Difference between revisions of "Cardioid"

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[[Image:cardioid 1.gif|left|thumb]]
 
[[Image:cardioid 1.gif|left|thumb]]
 
 
 
|ImageDesc=The curve is given by:
 
|ImageDesc=The curve is given by:
  
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<math>y = a  {\sin}  t (1 - {\cos t})</math>
 
<math>y = a  {\sin}  t (1 - {\cos t})</math>
  
==Properties==
 
*It has a cusp at the origin. [[Image:cardioidcremona2.gif|right|thumb]]
 
 
*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 2<math>a</math>
 
 
 
 
==Generating a Cardioid==
 
Draw a circle <math>C</math>, and pick a fixed point <math>A</math> on it. Then, draw a set of circles  centered on the circumference of <math>C</math> and passing through <math>A</math>. 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 [http://en.wikipedia.org/wiki/Lima%C3%A7on limacon]
 
 
 
==The Cardioid in Real Life==
 
An instance where one could see a cardioid is when looking into a cup of coffee. The [http://en.wikipedia.org/wiki/Caustic_(optics) 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.
 
 
|AuthorName=The Math Book
 
|AuthorName=The Math Book
 
|Field=Geometry
 
|Field=Geometry
 
|InProgress=Yes
 
|InProgress=Yes
 
}}
 
}}

Revision as of 10:36, 28 June 2010

Inprogress.png
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

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.

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})




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