Cissoid

File:Allgemeine zissoide english.svg

Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend

In geometry, a cissoid (Template:Ety) is a plane curve generated from two given curves Template:Math, Template:Math and a point Template:Mvar (the pole). Let Template:Mvar be a variable line passing through Template:Mvar and intersecting Template:Math at Template:Math and Template:Math at Template:Math. Let Template:Mvar be the point on Template:Mvar so that <math>\overline{OP} = \overline{P_1 P_2}.</math> (There are actually two such points but Template:Mvar is chosen so that Template:Mvar is in the same direction from Template:Mvar as Template:Math is from Template:Math.) Then the locus of such points Template:Mvar is defined to be the cissoid of the curves Template:Math, Template:Math 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 <math>\overline{OP} = \overline{OP_1} + \overline{OP_2}.</math> This is equivalent to the other definition if Template:Math is replaced by its reflection through Template:Mvar. Or Template:Mvar may be defined as the midpoint of Template:Math and Template:Math; this produces the curve generated by the previous curve scaled by a factor of 1/2.

Equations

If Template:Math and Template:Math are given in polar coordinates by <math>r=f_1(\theta)</math> and <math>r=f_2(\theta)</math> respectively, then the equation <math>r=f_2(\theta)-f_1(\theta)</math> describes the cissoid of Template:Math and Template:Math 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, Template:Math is also given by

<math> \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}</math> So the cissoid is actually the union of the curves given by the equations

<math>\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}</math> It can be determined on an individual basis depending on the periods of Template:Math and Template:Math, which of these equations can be eliminated due to duplication.

File:CissoidExample01.svg

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

For example, let Template:Math and Template:Math both be the ellipse

<math>r=\frac{1}{2-\cos \theta}.</math>

The first branch of the cissoid is given by

<math>r=\frac{1}{2-\cos \theta}-\frac{1}{2-\cos \theta}=0,</math>

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

<math>r=\frac{-1}{2+\cos \theta},</math>

so a second branch of the cissoid is given by

<math>r=\frac{1}{2-\cos \theta}+\frac{1}{2+\cos \theta}</math>

which is an oval shaped curve.

If each Template:Math and Template:Math are given by the parametric equations

<math>x = f_1(p),\ y = px</math>

and

<math>x = f_2(p),\ y = px,</math>

then the cissoid relative to the origin is given by

<math>x = f_2(p)-f_1(p),\ y = px.</math>

Specific cases

When Template:Math is a circle with center Template:Mvar then the cissoid is conchoid of Template:Math.

When Template:Math and Template:Math are parallel lines then the cissoid is a third line parallel to the given lines.

Hyperbolas

Let Template:Math and Template:Math be two non-parallel lines and let Template:Mvar be the origin. Let the polar equations of Template:Math and Template:Math be

<math>r=\frac{a_1}{\cos (\theta-\alpha_1)}</math>

and

<math>r=\frac{a_2}{\cos (\theta-\alpha_2)}.</math>

By rotation through angle <math>\tfrac{\alpha_1-\alpha_2}{2},</math> we can assume that <math>\alpha_1 = \alpha,\ \alpha_2 = -\alpha.</math> Then the cissoid of Template:Math and Template:Math relative to the origin is given by

<math>\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}</math> Combining constants gives

<math>r=\frac{b\cos\theta+c\sin\theta}{\cos^2\theta-m^2\sin^2\theta}</math>

which in Cartesian coordinates is

<math>x^2-m^2y^2=bx+cy.</math>

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

<math>2x(x^2+y^2)=a(3x^2-y^2)</math>

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

• The right strophoid

<math>y^2(a+x) = x^2(a-x)</math>

is the cissoid of the circle <math>(x+a)^2+y^2 = a^2</math> and the line <math>x=-a</math> relative to the origin.

File:Zissoide des diokles2.gif

Animation visualizing the Cissoid of Diocles

• The cissoid of Diocles

<math>x(x^2+y^2)+2ay^2=0</math>

is the cissoid of the circle <math>(x+a)^2+y^2 = a^2</math> and the line <math>x=-2a</math> 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 <math>(x+a)^2+y^2 = a^2</math> and the line <math>x=ka,</math> 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

<math>x^3+y^3=3axy</math>

is the cissoid of the ellipse <math>x^2-xy+y^2 = -a(x+y)</math> and the line <math>x+y=-a</math> relative to the origin. To see this, note that the line can be written

<math>x=-\frac{a}{1+p},\ y=px</math>

and the ellipse can be written

<math>x=-\frac{a(1+p)}{1-p+p^2},\ y=px.</math>

So the cissoid is given by

<math>x=-\frac{a}{1+p}+\frac{a(1+p)}{1-p+p^2} = \frac{3ap}{1+p^3},\ y=px</math>

which is a parametric form of the folium.

See also

• Conchoid
• Strophoid

References

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

External links

• Template:Springer
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Cissoid%7CCissoid.html}} |title = Cissoid |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• 2D Curves