Cissoid (original) (raw)

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Plane curve constructed from two other curves and a fixed point

Cissoid

Curve _C_1

Curve _C_2

Pole O

In geometry, a cissoid (; from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped') is a plane curve generated from two given curves _C_1, _C_2 and a point O (the pole). Let L be a variable line passing through O and intersecting _C_1 at _P_1 and _C_2 at _P_2. Let P be the point on L so that O P ¯ = P 1 P 2 ¯ . {\displaystyle {\overline {OP}}={\overline {P_{1}P_{2}}}.} {\displaystyle {\overline {OP}}={\overline {P_{1}P_{2}}}.} (There are actually two such points but P is chosen so that P is in the same direction from O as _P_2 is from _P_1.) Then the locus of such points P is defined to be the cissoid of the curves _C_1, _C_2 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 O P ¯ = O P 1 ¯ + O P 2 ¯ . {\displaystyle {\overline {OP}}={\overline {OP_{1}}}+{\overline {OP_{2}}}.} {\displaystyle {\overline {OP}}={\overline {OP_{1}}}+{\overline {OP_{2}}}.} This is equivalent to the other definition if _C_1 is replaced by its reflection through O. Or P may be defined as the midpoint of _P_1 and _P_2; this produces the curve generated by the previous curve scaled by a factor of 1/2.

If _C_1 and _C_2 are given in polar coordinates by r = f 1 ( θ ) {\displaystyle r=f_{1}(\theta )} {\displaystyle r=f_{1}(\theta )} and r = f 2 ( θ ) {\displaystyle r=f_{2}(\theta )} {\displaystyle r=f_{2}(\theta )} respectively, then the equation r = f 2 ( θ ) − f 1 ( θ ) {\displaystyle r=f_{2}(\theta )-f_{1}(\theta )} {\displaystyle r=f_{2}(\theta )-f_{1}(\theta )} describes the cissoid of _C_1 and _C_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, _C_1 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 ) ⋮ {\displaystyle {\begin{aligned}&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{aligned}}} {\displaystyle {\begin{aligned}&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{aligned}}}

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 ) ⋮ {\displaystyle {\begin{aligned}&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{aligned}}} {\displaystyle {\begin{aligned}&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{aligned}}}

It can be determined on an individual basis depending on the periods of _f_1 and _f_2, which of these equations can be eliminated due to duplication.

Ellipse r = 1 2 − cos ⁡ θ {\displaystyle r={\frac {1}{2-\cos \theta }}} {\displaystyle r={\frac {1}{2-\cos \theta }}} in red, with its two cissoid branches in black and blue (origin)

For example, let _C_1 and _C_2 both be the ellipse

r = 1 2 − cos ⁡ θ . {\displaystyle r={\frac {1}{2-\cos \theta }}.} {\displaystyle r={\frac {1}{2-\cos \theta }}.}

The first branch of the cissoid is given by

r = 1 2 − cos ⁡ θ − 1 2 − cos ⁡ θ = 0 , {\displaystyle r={\frac {1}{2-\cos \theta }}-{\frac {1}{2-\cos \theta }}=0,} {\displaystyle r={\frac {1}{2-\cos \theta }}-{\frac {1}{2-\cos \theta }}=0,}

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

r = − 1 2 + cos ⁡ θ , {\displaystyle r={\frac {-1}{2+\cos \theta }},} {\displaystyle r={\frac {-1}{2+\cos \theta }},}

so a second branch of the cissoid is given by

r = 1 2 − cos ⁡ θ + 1 2 + cos ⁡ θ {\displaystyle r={\frac {1}{2-\cos \theta }}+{\frac {1}{2+\cos \theta }}} {\displaystyle r={\frac {1}{2-\cos \theta }}+{\frac {1}{2+\cos \theta }}}

which is an oval shaped curve.

If each _C_1 and _C_2 are given by the parametric equations

x = f 1 ( p ) , y = p x {\displaystyle x=f_{1}(p),\ y=px} {\displaystyle x=f_{1}(p),\ y=px}

and

x = f 2 ( p ) , y = p x , {\displaystyle x=f_{2}(p),\ y=px,} {\displaystyle x=f_{2}(p),\ y=px,}

then the cissoid relative to the origin is given by

x = f 2 ( p ) − f 1 ( p ) , y = p x . {\displaystyle x=f_{2}(p)-f_{1}(p),\ y=px.} {\displaystyle x=f_{2}(p)-f_{1}(p),\ y=px.}

When _C_1 is a circle with center O then the cissoid is conchoid of _C_2.

When _C_1 and _C_2 are parallel lines then the cissoid is a third line parallel to the given lines.

Let _C_1 and _C_2 be two non-parallel lines and let O be the origin. Let the polar equations of _C_1 and _C_2 be

r = a 1 cos ⁡ ( θ − α 1 ) {\displaystyle r={\frac {a_{1}}{\cos(\theta -\alpha _{1})}}} {\displaystyle r={\frac {a_{1}}{\cos(\theta -\alpha _{1})}}}

and

r = a 2 cos ⁡ ( θ − α 2 ) . {\displaystyle r={\frac {a_{2}}{\cos(\theta -\alpha _{2})}}.} {\displaystyle r={\frac {a_{2}}{\cos(\theta -\alpha _{2})}}.}

By rotation through angle α 1 − α 2 2 , {\displaystyle {\tfrac {\alpha _{1}-\alpha _{2}}{2}},} {\displaystyle {\tfrac {\alpha _{1}-\alpha _{2}}{2}},} we can assume that α 1 = α , α 2 = − α . {\displaystyle \alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha .} {\displaystyle \alpha _{1}=\alpha ,\ \alpha _{2}=-\alpha .} Then the cissoid of _C_1 and _C_2 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 . {\displaystyle {\begin{aligned}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{aligned}}} {\displaystyle {\begin{aligned}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{aligned}}}

Combining constants gives

r = b cos ⁡ θ + c sin ⁡ θ cos 2 ⁡ θ − m 2 sin 2 ⁡ θ {\displaystyle r={\frac {b\cos \theta +c\sin \theta }{\cos ^{2}\theta -m^{2}\sin ^{2}\theta }}} {\displaystyle r={\frac {b\cos \theta +c\sin \theta }{\cos ^{2}\theta -m^{2}\sin ^{2}\theta }}}

which in Cartesian coordinates is

x 2 − m 2 y 2 = b x + c y . {\displaystyle x^{2}-m^{2}y^{2}=bx+cy.} {\displaystyle x^{2}-m^{2}y^{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.

Cissoids of Zahradnik

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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:

2 x ( x 2 + y 2 ) = a ( 3 x 2 − y 2 ) {\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})} {\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})}

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

y 2 ( a + x ) = x 2 ( a − x ) {\displaystyle y^{2}(a+x)=x^{2}(a-x)} {\displaystyle y^{2}(a+x)=x^{2}(a-x)}

is the cissoid of the circle ( x + a ) 2 + y 2 = a 2 {\displaystyle (x+a)^{2}+y^{2}=a^{2}} {\displaystyle (x+a)^{2}+y^{2}=a^{2}} and the line x = − a {\displaystyle x=-a} {\displaystyle x=-a} relative to the origin.

Animation visualizing the Cissoid of Diocles

x ( x 2 + y 2 ) + 2 a y 2 = 0 {\displaystyle x(x^{2}+y^{2})+2ay^{2}=0} {\displaystyle x(x^{2}+y^{2})+2ay^{2}=0}

is the cissoid of the circle ( x + a ) 2 + y 2 = a 2 {\displaystyle (x+a)^{2}+y^{2}=a^{2}} {\displaystyle (x+a)^{2}+y^{2}=a^{2}} and the line x = − 2 a {\displaystyle x=-2a} {\displaystyle 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.

x 3 + y 3 = 3 a x y {\displaystyle x^{3}+y^{3}=3axy} {\displaystyle x^{3}+y^{3}=3axy}

is the cissoid of the ellipse x 2 − x y + y 2 = − a ( x + y ) {\displaystyle x^{2}-xy+y^{2}=-a(x+y)} {\displaystyle x^{2}-xy+y^{2}=-a(x+y)} and the line x + y = − a {\displaystyle x+y=-a} {\displaystyle x+y=-a} relative to the origin. To see this, note that the line can be written

x = − a 1 + p , y = p x {\displaystyle x=-{\frac {a}{1+p}},\ y=px} {\displaystyle x=-{\frac {a}{1+p}},\ y=px}

and the ellipse can be written

x = − a ( 1 + p ) 1 − p + p 2 , y = p x . {\displaystyle x=-{\frac {a(1+p)}{1-p+p^{2}}},\ y=px.} {\displaystyle x=-{\frac {a(1+p)}{1-p+p^{2}}},\ y=px.}

So the cissoid is given by

x = − a 1 + p + a ( 1 + p ) 1 − p + p 2 = 3 a p 1 + p 3 , y = p x {\displaystyle x=-{\frac {a}{1+p}}+{\frac {a(1+p)}{1-p+p^{2}}}={\frac {3ap}{1+p^{3}}},\ y=px} {\displaystyle 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.