Cissoid

In geometry, a cissoid (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') 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 P₁ and C₂ at P₂. Let P be the point on L so that ${\displaystyle {\overline {OP}}={\overline {P_{1}P_{2}}}.}$ (There are actually two such points but P is chosen so that P is in the same direction from O as 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 ${\displaystyle {\overline {OP}}={\overline {OP_{1}}}+{\overline {OP_{2}}}.}$ 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 P₁ and P₂; this produces the curve generated by the previous curve scaled by a factor of 1/2.

Equations

If C₁ and C₂ are given in polar coordinates by ${\displaystyle r=f_{1}(\theta )}$ and ${\displaystyle r=f_{2}(\theta )}$ respectively, then the equation ${\displaystyle r=f_{2}(\theta )-f_{1}(\theta )}$ 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

${\displaystyle {\begin{aligned}&r=-f_{1}(\theta +\pi )\\&r=-f_{1}(\theta -\pi )\\&r=f_{1}(\theta +2\pi )\\&r=f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}}$

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

${\displaystyle {\begin{aligned}&r=f_{2}(\theta )-f_{1}(\theta )\\&r=f_{2}(\theta )+f_{1}(\theta +\pi )\\&r=f_{2}(\theta )+f_{1}(\theta -\pi )\\&r=f_{2}(\theta )-f_{1}(\theta +2\pi )\\&r=f_{2}(\theta )-f_{1}(\theta -2\pi )\\&\qquad \qquad \vdots \end{aligned}}}$

It can be determined on an individual basis depending on the periods of f₁ and f₂, which of these equations can be eliminated due to duplication.

For example, let C₁ and C₂ both be the ellipse

${\displaystyle r={\frac {1}{2-\cos \theta }}.}$

The first branch of the cissoid is given by

${\displaystyle r={\frac {1}{2-\cos \theta }}-{\frac {1}{2-\cos \theta }}=0,}$

which is simply the origin. The ellipse is also given by

${\displaystyle r={\frac {-1}{2+\cos \theta }},}$

so a second branch of the cissoid is given by

${\displaystyle r={\frac {1}{2-\cos \theta }}+{\frac {1}{2+\cos \theta }}}$

which is an oval shaped curve.

If each C₁ and C₂ are given by the parametric equations

${\displaystyle x=f_{1}(p),\ y=px}$

and

${\displaystyle x=f_{2}(p),\ y=px,}$

then the cissoid relative to the origin is given by

${\displaystyle x=f_{2}(p)-f_{1}(p),\ y=px.}$

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

${\displaystyle r={\frac {a_{1}}{\cos(\theta -\alpha _{1})}}}$

and

${\displaystyle r={\frac {a_{2}}{\cos(\theta -\alpha _{2})}}.}$

By rotation through angle ${\displaystyle {\tfrac {\alpha _{1}-\alpha _{2}}{2}},}$ we can assume that ${\displaystyle \alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha .}$ Then the cissoid of C₁ and C₂ relative to the origin is given by

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

Combining constants gives

${\displaystyle r={\frac {b\cos \theta +c\sin \theta }{\cos ^{2}\theta -m^{2}\sin ^{2}\theta }}}$

which in Cartesian coordinates is

${\displaystyle x^{2}-m^{2}y^{2}=bx+cy.}$

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

${\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})}$

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

• The right strophoid

${\displaystyle y^{2}(a+x)=x^{2}(a-x)}$

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

• The cissoid of Diocles

${\displaystyle x(x^{2}+y^{2})+2ay^{2}=0}$

is the cissoid of the circle ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=-2a}$ 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 ${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$ and the line ${\displaystyle x=ka,}$ 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

${\displaystyle x^{3}+y^{3}=3axy}$

is the cissoid of the ellipse ${\displaystyle x^{2}-xy+y^{2}=-a(x+y)}$ and the line ${\displaystyle x+y=-a}$ relative to the origin. To see this, note that the line can be written

${\displaystyle x=-{\frac {a}{1+p}},\ y=px}$

and the ellipse can be written

${\displaystyle x=-{\frac {a(1+p)}{1-p+p^{2}}},\ y=px.}$

So the cissoid is given by

${\displaystyle x=-{\frac {a}{1+p}}+{\frac {a(1+p)}{1-p+p^{2}}}={\frac {3ap}{1+p^{3}}},\ y=px}$

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

• "Cissoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Cissoid". MathWorld.
• 2D Curves