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.

Ellipse

r={\frac {1}{2-\cos \theta }}

in red, with its two cissoid branches in black and blue (origin)

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.

Animation visualizing the Cissoid of Diocles

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