View source for Cardioid

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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:

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

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


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