Cissoid of Diocles

 

Here is the definition of cissoid of two curves.
Cissoid[4]. Let O be a fixed point and let L be a line through O intersecting the curves C1 and C2 at Q1 and Q2. The locus of points P1 and P2 on L such that OP = OQ2 - OQ1 = Q1Q2is the cissoid of C1 and C2 with respect to O.
The cissoid of Diocles is the cissoid of a circle and a tangent line, with respect to a fixed point O on the circumference opposite the point of tangency A. The screenshot below shows the cissoid drawn using Jeometry.

Let O be the origin (0, 0) and x = a be the line tangent to the circle. Let Ô be the angle BÔA in the picture above. Considering the right triangles OBA and CAO, we have

OP = OC - OB = a secÔ - a cosÔ = a sinÔ tanÔ.

Hence the polar equation of the cissoid is

r = a sinÔ tanÔ.

Then the Cartesian equation follows immediately by substitution,

y2(a - x) = x3.

This is the same equation we found when considering the pedal of a parabola with respect to its vertex (let a = -1/4k).

We would like to find a parametric rapresentation of the curve. To do that, we note that the cissoid of Diocles is a cubic curve with a cusp in the origin, so we can find a rational parametrization by intersecting the cissoid with the line y = tx where t is a parameter. Then we have to eliminate x and y one at a time from the equations

y2(a - x) = x3
y = tx

We can easily carry out the calculation by ourselves, but we can use the Groebner applet as well, if we like.
To eliminate y we find the 1st elimination ideal <f1> of <y2(a - x) = x3, y = tx> by computing the Groebner basis w.r.t. lex order with y>x>t>a, then solve f1 = 0 for x.
To eliminate x we find the 1st elimination ideal <g1> of <y2(a - x) = x3, y = tx> by computing the Groebner basis w.r.t. lex order with x>y>t>a, then solve g1 = 0 for y.

We get

  a
x= ------------
  t(t2 + 1)
   
  a
y= ------------.
  (t2 + 1)

Around 180 B.C., Diocles used this curve to solve the problem of the duplication of the cube, i.e. of trying to construct using ruler and compass. It is known that this is impossible given just a ruler and compass. Diocles showed that if in addition you allow the use of the cissoid, then one can construct . Here is how it works. The line x = -2y + a, connecting A(a, 0) to (a/2, a/4), meet the cissoid at a point (x, y) such that (x/y)3 = 2.

 

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