Hyperbolas

Let C₁ and C₂ be two non-parallel lines and let O be the origin. Let the polar equations of C₁ and C₂ be

${\displaystyle r={\frac {a_{1}}{\cos(\theta -\alpha _{1})}}}$

and

${\displaystyle r={\frac {a_{2}}{\cos(\theta -\alpha _{2})}}.}$

By rotation through angle ${\displaystyle {\tfrac {\alpha _{1}-\alpha _{2}}{2}},}$ we can assume that ${\displaystyle \alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha .}$ Then the cissoid of C₁ and C₂ relative to the origin is given by

${\displaystyle {\begin{aligned}r&={\frac {a_{2}}{\cos(\theta +\alpha )}}-{\frac {a_{1}}{\cos(\theta -\alpha )}}\\&={\frac {a_{2}\cos(\theta -\alpha )-a_{1}\cos(\theta +\alpha )}{\cos(\theta +\alpha )\cos(\theta -\alpha )}}\\&={\frac {(a_{2}\cos \alpha -a_{1}\cos \alpha )\cos \theta -(a_{2}\sin \alpha +a_{1}\sin \alpha )\sin \theta }{\cos ^{2}\alpha \ \cos ^{2}\theta -\sin ^{2}\alpha \ \sin ^{2}\theta }}.\end{aligned}}}$

Combining constants gives

${\displaystyle r={\frac {b\cos \theta +c\sin \theta }{\cos ^{2}\theta -m^{2}\sin ^{2}\theta }}}$

which in Cartesian coordinates is

${\displaystyle x^{2}-m^{2}y^{2}=bx+cy.}$

This is a hyperbola passing through the origin. So the cissoid of two non-parallel lines is a hyperbola containing the pole. A similar derivation show that, conversely, any hyperbola is the cissoid of two non-parallel lines relative to any point on it.

Cissoids of Zahradnik

A cissoid of Zahradnik (named after Karel Zahradnik) is defined as the cissoid of a conic section and a line relative to any point on the conic. This is a broad family of rational cubic curves containing several well-known examples. Specifically:

• The Trisectrix of Maclaurin given by

${\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})}$

is the cissoid of the circle ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=-{\tfrac {a}{2}}}$ relative to the origin.

• The right strophoid

${\displaystyle y^{2}(a+x)=x^{2}(a-x)}$

is the cissoid of the circle ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=-a}$ relative to the origin.

Animation visualizing the Cissoid of Diocles

• The cissoid of Diocles

${\displaystyle x(x^{2}+y^{2})+2ay^{2}=0}$

is the cissoid of the circle ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=-2a}$ relative to the origin. This is, in fact, the curve for which the family is named and some authors refer to this as simply as cissoid.

• The cissoid of the circle ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=ka,}$ where k is a parameter, is called a Conchoid of de Sluze. (These curves are not actually conchoids.) This family includes the previous examples.
• The folium of Descartes

${\displaystyle x^{3}+y^{3}=3axy}$

is the cissoid of the ellipse ${\displaystyle x^{2}-xy+y^{2}=-a(x+y)}$ and the line ${\displaystyle x+y=-a}$ relative to the origin. To see this, note that the line can be written

${\displaystyle x=-{\frac {a}{1+p}},\ y=px}$

and the ellipse can be written

${\displaystyle x=-{\frac {a(1+p)}{1-p+p^{2}}},\ y=px.}$

So the cissoid is given by

${\displaystyle x=-{\frac {a}{1+p}}+{\frac {a(1+p)}{1-p+p^{2}}}={\frac {3ap}{1+p^{3}}},\ y=px}$

which is a parametric form of the folium.