Cissoid

Equations

If C₁ and C₂ are given in polar coordinates by r = f 1 ( θ ) and r = f 2 ( θ ) respectively, then the equation r = f 2 ( θ ) − f 1 ( θ ) 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 ( θ + π ) ,   r = − f 1 ( θ − π ) ,   r = f 1 ( θ + 2 π ) ,   r = f 1 ( θ − 2 π ) ,   … .

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

r = f 2 ( θ ) − f 1 ( θ ) ,   r = f 2 ( θ ) + f 1 ( θ + π ) ,   r = f 2 ( θ ) + f 1 ( θ − π ) ,   r = f 2 ( θ ) − f 1 ( θ + 2 π ) ,   r = f 2 ( θ ) − f 1 ( θ − 2 π ) ,   … .

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.

For example, let C₁ and C₂ both be the ellipse

r = 1 2 − cos ⁡ θ .

The first branch of the cissoid is given by

r = 1 2 − cos ⁡ θ − 1 2 − cos ⁡ θ = 0 ,

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

r = − 1 2 + cos ⁡ θ ,

so a second branch of the cissoid is given by

r = 1 2 − cos ⁡ θ + 1 2 + cos ⁡ θ

which is an oval shaped curve.

If each C₁ and C₂ are given by the parametric equations

x = f 1 ( p ) ,   y = p x

and

x = f 2 ( p ) ,   y = p x ,

then the cissoid relative to the origin is given by

x = f 2 ( p ) − f 1 ( p ) ,   y = p x .

Specific cases

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

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 = a 1 cos ⁡ ( θ − α 1 )

and

r = a 2 cos ⁡ ( θ − α 2 ) .

By rotation through angle ( α 1 − α 2 ) / 2 , we can assume that α 1 = α ,   α 2 = − α . Then the cissoid of C₁ and C₂ relative to the origin is given by

r = a 2 cos ⁡ ( θ + α ) − a 1 cos ⁡ ( θ − α ) = a 2 cos ⁡ ( θ − α ) − a 1 cos ⁡ ( θ + α ) cos ⁡ ( θ + α ) cos ⁡ ( θ − α ) = ( a 2 cos ⁡ α − a 1 cos ⁡ α ) cos ⁡ θ − ( a 2 sin ⁡ α + a 1 sin ⁡ α ) sin ⁡ θ cos 2 ⁡ α   cos 2 ⁡ θ − sin 2 ⁡ α   sin 2 ⁡ θ .

Combining constants gives

r = b cos ⁡ θ + c sin ⁡ θ cos 2 ⁡ θ − m 2 sin 2 ⁡ θ

which in Cartesian coordinates is

x 2 − m 2 y 2 = b x + c y .

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

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:

The Trisectrix of Maclaurin given by

is the cissoid of the circle ( x + a ) 2 + y 2 = a 2 and the line x = − a 2 relative to the origin.

The right strophoid

is the cissoid of the circle ( x + a ) 2 + y 2 = a 2 and the line x = − a relative to the origin.

The cissoid of Diocles

is the cissoid of the circle ( x + a ) 2 + y 2 = a 2 and the line x = − 2 a 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 = k a , where k is a parameter, is called a Conchoid of de Sluze. (These curves are not actually concoids.) This family includes the previous examples.

The folium of Descartes

is the cissoid of the ellipse x 2 − x y + 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 = − a 1 + p ,   y = p x and the ellipse can be written So the cissoid is given by which is a parametric form of the folium.