Difference between revisions of "Cardioid"

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|ImageName=Cardioid
 
|ImageName=Cardioid
 
|Image=CardioidEnvelope 400.gif
 
|Image=CardioidEnvelope 400.gif
|ImageIntro=The Cardioid an [[epicycloid]] with one cusp.
+
|ImageIntro=The Cardioid a [[Roulette|roulette]], more specifically an [[epicycloid]] with one cusp.
 
|ImageDescElem=The image is formed by tracing a point on the circumference of a circle of radius <math>a</math>, without slipping, on another stationery circle, as in the image below[[Image:cardioid 1.gif|left|thumb]].
 
|ImageDescElem=The image is formed by tracing a point on the circumference of a circle of radius <math>a</math>, without slipping, on another stationery circle, as in the image below[[Image:cardioid 1.gif|left|thumb]].
 
|ImageDesc=
 
|ImageDesc=

Revision as of 12:23, 3 June 2009

Inprogress.png
Cardioid
CardioidEnvelope 400.gif
Field: Geometry
Image Created By: Wolfram Math World

Cardioid

The Cardioid 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.

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 a roulette, more specifically an epicycloid with one cusp.|]]