View source for Cissoid

{{short description|Plane curve constructed from two other curves and a fixed point}}
{{Distinguish|Cisoidal oscillation|Cisoid (disambiguation)}}
[[File:Allgemeine zissoide_english.svg|thumb|upright=1.5|
{{legend-line|solid red|Cissoid}}
{{legend-line|solid green|Curve {{math|''C''{{sub|1}}}}}}
{{legend-line|solid blue|Curve {{math|''C''{{sub|2}}}}}}
{{legend|black|Pole {{mvar|O}}}}]]

In [[geometry]], a '''cissoid''' ({{ety|grc|''κισσοειδής'' (kissoeidēs)|[[ivy]]-shaped}}) is a [[plane curve]] generated from two given curves {{math|''C''{{sub|1}}}}, {{math|''C''{{sub|2}}}} and a point {{mvar|O}} (the '''pole'''). Let {{mvar|L}} be a variable line passing through {{mvar|O}} and intersecting {{math|''C''{{sub|1}}}} at {{math|''P''{{sub|1}}}} and {{math|''C''{{sub|2}}}} at {{math|''P''{{sub|2}}}}. Let {{mvar|P}} be the point on {{mvar|L}} so that <math>\overline{OP} = \overline{P_1 P_2}.</math> (There are actually two such points but {{mvar|P}} is chosen so that {{mvar|P}} is in the same direction from {{mvar|O}} as {{math|''P''{{sub|2}}}} is from {{math|''P''{{sub|1}}}}.) Then the locus of such points {{mvar|P}} is defined to be the cissoid of the curves {{math|''C''{{sub|1}}}}, {{math|''C''{{sub|2}}}} relative to {{mvar|O}}.

Slightly different but essentially equivalent definitions are used by different authors. For example, {{mvar|P}} 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 {{math|''C''{{sub|1}}}} is replaced by its [[Point reflection|reflection]] through {{mvar|O}}. Or {{mvar|P}} may be defined as the midpoint of {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}}; this produces the curve generated by the previous curve scaled by a factor of 1/2.

==Equations==
If {{math|''C''{{sub|1}}}} and {{math|''C''{{sub|2}}}} 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 {{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 
:<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 {{math|''f''{{sub|1}}}} and {{math|''f''{{sub|2}}}}, which of these equations can be eliminated due to duplication.

[[File:CissoidExample01.svg|thumb|Ellipse <math>r=\frac{1}{2-\cos \theta}</math> in red, with its two cissoid branches in black and blue (origin)]]
For example, let {{math|''C''{{sub|1}}}} and {{math|''C''{{sub|2}}}} 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 {{math|''C''{{sub|1}}}} and {{math|''C''{{sub|2}}}} 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 {{math|''C''{{sub|1}}}} is a circle with center {{mvar|O}} then the cissoid is [[conchoid (mathematics)|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.

===Hyperbolas===
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 
:<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 {{math|''C''{{sub|1}}}} and {{math|''C''{{sub|2}}}} 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|thumb|upright=1.25|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 {{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]]
::<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 (mathematics)|Conchoid]]
*[[Strophoid]]

==References==
* {{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.]

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

[[Category:Curves]]
[[Category:Algebraic curves]]

[[zh:蔓叶线]]