# Cissoid

In geometry, a cissoid (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves C$_1$, C$_2$ and a point O (the pole). Let L be a variable line passing through O and intersecting C$_1$ at P$_1$ and C$_2$ at P$_2$. Let P be the point on L so that ${\overline {OP}}={\overline {P_{1}P_{2}}}.$ (There are actually two such points but P is chosen so that P is in the same direction from O as P$_2$ is from P$_1$.) Then the locus of such points P is defined to be the cissoid of the curves C$_1$, C$_2$ relative to O.
Slightly different but essentially equivalent definitions are used by different authors. For example, P may be defined to be the point so that ${\overline {OP}}={\overline {OP_{1}}}+{\overline {OP_{2}}}.$ This is equivalent to the other definition if C$_1$ is replaced by its reflection through O. Or P may be defined as the midpoint of P$_1$ and P$_2$; this produces the curve generated by the previous curve scaled by a factor of 1/2.

## Equations

If C$_1$ and C$_2$ are given in polar coordinates by $r=f_{1}(\theta )$ and $r=f_{2}(\theta )$ respectively, then the equation $r=f_{2}(\theta )-f_{1}(\theta )$ describes the cissoid of C$_1$ and C$_2$ 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, C$_1$ is also given by
    ${\begin{aligned}&r=-f_{1}(\theta +\pi )\\&r=-f_{1}(\theta -\pi )\\&r=f_{1}(\theta +2\pi )\\&r=f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}$
So the cissoid is actually the union of the curves given by the equations
    ${\begin{aligned}&r=f_{2}(\theta )-f_{1}(\theta )\\&r=f_{2}(\theta )+f_{1}(\theta +\pi )\\&r=f_{2}(\theta )+f_{1}(\theta -\pi )\\&r=f_{2}(\theta )-f_{1}(\theta +2\pi )\\&r=f_{2}(\theta )-f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}$
It can be determined on an individual basis depending on the periods of f$_1$ and f$_2$, which of these equations can be eliminated due to duplication.
For example, let C$_1$ and C$_2$ both be the ellipse
    $r={\frac {1}{2-\cos \theta }}.$
The first branch of the cissoid is given by
    $r={\frac {1}{2-\cos \theta }}-{\frac {1}{2-\cos \theta }}=0,$
which is simply the origin. The ellipse is also given by
    $r={\frac {-1}{2+\cos \theta }},$
so a second branch of the cissoid is given by
    $r={\frac {1}{2-\cos \theta }}+{\frac {1}{2+\cos \theta }}$
which is an oval shaped curve.
If each C$_1$ and C$_2$ are given by the parametric equations
    $x=f_{1}(p),\ y=px$
and
    $x=f_{2}(p),\ y=px,$
then the cissoid relative to the origin is given by
    $x=f_{2}(p)-f_{1}(p),\ y=px.$

## Specific cases

When C$_1$ is a circle with center O then the cissoid is conchoid of C$_2$.
When C$_1$ and C$_2$ are parallel lines then the cissoid is a third line parallel to the given lines.

### Hyperbolas

Let C$_1$ and C$_2$ be two non-parallel lines and let O be the origin. Let the polar equations of C$_1$ and C$_2$ be
    $r={\frac {a_{1}}{\cos(\theta -\alpha _{1})}}$
and
    $r={\frac {a_{2}}{\cos(\theta -\alpha _{2})}}.$
By rotation through angle ${\tfrac {\alpha _{1}-\alpha _{2}}{2}},$ we can assume that $\alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha .$ Then the cissoid of C$_1$ and C$_2$ relative to the origin is given by
    ${\begin{aligned}r&={\frac {a_{2}}{\cos(\theta +\alpha )}}-{\frac {a_{1}}{\cos(\theta -\alpha )}}\\&={\frac {a_{2}\cos(\theta -\alpha )-a_{1}\cos(\theta +\alpha )}{\cos(\theta +\alpha )\cos(\theta -\alpha )}}\\&={\frac {(a_{2}\cos \alpha -a_{1}\cos \alpha )\cos \theta -(a_{2}\sin \alpha +a_{1}\sin \alpha )\sin \theta }{\cos ^{2}\alpha \ \cos ^{2}\theta -\sin ^{2}\alpha \ \sin ^{2}\theta }}.\end{aligned}}$
Combining constants gives
    $r={\frac {b\cos \theta +c\sin \theta }{\cos ^{2}\theta -m^{2}\sin ^{2}\theta }}$
which in Cartesian coordinates is
    $x^{2}-m^{2}y^{2}=bx+cy.$
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
       $2x(x^{2}+y^{2})=a(3x^{2}-y^{2})$
    is the cissoid of the circle $(x+a)^{2}+y^{2}=a^{2}$ and the line $x=-{\tfrac {a}{2}}$ relative to the origin.
* The right strophoid
       $y^{2}(a+x)=x^{2}(a-x)$
    is the cissoid of the circle $(x+a)^{2}+y^{2}=a^{2}$ and the line $x=-a$ relative to the origin.
* The cissoid of Diocles
       $x(x^{2}+y^{2})+2ay^{2}=0$
    is the cissoid of the circle $(x+a)^{2}+y^{2}=a^{2}$ and the line $x=-2a$ 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 $(x+a)^{2}+y^{2}=a^{2}$ and the line $x=ka,$ 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
       $x^{3}+y^{3}=3axy$
    is the cissoid of the ellipse $x^{2}-xy+y^{2}=-a(x+y)$ and the line $x+y=-a$ relative to the origin. To see this, note that the line can be written
       $x=-{\frac {a}{1+p}},\ y=px$
    and the ellipse can be written
       $x=-{\frac {a(1+p)}{1-p+p^{2}}},\ y=px.$
    So the cissoid is given by
       $x=-{\frac {a}{1+p}}+{\frac {a(1+p)}{1-p+p^{2}}}={\frac {3ap}{1+p^{3}}},\ y=px$
    which is a parametric form of the folium.