In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a cissoid (() is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that
(There are actually two such points but is chosen so that is in the same direction from as is from .) Then the locus of such points is defined to be the cissoid of the curves , relative to .
Slightly different but essentially equivalent definitions are used by different authors. For example, may be defined to be the point so that
This is equivalent to the other definition if is replaced by its
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
through . Or may be defined as the midpoint of and ; this produces the curve generated by the previous curve scaled by a factor of 1/2.
Equations
If and are given in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
by
and
respectively, then the equation
describes the cissoid of and 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, is 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 of and , which of these equations can be eliminated due to duplication.
For example, let and both 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 each and are given by the parametric equations
:
and
:
then the cissoid relative to the origin is given by
:
Specific cases
When is a circle with center then the cissoid is
conchoid of .
When and are parallel lines then the cissoid is a third line parallel to the given lines.
Hyperbolas
Let and be two non-parallel lines and let be the origin. Let the polar equations of and be
:
and
:
By rotation through angle
we can assume that
Then the cissoid of and relative 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.
Cissoids of Zahradnik
A cissoid of Zahradnik (named after
Karel Zahradnik
Karel Zahradnik (1848–1916) was a renowned Czech mathematician at the University of Zagreb. In his 23 years of productive activity in Zagreb he wrote several significant scholarly works, mainly concerned with algebraic curves
In mathematics, ...
) is defined as the cissoid of a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
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
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a ...
given by
::
:is the cissoid of the circle
and the line
relative to the origin.
* The
right strophoid
In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from ...
::
:is the cissoid of the circle
and the line
relative to the origin.
* The
cissoid of Diocles
In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be de ...
::
:is the cissoid of the circle
and the line
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
and the line
where is a parameter, is called a
Conchoid of de Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze..
The curves are defined by the polar equation
:r=\sec\theta+a\cos\theta \,.
In cartes ...
. (These curves are not actually conchoids.) This family includes the previous examples.
*The
folium of Descartes
In geometry, the folium of Descartes (; named for René Decartes) is an algebraic curve defined by the implicit equation
:x^3 + y^3 - 3 a x y = 0.
History
The curve was first proposed and studied by René Descartes in 1638. Its claim to fam ...
::
:is the cissoid of the
ellipse and the line
relative to the origin. To see this, note that the line can be written
::
:and the ellipse can be written
::
:So the cissoid is given by
::
:which is a parametric form of the folium.
See also
*
Conchoid
*
Strophoid
References
*
C. A. Nelson "Note on rational plane cubics" ''Bull. Amer. Math. Soc.'' Volume 32, Number 1 (1926), 71-76.
External links
*
* {{MathWorld, urlname=Cissoid, title=Cissoid
Curves
Algebraic curves
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