Quadratrix of Hippias
Where a falling line meets a turning radius: one of the oldest named curves, used to trisect angles and square the circle.
Adjust parameters
Export
Formulas and key values
- Cartesian equation
- Polar equation
- Meets the x-axis atIn the graph ≈ 0.6366
- Asymptote
a is the side of Hippias’ square; the curve meets its base at x = 2a/π.
Explainer video
Make a video of the Quadratrix of Hippias
Opens the Creator with a ready-made brief for this curve. Free-form generation is in beta.
Start the video01 · Definition
What is the Quadratrix of Hippias?
Around 420 BC Hippias of Elis imagined two motions in a square of side a: the top side slides down to the bottom at constant speed while a radius from the corner turns from vertical to horizontal, also at constant speed, and both finish together. Their meeting point draws the quadratrix.
Because the descent is proportional to the angle, the curve divides angles in any ratio, so it trisects them. Dinostratus later noticed that it meets the base at x = 2a/π, which turns a length into π and squares the circle. The curve cannot be drawn with straightedge and compass, which is why it does not contradict the impossibility of those constructions.
02 · Properties
Key properties
- Cartesian form x = y·cot(πy/2a); polar form r = 2aθ/(π·sin θ).
- Meets the x-axis at x = 2a/π (Dinostratus).
- Horizontal asymptotes y = ±2a.
- Divides an angle in any ratio, so it trisects angles.
- Its inverse in a circle of radius a·√(2/π) is the cochleoid.
03 · Questions
Quadratrix of Hippias: common questions
- What is the equation of the Quadratrix of Hippias?
- Cartesian equation: x = y cot (π y)/2a; Polar equation: r = (2a θ)/(π sin θ). a is the side of Hippias’ square; the curve meets its base at x = 2a/π.
04 · Keep exploring