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Difference between revisions of "Cardioid"

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A plane algebraic curve of order four which is described by a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203901.png" /> of a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203902.png" /> rolling on a circle with the same radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203903.png" />; an [[Epicycloid|epicycloid]] with modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203904.png" />. The equation of the cardioid in polar coordinates is:
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A plane algebraic curve of order four which is described by a point $M$ of a circle of radius $r$ rolling on a circle with the same radius $r$; an [[Epicycloid|epicycloid]] with modulus $m=1$. The equation of the cardioid in polar coordinates is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203905.png" /></td> </tr></table>
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$$\rho=2r(1-\cos\phi),$$
  
 
In Cartesian coordinates it is:
 
In Cartesian coordinates it is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203906.png" /></td> </tr></table>
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$$(x^2+y^2+2rx)^2=4r^2(x^2+y^2).$$
  
 
The arc length from the cusp is:
 
The arc length from the cusp is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203907.png" /></td> </tr></table>
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$$l=16r\sin^2\frac\phi4.$$
  
 
The radius of curvature is:
 
The radius of curvature is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203908.png" /></td> </tr></table>
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$$r_k=\frac{8r}{3}\sin\frac\phi2.$$
  
The area bounded by the curve equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c0203909.png" />. The length of the curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020390/c02039010.png" />. The cardioid is a [[Conchoid|conchoid]] of the circle, a special case of a [[Pascal limaçon|Pascal limaçon]] and a [[Sinusoidal spiral|sinusoidal spiral]].
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The area bounded by the curve equals $S=6\pi r^2$. The length of the curve is $16r$. The cardioid is a [[Conchoid|conchoid]] of the circle, a special case of a [[Pascal limaçon|Pascal limaçon]] and a [[Sinusoidal spiral|sinusoidal spiral]].
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c020390a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c020390a.gif" />

Revision as of 21:00, 11 April 2014

A plane algebraic curve of order four which is described by a point $M$ of a circle of radius $r$ rolling on a circle with the same radius $r$; an epicycloid with modulus $m=1$. The equation of the cardioid in polar coordinates is:

$$\rho=2r(1-\cos\phi),$$

In Cartesian coordinates it is:

$$(x^2+y^2+2rx)^2=4r^2(x^2+y^2).$$

The arc length from the cusp is:

$$l=16r\sin^2\frac\phi4.$$

The radius of curvature is:

$$r_k=\frac{8r}{3}\sin\frac\phi2.$$

The area bounded by the curve equals $S=6\pi r^2$. The length of the curve is $16r$. The cardioid is a conchoid of the circle, a special case of a Pascal limaçon and a sinusoidal spiral.

Figure: c020390a

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Cardioid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardioid&oldid=13869
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article