Cissoid

[[File:Allgemeine zissoide_english.svg|thumb|upright=1.5|

{{legend|black|Pole $O$}}]]

In geometry, a cissoid is a plane curve generated from two given curves $C1$, $C2$ and a point $O$ (the pole). Let $L$ be a variable line passing through $O$ and intersecting $C1$ at $C2$ and $C1$ at $P1$. 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 $C2$ is from $P2$.) Then the locus of such points $P$ is defined to be the cissoid of the curves $P2$, $P1$ 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 $C1$ is replaced by its reflection through $O$. Or $P$ may be defined as the midpoint of $C2$ and $C1$; this produces the curve generated by the previous curve scaled by a factor of 1/2.

Equations
If $P1$ and $P2$ 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 $C1$ 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
 * $$ \begin{align}

& 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{align}$$ So the cissoid is actually the union of the curves given by the equations
 * $$\begin{align}

& 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{align}$$ It can be determined on an individual basis depending on the periods of $C2$ and $C1$, which of these equations can be eliminated due to duplication.

For example, let $f1$ and $f2$ 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 $C1$ and $C2$ 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 $C1$ 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
Let $C1$ and $C2$ be two non-parallel lines and let $O$ be the origin. Let the polar equations of $C1$ and $C2$ 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 $C1$ and $C2$ relative to the origin is given by
 * $$\begin{align}

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{align}$$ Combining constants gives
 * $$r=\frac{b\cos\theta+c\sin\theta}{\cos^2\theta-m^2\sin^2\theta}$$

which in Cartesian coordinates is
 * $$x^2-m^2y^2=bx+cy.$$

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.