Cissoid of Diocles
Polar curvesThe cusped curve r = 2a·sin θ·tan θ invented around 180 BC to solve the ancient problem of doubling the cube.
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Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Cissoid of Diocles?
The cissoid of Diocles — from the Greek for “ivy-shaped” — was constructed around 180 BC as a tool for doubling the cube: finding a cube with exactly twice the volume of a given one. Compass and straightedge cannot do it, but intersecting this curve with a line yields the required ∛2 ratio exactly.
The curve has a cusp at the origin and hugs the vertical asymptote x = 2a. Geometrically it is generated from a circle of diameter 2a: for each ray from the origin, the cissoid point copies the distance between where the ray leaves the circle and where it meets the tangent line at the far end.
Key properties
- Cusp at the origin; vertical asymptote at x = 2a.
- The area between the curve and its asymptote is 3πa².
- Cartesian form: y²(2a − x) = x³, a cubic curve.
- It is the pedal-related cissoid of a circle and its tangent line, taken from the point opposite the tangency.
- Newton later showed how to draw it with two rulers and a fixed right angle.
Where you'll see it
The history of the three classical Greek construction problems, cubic-curve galleries, and mechanism design exercises.