Cissoid

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CissoidConstruction.svg

In geometry, a cissoid is a 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 OP = P1P2. (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 OP = OP1 + OP2. 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.

The word "cissoid" comes from the Greek kissoeidēs "ivy shaped" from kissos "ivy" and -oeidēs "having the likeness of".

Equations[edit]

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

r=-f_1(\theta+\pi),\ r=-f_1(\theta-\pi),\ r=f_1(\theta+2\pi),\ r=f_1(\theta-2\pi),\ \dots.

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

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),\ \dots.

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.

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[edit]

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[edit]

Let C1 and C2 be two non-parallel lines and let O 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 (\alpha_1-\alpha2)/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

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}.

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^2y^2=bx+cy.

This is a hyperbola passing though 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[edit]

A cissoid of Zahradnik (name 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:

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={-{a \over 2}} relative to the origin.
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.
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 k is a parameter, is called a Conchoid of de Sluze. (These curves are not actually concoids.) This family includes the previous examples.
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 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[edit]

References[edit]

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