Cissoid

File:Allgemeine zissoide english.svg

  Cissoid

  Curve C₁

  Curve C₂

  Pole Template:Mvar

In geometry, a cissoid (; Template:Ety) is a plane curve generated from two given curves C₁, C₂ and a point Template:Mvar (the pole). Let Template:Mvar be a variable line passing through Template:Mvar and intersecting C₁ at P₁ and C₂ at P₂. Let Template:Mvar be the point on Template:Mvar so that $\overline{OP}=\overline{P_1{P}_2}.$ (There are actually two such points but Template:Mvar is chosen so that Template:Mvar is in the same direction from Template:Mvar as P₂ is from P₁.) Then the locus of such points Template:Mvar is defined to be the cissoid of the curves C₁, C₂ relative to Template:Mvar.

Slightly different but essentially equivalent definitions are used by different authors. For example, Template:Mvar may be defined to be the point so that $\overline{OP}=\overline{O{P}_1}+\overline{O{P}_2}.$ This is equivalent to the other definition if C₁ is replaced by its reflection through Template:Mvar. Or Template:Mvar 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 $r=f_1(\theta )$ 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

$\begin{array}{rl}& r=-f_1(\theta +\pi )\\ & r=-f_1(\theta -\pi )\\ & r=f_1(\theta +2\pi )\\ & r=f_1(\theta -2\pi )\\ & ⋮\end{array}$

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

$\begin{array}{rl}& r=f_2(\theta )-f_1(\theta )\\ & r=f_2(\theta )+f_1(\theta +\pi )\\ & r=f_2(\theta )+f_1(\theta -\pi )\\ & r=f_2(\theta )-f_1(\theta +2\pi )\\ & r=f_2(\theta )-f_1(\theta -2\pi )\\ & ⋮\end{array}$

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.

File:CissoidExample01.svg

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 Template:Mvar 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 Template:Mvar 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 $\frac{{\alpha }_1-{\alpha }_2}{2},$ we can assume that ${\alpha }_1=\alpha ,{\alpha }_2=-\alpha .$ Then the cissoid of C₁ and C₂ relative to the origin is given by

$\begin{array}{rl}r& =\frac{a_2}{\cos (\theta +\alpha )}-\frac{a_1}{\cos (\theta -\alpha )}\\ & =\frac{a_2\cos (\theta -\alpha )-a_1\cos (\theta +\alpha )}{\cos (\theta +\alpha )\cos (\theta -\alpha )}\\ & =\frac{(a_2\cos \alpha -a_1\cos \alpha )\cos \theta -(a_2\sin \alpha +a_1\sin \alpha )\sin \theta }{{\cos }^2\alpha {\cos }^2\theta -{\sin }^2\alpha {\sin }^2\theta }.\end{array}$

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 $x=-\frac{a}{2}$ 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.

File:Zissoide des diokles2.gif

Animation visualizing the Cissoid of Diocles

• The cissoid of Diocles

$x(x^2+y^2)+2a{y}^2=0$

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 Template:Mvar 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.

See also

• Conchoid
• Strophoid

References

• C. A. Nelson "Note on rational plane cubics" Bull. Amer. Math. Soc. Volume 32, Number 1 (1926), 71-76.

External links

• Template:Springer
• 2D Curves