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 defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.
Let the radius of be . By translation and rotation, we may take to be the origin and the center of the circle to be (a, 0), so is . Then the polar equations of and are:
\begin{align} &r=2a\sec\theta\\ &r=2a\cos\theta. \end{align}
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 on and . In other words, the polar equation of the cissoid is
r=2a\sec\theta-2a\cos\theta=2a(\sec\theta-\cos\theta).
r=2a\sin2\thetan/\cos\theta=2a\sin\theta\tan\theta.
Let in the above equation. Then
\begin{align} &x=r\cos\theta=2a\sin2\theta=
| 2a\tan2\theta | |
| \sec2\theta |
=
| 2at2 | |
| 1+t2 |
\\ &y=tx=
| 2at3 | |
| 1+t2 |
\end{align}
Converting the polar form to Cartesian coordinates produces
(x2+y2)x=2ay2
A compass-and-straightedge construction of various points on the cissoid proceeds as follows. Given a line and a point not on, construct the line through parallel to . Choose a variable point on, and construct, the orthogonal projection of on, then, the orthogonal projection of on . Then the cissoid is the locus of points .
To see this, let be the origin and the line as above. Let be the point ; then is and the equation of the line is . The line through perpendicular to is
t(y-2at)+x=0.
To find the point of intersection, set in this equation to get
\begin{align} &t(tx-2at)+x=0, x(t2+1)=2at
| ||||
\\ &y=tx=
| 2at3 | |
| t2+1 |
\end{align}
While this construction produces arbitrarily many points on the cissoid, it cannot trace any continuous segment of the curve.
The following construction was given by Isaac Newton. Let be a line and a point not on . Let be a right angle which moves so that equals the distance from to and remains on, while the other leg slides along . Then the midpoint of describes the curve.
To see this,[1] let the distance between and be . By translation and rotation, take and the line . Let and let be the angle between and the -axis; this is equal to the angle between and . By construction,, so the distance from to is . In other words . Also, is the -coordinate of if it is rotated by angle, so . After simplification, this produces parametric equations
| x=a(1-\sin\psi),y=a | (1-\sin\psi)2 |
| \cos\psi |
.
| x=a(1-\cos\psi),y=a | (1-\cos\psi)2 |
| \sin\psi |
x=2a\sin2{\psi\over2},y=a
| 4\sin4{\psi\over2 | |
| r=2a | \sin2\theta |
| \cos\theta |
Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve.
The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratio. This means that given lengths and, the curve can be used to find and so that is to as is to as is to, i.e., as discovered by Hippocrates of Chios. As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume? Specifically, if is the side of a cube, and, then the volume of a cube of side is
u3=a
| ||||
\right)3=a
| |||||
| \right)\left( |
| v | \right)\left( | |
| u |
| b | |
| v |
| ||||
| \right)=a |
\right)=2a3
Let and be given. It is required to find so that, giving and as the mean proportionals. Let the cissoid
(x2+y2)x=2ay2
To see this,[2] rewrite the equation of the curve as
| ||||
| y |
| ||||
| \overline{PN} |
| \overline{PN | |
| 3}{\overline{ON} |
| ||||
| \overline{UC | |
| 3}{\overline{OC} |
| ||||
| u3 | = | |
| a3 |
| b | |
| a |
,u3=a2b
Once the finite set of points on the cissoid have been drawn, then line will 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 line, 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 by Zeno'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 new axioms, 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.
The pedal curve of 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 congruent parabolas are set vertex-to-vertex and one is rolled along the other; the vertex of the rolling parabola will trace the cissoid.
The cissoid of Diocles can also be defined as the inverse curve of a parabola with the center of inversion at the vertex. To see this, take the parabola to be x = y2, in polar coordinate
r\cos\theta=(r\sin\theta)2
| r= | \cos\theta |
| \sin2\theta |
.
The inverse curve is thus:
| r= | \sin2\theta |
| \cos\theta |
=\sin\theta\tan\theta,
which agrees with the polar equation of the cissoid above.