If C₁ and C₂ are given in polar coordinates by ${\displaystyle r=f_{1}(\theta )}$  and ${\displaystyle r=f_{2}(\theta )}$  respectively, then the equation ${\displaystyle r=f_{2}(\theta )-f_{1}(\theta )}$  describes the cissoid of C₁ and C₂ relative to the origin. However, because a point may be represented in multiple ways in polar coordinates, there may be other branches of the cissoid which have a different equation. Specifically, C₁ is also given by

${\displaystyle {\begin{aligned}&r=-f_{1}(\theta +\pi )\\&r=-f_{1}(\theta -\pi )\\&r=f_{1}(\theta +2\pi )\\&r=f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}}$ 

So the cissoid is actually the union of the curves given by the equations

${\displaystyle {\begin{aligned}&r=f_{2}(\theta )-f_{1}(\theta )\\&r=f_{2}(\theta )+f_{1}(\theta +\pi )\\&r=f_{2}(\theta )+f_{1}(\theta -\pi )\\&r=f_{2}(\theta )-f_{1}(\theta +2\pi )\\&r=f_{2}(\theta )-f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}}$ 

It can be determined on an individual basis depending on the periods of f₁ and f₂, which of these equations can be eliminated due to duplication.

For example, let C₁ and C₂ both be the ellipse

${\displaystyle r={\frac {1}{2-\cos \theta }}.}$ 

The first branch of the cissoid is given by

${\displaystyle r={\frac {1}{2-\cos \theta }}-{\frac {1}{2-\cos \theta }}=0,}$ 

which is simply the origin. The ellipse is also given by

${\displaystyle r={\frac {-1}{2+\cos \theta }},}$ 

so a second branch of the cissoid is given by

${\displaystyle r={\frac {1}{2-\cos \theta }}+{\frac {1}{2+\cos \theta }}}$ 

which is an oval shaped curve.

If each C₁ and C₂ are given by the parametric equations

${\displaystyle x=f_{1}(p),\ y=px}$ 

and

${\displaystyle x=f_{2}(p),\ y=px,}$ 

then the cissoid relative to the origin is given by

${\displaystyle x=f_{2}(p)-f_{1}(p),\ y=px.}$ 

When C₁ is a circle with center O then the cissoid is conchoid of C₂.

When C₁ and C₂ are parallel lines then the cissoid is a third line parallel to the given lines.

Hyperbolas edit

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 edit

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.

• 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.

• Conchoid
• Strophoid

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 53–56. ISBN 0-486-60288-5.
• C. A. Nelson "Note on rational plane cubics" Bull. Amer. Math. Soc. Volume 32, Number 1 (1926), 71-76.

• "Cissoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Cissoid". MathWorld.
• 2D Curves