cardioid microphone
- Lemniscate of Bernoulli
- Loop antenna
- Radio direction finder
- Radio direction finding
- Yagi antenna
- Giovanni Salvemini
- Bonne's Projection
- Weisstein, Eric W., "Cardioid", MathWorld.
- R.C. Yates (1952). "Cardioid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 4 ff.
- Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 24–25. ISBN 0-14-011813-6.
|Wikimedia Commons has media related to Cardioids.|
- O'Connor, John J.; Robertson, Edmund F., "Cardioid", MacTutor History of Mathematics archive, University of St Andrews.
- Hearty Munching on Cardioids at cut-the-knot
- Weisstein, Eric W., "Epicycloid--1-Cusped", MathWorld.
- Weisstein, Eric W., "Heart Curve", MathWorld.
- "Cardioid" at Encyclopédie des Formes Mathématiques Remarquables
- Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).
- Jan Wassenaar, Cardioid, (2005)