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Summary Cissoid(red) based on curves C1, C2 and pole O

In geometry, a cissoid is a 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 P1 and C2 at P2. Let P be the point on L so that OP = P1P2. (There are actually two such points but P is chosen so that P is in the same direction from O as P2 is from P1.) Then the locus of such points P is defined to be the cissoid of the curves C1, C2 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 OP = OP1 + OP2. 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 P1 and P2; this produces the curve generated by the previous curve scaled by a factor of 1/2.

The word "cissoid" comes from the Greek: κισσοειδής, lit.'ivy shaped' from κισσός, 'ivy', and -οειδής, 'having the likeness of'.

Equations

If C1 and C2 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

$r=-f_{1}(\theta +\pi ),\ r=-f_{1}(\theta -\pi ),\ r=f_{1}(\theta +2\pi ),\ r=f_{1}(\theta -2\pi ),\ \dots$ .

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

$r=f_{2}(\theta )-f_{1}(\theta ),\ r=f_{2}(\theta )+f_{1}(\theta +\pi ),\ r=f_{2}(\theta )+f_{1}(\theta -\pi ),\$ $r=f_{2}(\theta )-f_{1}(\theta +2\pi ),\ r=f_{2}(\theta )-f_{1}(\theta -2\pi ),\ \dots$ .

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. Ellipse $r={\frac {1}{2-\cos \theta }}$ in red, with its two cissoid branches in black and blue (origin)

For example, let C1 and C2 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 $(\alpha _{1}-\alpha 2)/2$ , we can assume that $\alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha$ . Then the cissoid of C1 and C2 relative to the origin is given by

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

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.

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:

$2x(x^{2}+y^{2})=a(3x^{2}-y^{2})$ is the cissoid of the circle $(x+a)^{2}+y^{2}=a^{2}$ and the line $x={-{a \over 2}}$ relative to the origin.
$y^{2}(a+x)=x^{2}(a-x)$ 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
$x(x^{2}+y^{2})+2ay^{2}=0$ 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.