Cochleoid
r = a·sin θ / θ: a snail-shell spiral that passes through the pole at every multiple of π.
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Formulas and key values
- Polar equation
- Cartesian equation
- Returns to the pole at
a is the radius at θ = 0 (a = 1 in the graph); θ runs both ways from 0.
Explainer video
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Start the video01 · Definition
What is the Cochleoid?
The cochleoid (Latin cochlea, snail shell) is the polar graph of sin θ / θ, the function known elsewhere as sinc. At θ = 0 the radius is a; every time θ reaches a multiple of π the radius is zero, so the curve keeps returning to the pole in ever smaller loops.
It is the inverse of the quadratrix of Hippias: multiplying the two radii, a·sin θ/θ and 2aθ/(π sin θ), gives the constant 2a²/π. Its loops are all tangent to one another at the pole.
02 · Properties
Key properties
- Polar form r = a·sin θ / θ, with r = a at θ = 0.
- Passes through the pole whenever θ is a nonzero multiple of π.
- Cartesian form: (x² + y²)·arctan(y/x) = ay.
- Inversion turns it into the quadratrix of Hippias (product of radii 2a²/π).
- Symmetric about the x-axis.
03 · Questions
Cochleoid: common questions
- What is the equation of the Cochleoid?
- Polar equation: r = (a sin θ)/θ; Cartesian equation: (x^2 + y^2) arctan y/x = ay. a is the radius at θ = 0 (a = 1 in the graph); θ runs both ways from 0.
04 · Keep exploring