cissoid

If {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} are given in polar coordinates by $r=f\_1(\backslash theta)$ and $r=f\_2(\backslash theta)$ respectively, then the equation $r=f\_2(\backslash theta)-f\_1(\backslash theta)$ describes the cissoid of {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} 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, {{math|C{{sub|1}}}} is also given by

:$\backslash begin\{align\}$

& 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}

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

:$\backslash begin\{align\}$

& 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}

It can be determined on an individual basis depending on the periods of {{math|f{{sub|1}}}} and {{math|f{{sub|2}}}}, which of these equations can be eliminated due to duplication.

File:CissoidExample01.svg

For example, let {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} both be the ellipse

:$r=\backslash frac\{1\}\{2-\backslash cos\; \backslash theta\}.$

The first branch of the cissoid is given by

:$r=\backslash frac\{1\}\{2-\backslash cos\; \backslash theta\}-\backslash frac\{1\}\{2-\backslash cos\; \backslash theta\}=0,$

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

:$r=\backslash frac\{-1\}\{2+\backslash cos\; \backslash theta\},$

so a second branch of the cissoid is given by

:$r=\backslash frac\{1\}\{2-\backslash cos\; \backslash theta\}+\backslash frac\{1\}\{2+\backslash cos\; \backslash theta\}$

which is an oval shaped curve.

If each {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} are given by the parametric equations

:$x\; =\; f\_1(p),\backslash \; y\; =\; px$

and

:$x\; =\; f\_2(p),\backslash \; y\; =\; px,$

then the cissoid relative to the origin is given by

:$x\; =\; f\_2(p)-f\_1(p),\backslash \; y\; =\; px.$

When {{math|C{{sub|1}}}} is a circle with center {{mvar|O}} then the cissoid is conchoid of {{math|C{{sub|2}}}}.

When {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} are parallel lines then the cissoid is a third line parallel to the given lines.

Let {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} be two non-parallel lines and let {{mvar|O}} be the origin. Let the polar equations of {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} be

:$r=\backslash frac\{a\_1\}\{\backslash cos\; (\backslash theta-\backslash alpha\_1)\}$

and

:$r=\backslash frac\{a\_2\}\{\backslash cos\; (\backslash theta-\backslash alpha\_2)\}.$

By rotation through angle $\backslash tfrac\{\backslash alpha\_1-\backslash alpha\_2\}\{2\},$ we can assume that $\backslash alpha\_1\; =\; \backslash alpha,\backslash \; \backslash alpha\_2\; =\; -\backslash alpha.$ Then the cissoid of {{math|C{{sub|1}}}} and {{math|C{{sub|2}}}} relative to the origin is given by

:$\backslash begin\{align\}$

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}

Combining constants gives

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

which in Cartesian coordinates is

:$x^2-m^2y^2=bx+cy.$

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:

• 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 $x=-\backslash tfrac\{a\}\{2\}$ 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

• 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 {{mvar|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

::$x=-\backslash frac\{a\}\{1+p\},\backslash \; y=px$

:and the ellipse can be written

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

:So the cissoid is given by

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

:which is a parametric form of the folium.

• Conchoid
• Strophoid

• {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/53 53–56] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/53 }}
• [http://projecteuclid.org/euclid.bams/1183486856 C. A. Nelson "Note on rational plane cubics" Bull. Amer. Math. Soc. Volume 32, Number 1 (1926), 71-76.]

• {{springer|title=Cissoid|id=p/c022340}}
• {{MathWorld|urlname=Cissoid|title=Cissoid}}
• [http://2dcurves.com/derived/cissoid.html 2D Curves]

Category:Curves

Category:Algebraic curves

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