View source for Strophoid

{{TEX|done}}
A third-order plane algebraic curve whose equation takes the form

$$y^2=x^2\frac{d+x}{d-x}$$

in Cartesian coordinates, and

$$\rho=-d\frac{\cos2\phi}{\cos\phi}$$

in polar coordinates. The coordinate origin is a node with tangents $y=\pm x$ (see Fig.). The asymptote is $x=d$. The area of the loop is

$$S=2d^2-\frac{1}{2\pi d^2}.$$

The area between the curve and the asymptote is

$$S_2=2d^2+\frac{1}{2\pi d^2}.$$

A strophoid is related to the so-called cusps (cf. [[Cusp(2)|Cusp]]).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s090630a.gif" />

Figure: s090630a

====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">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,   E.S. Stolova,   "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Gomes Teixeira,   "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special planar curves" , Dover, reprint  (1972)</TD></TR></table>

[[Category:Geometry]]