Difference between revisions of "Cardioid"

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|Image=CardioidEnvelope 400.gif
 
|Image=CardioidEnvelope 400.gif
 
|ImageIntro=The Cardioid an [[epicycloid]] with one cusp.
 
|ImageIntro=The Cardioid an [[epicycloid]] with one cusp.
|ImageDescElem=The shape 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[[Image:cardioid 1.gif|left|thumb]].  
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|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]].
 
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|ImageDesc=
 
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The curve is given by:
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
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*Cartesian equation <math>({x^2} + {y^2} - 2ax)^2 = 4{a^2}({x^2} + {y^2})</math>, where <math>a</math> is the radius of the moving circle.
  
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*Polar equation  <math>r = a (1 - {\cos} {\theta})</math>
  
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*Parametric equations
  
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<math>x = a  {\cos}  t (1 - {\cos t})</math>
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<math>y = a  {\sin}  t (1 - {\cos t})</math>
  
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==Properties==
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*It has a cusp at the origin.
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*
  
 
|AuthorName=Wolfram Math World
 
|AuthorName=Wolfram Math World

Revision as of 13:31, 2 June 2009

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

Cardioid

The Cardioid 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 equations

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

Properties

  • It has a cusp at the origin.




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  [[Description::The Cardioid an epicycloid with one cusp.|]]