Cardioid

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Cardioid
Cardioidmainimage.jpg
Field: Geometry
Image Created By: The Math Book

Cardioid

A cardioid is a curve which resembles a heart. Its name is derived from Greek where kardia means heart and eidos means shape, though it is actually shaped more like the outline of the cross section of an apple.


Basic Description

Cardioid 1.gif


In geometry, a cardioid is the curve traced by a point on the circumference of a circle that rolls around the circumference of another equal circle.

A More Mathematical Explanation

Generating a Cardioid

When a is the radius of the moving circ [...]

Generating a Cardioid

When a is the radius of the moving circle, the Cardioid curve is given by:

  • Cartesian equation ({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2}).
  • Polar equation r = a (1 - {\cos} {\theta})
  • Parametric equation

x = a  {\cos}  t (1 - {\cos t})

y = a  {\sin}  t (1 - {\cos t})

Properties of Cardioids

The evolute of a cardioid is equal to itself.

Cardioidevolute1.png

Cardioid is the pedal of a circle with respect to a fixed point on the circle. Cardioidpedal1.png

Cardioid is the catacaustic of a circle with light source on the circle.

Cardioid is also the envelope of circles with centers on a fixed base circle C and each circle passing through a fixed point P on the base circle C. Cardioidenvelope.png

Cardioid is the inverse of parabola with respect to its focus.

Cardioid is also the conchoid of a circle of radius r with respect to a fixed point on the circle, and offset 2 r. Cardioidconchoid1.png


Why It's Interesting

====Cardioid Microphone====

The cardioid microphone...

Cardioid's in the Mandelbrot Set

Show image

Fibonacci and Cardioids

x

Caustics

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