Ingeometry,acissoid(fromAncient Greekκισσοειδής(kissoeidēs)'ivy-shaped') is aplane curvegenerated from two given curvesC1,C2and a pointO(thepole). LetLbe a variable line passing throughOand intersectingC1atP1andC2atP2.LetPbe the point onLso that(There are actually two such points butPis chosen so thatPis in the same direction fromOasP2is fromP1.) Then the locus of such pointsPis defined to be the cissoid of the curvesC1,C2relative toO.
Slightly different but essentially equivalent definitions are used by different authors. For example,Pmay be defined to be the point so thatThis is equivalent to the other definition ifC1is replaced by itsreflectionthroughO.OrPmay be defined as the midpoint ofP1andP2;this produces the curve generated by the previous curve scaled by a factor of 1/2.
IfC1andC2are given inpolar coordinatesbyandrespectively, then the equationdescribes the cissoid ofC1andC2relative 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,C1is also given by
So the cissoid is actually the union of the curves given by the equations
It can be determined on an individual basis depending on the periods off1andf2,which of these equations can be eliminated due to duplication.
For example, letC1andC2both be the ellipse
The first branch of the cissoid is given by
which is simply the origin. The ellipse is also given by
so a second branch of the cissoid is given by
which is an oval shaped curve.
If eachC1andC2are given by the parametric equations
and
then the cissoid relative to the origin is given by
LetC1andC2be two non-parallel lines and letObe the origin. Let the polar equations ofC1andC2be
and
By rotation through anglewe can assume thatThen the cissoid ofC1andC2relative to the origin is given by
Combining constants gives
which in Cartesian coordinates is
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.
Acissoid of Zahradnik(named afterKarel Zahradnik) is defined as the cissoid of aconic sectionand a line relative to any point on the conic. This is a broad family of rational cubic curves containing several well-known examples. Specifically:
is the cissoid of the circleand the linerelative 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 circleand the linewherekis a parameter, is called aConchoid of de Sluze.(These curves are not actually conchoids.) This family includes the previous examples.