Cissoid of Diocles
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![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Cissoide2.svg/220px-Cissoide2.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Cissoid_of_Diocles.gif/220px-Cissoid_of_Diocles.gif)
Ingeometry,thecissoid of Diocles(fromAncient Greekκισσοειδής(kissoeidēs)'ivy-shaped'; named forDiocles) is acubic plane curvenotable for the property that it can be used to construct twomean proportionalsto a givenratio.In particular, it can be used todouble a cube.It can be defined as thecissoidof acircleand a linetangentto it with respect to the point on the circle opposite to the point of tangency. In fact, thecurve familyof cissoids is named for this example and some authors refer to it simply asthecissoid. It has a singlecuspat the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is anasymptote.It is a member of theconchoid of de Sluzefamily of curves and in form it resembles atractrix.
Construction and equations[edit]
Let the radius ofCbea.By translation and rotation, we may takeOto be the origin and the center of the circle to be (a,0), soAis(2a,0).Then the polar equations ofLandCare:
By construction, the distance from the origin to a point on the cissoid is equal to the difference between the distances between the origin and the corresponding points onLandC.In other words, the polar equation of the cissoid is
Applying some trigonometric identities, this is equivalent to
Lett= tanθin the above equation. Then
are parametric equations for the cissoid.
Converting the polar form to Cartesian coordinates produces
Construction by double projection[edit]
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Cissoide_mecanique.svg/220px-Cissoide_mecanique.svg.png)
Acompass-and-straightedgeconstruction of various points on the cissoid proceeds as follows. Given a lineLand a pointOnot onL,construct the lineL'throughOparallel toL.Choose a variable pointPonL,and constructQ,the orthogonal projection ofPonL',thenR,the orthogonal projection ofQonOP.Then the cissoid is the locus of pointsR.
To see this, letObe the origin andLthe linex= 2aas above. LetPbe the point(2a,2at);thenQis(0, 2at)and the equation of the lineOPisy=tx.The line throughQperpendicular toOPis
To find the point of intersectionR,sety=txin this equation to get
which are the parametric equations given above.
While this construction produces arbitrarily many points on the cissoid, it cannot trace any continuous segment of the curve.
Newton's construction[edit]
![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/32/CissoidConstructionByNewton.svg/150px-CissoidConstructionByNewton.svg.png)
The following construction was given byIsaac Newton.LetJbe a line andBa point not onJ.Let∠BSTbe a right angle which moves so thatSTequals the distance fromBtoJandTremains onJ,while the other legBSslides alongB.Then the midpointPofSTdescribes the curve.
To see this,[1]let the distance betweenBandJbe2a.By translation and rotation, takeB= (–a, 0)andJthe linex=a.LetP= (x,y)and letψbe the angle betweenSBand thex-axis; this is equal to the angle betweenSTandJ.By construction,PT=a,so the distance fromPtoJisasinψ.In other wordsa–x=asinψ.Also,SP=ais they-coordinate of(x,y)if it is rotated by angleψ,soa= (x+a) sinψ+ycosψ.After simplification, this produces parametric equations
Change parameters by replacingψwith its complement to get
or, applying double angle formulas,
But this is polar equation
given above withθ=ψ/2.
Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve.
Delian problem[edit]
The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a givenratio.This means that given lengthsaandb,the curve can be used to finduandvso thatais touasuis tovasvis tob,i.e.a/u=u/v=v/b,as discovered byHippocrates of Chios.As a special case, this can be used to solve the Delian problem: how much must the length of acubebe increased in order todoubleitsvolume?Specifically, ifais the side of a cube, andb= 2a,then the volume of a cube of sideuis
souis the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules ofcompass and straightedge constructionsince it relies on the existence of the cissoid.
Letaandbbe given. It is required to finduso thatu3=a2b,givinguandv=u2/aas the mean proportionals. Let the cissoid
be constructed as above, withOthe origin,Athe point(2a,0),andJthe linex=a,also as given above. LetCbe the point of intersection ofJwithOA.From the given lengthb,markBonJso thatCB=b.DrawBAand letP= (x,y)be the point where it intersects the cissoid. DrawOPand let it intersectJatU.Thenu=CUis the required length.
To see this,[2]rewrite the equation of the curve as
and letN= (x,0),soPNis the perpendicular toOAthroughP. From the equation of the curve,
From this,
By similar trianglesPN/ON=UC/OCandPN/NA=BC/CA.So the equation becomes
so
as required.
![](https://upload.wikimedia.org/wikipedia/commons/2/20/Cissoid_500points_220x220.gif)
Diocles did not really solve the Delian problem. The reason is that the cissoid of Diocles cannot be constructed perfectly, at least not with compass and straightedge. To construct the cissoid of Diocles, one would construct a finite number of its individual points, then connect all these points to form a curve. (An example of this construction is shown on the right.) The problem is that there is no well-defined way to connect the points. If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation. Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect. Or one could simply draw a curve directly, trying to eyeball the shape of the curve, but the result would only be imprecise guesswork.
Once the finite set of points on the cissoid have been drawn, then linePCwill probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with linePC,but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled byZeno's paradoxes).
One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical — axiomatic — consistency. Allowing construction by new tools would be like adding newaxioms,but axioms are supposed to be simple and self-evident, but such tools are not. So by the rules of classical,synthetic geometry,Diocles did not solve the Delian problem, which actually can not be solved by such means.
As a pedal curve[edit]
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/RouletteAnim2.gif/220px-RouletteAnim2.gif)
Thepedal curveof a parabola with respect to its vertex is a cissoid of Diocles.[3]The geometrical properties of pedal curves in general produce several alternate methods of constructing the cissoid. It is the envelopes of circles whose centers lie on a parabola and which pass through the vertex of the parabola. Also, if two congruentparabolasare set vertex-to-vertex and one is rolled along the other; the vertex of the rolling parabola will trace the cissoid.
Inversion[edit]
The cissoid of Diocles can also be defined as theinverse curveof a parabola with the center of inversion at the vertex. To see this, take the parabola to bex=y2,in polar coordinateor:
The inverse curve is thus:
which agrees with the polar equation of the cissoid above.
References[edit]
- ^See Basset for the derivation, many other sources give the construction.
- ^Proof is a slightly modified version of that given in Basset.
- ^J. Edwards (1892).Differential Calculus.London: MacMillan and Co. p.166,Example 3.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png)
- J. Dennis Lawrence (1972).A catalog of special plane curves.Dover Publications. pp.95, 98–100.ISBN0-486-60288-5.
- Weisstein, Eric W."Cissoid of Diocles".MathWorld.
- "Cissoid of Diocles" at Visual Dictionary Of Special Plane Curves
- "Cissoid of Diocles" at MacTutor's Famous Curves Index
- "Cissoid" on 2dcurves.com
- "Cissoïde de Dioclès ou Cissoïde Droite" at Encyclopédie des Formes Mathématiques Remarquables(in French)
- "The Cissoid"An elementary treatise on cubic and quartic curvesAlfred Barnard Basset (1901) Cambridge pp. 85ff