Diocles ( 250 – ∼100 BC) invented this curve to solve the doubling of the cube problem (also know as the the Delian problem). The name cissoid (ivy-shaped) derives from the shape of the curve. Later the method used to generate this curve was generalized, and we call all curves generated in a similar way cissoids. The name first appears in the work of Geminus about 100 years later. Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was $3\pi a^2$. From a given point there are either one or three tangents to the cissoid. The Cissoid of Diocles is a special case of the general cissoid. It is a cissoid of a circle and a line tangent to the circle with respect to a point on the circle opposite to the tangent point. Here is a step-by-step description of the construction: - Let there be given a circle C and a line L tangent to this circle.
- Let O be the point on the circle opposite to the tangent point.
- Let $P_1$ be a point on the circle C.
- Let $P_2$ be the intersection of line and $\left [ O,P_1 \right ]$ and L.
- Let Q be a point on line $\left [ O,P_1 \right ]$ such that $dist\left [ O,Q \right ]=dist\left [ P_1,P_2 \right ]$.
- The locus of Q (as $P_1$ moves on C) is the cissoid of Diocles.
**Click "Read More" to see interactive graphic ** | Equations: Cartesian Equation: $\displaystyle y^2=\frac{x^3}{2a-x}$ Parametric Equations: $\displaystyle \begin{matrix} x=2a\sin^2{t}\\ y=\frac{2a\sin^3{t}}{\cos{t}} \end{matrix}$ from $\displaystyle -\frac{\pi}{2}<t<\frac{\pi}{2}$ and/or $\displaystyle \begin{matrix} x=\frac{at^2}{1+t^2}\\ y=\frac{at^3}{1+t^2} \end{matrix}$ from $\displaystyle -\infty <t<\infty$ Polar Equation: $\displaystyle r=2a\tan{\theta}\sin{\theta}$ Graphs: |

**Interactive Graphic:**

**References:**

- "Cissoid of Diocles."
*Wikipedia*. Wikimedia Foundation, 18 Mar. 2014. Web. 22 Mar. 2014. - Weisstein, Eric W. "Cissoid of Diocles."
*Wolfram MathWorld*. Wolfram Research, Inc., 26 Oct. 2012. Web. 22 Mar. 2014.