Cardioid

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

Cardioid

The Cardioid, more commonly referred to as the heart curve is a roulette, more specifically an epicycloid with one cusp.


Basic Description

The image is formed by tracing a point on the circumference of a circle of radius a, without slipping, on another stationery circle, as in the image below

Cardioid 1.gif

.

A More Mathematical Explanation

The curve is given by:

  • Cartesian equation ({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2}), where '"`UNIQ--math-00 [...]

The curve is given by:

  • Cartesian equation ({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2}), where a is the radius of the moving circle.
  • Polar equation r = a (1 - {\cos} {\theta})
  • Parametric equation

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

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

Properties

  • It has a cusp at the origin.
    Cardioidcremona2.gif
  • There are exactly three tangents to the cardioid with any given gradient
  • 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


Generating a Cardioid

Draw a circle C, and pick a fixed point A on it. Then, draw a set of circles centered on the circumference of C and passing through A. The envelop of the chords of these circles is a cardioid, as in the main image. If the fixed point A is not on the circle, then the figure becomes a limacon


The Cardioid in Real Life

An instance where one could see a cardioid is when looking into a cup of coffee. The caustic seen at the bottom of a cup of coffee could be a cardioid, depending on the angle of light relative to the bottom of the cup.

Also, all unidirectional microphones are cardioid-shaped.




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  [[Description::The Cardioid, more commonly referred to as the heart curve is a roulette, more specifically an epicycloid with one cusp.|]]