View source for Cardioid

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:

<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>

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>

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>

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>

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]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c020390a.gif" />

Figure: c020390a

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR></table>