Cissoid

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 kissoeidēs "ivy shaped" from kissos "ivy" and -oeidēs "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 though 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 (name 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 $x={-{a \over 2}}$ 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 concoids.) 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.

See also

• Conchoid
• Strophoid

References

• 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.

External links

• Hazewinkel, Michiel, ed. (2001), "Cissoid", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Cissoid". MathWorld.

• 2D Curves
• "Courbe Cissoïdale" at Encyclopédie des Formes Mathématiques Remarquables (in French)
• "Cissoïdales de Zahradnik" at Encyclopédie des Formes Mathématiques Remarquables (in French)