Hyperbolic spiral
r = a/θ: it winds in towards the pole forever while its far end runs along the line y = a.
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Formulas and key values
- Polar equation
- Asymptote
a sets the scale (a = 1 in the graph) and is also the height of the asymptote.
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Start the video01 · Definition
What is the Hyperbolic spiral?
Pierre Varignon studied the hyperbolic spiral in 1704. Its equation r = a/θ is the Archimedean spiral turned inside out: the radius shrinks in proportion to 1/θ, so the curve circles the pole infinitely many times without ever reaching it.
At the other end something unexpected happens. As θ goes to 0 the radius grows without bound, but the height r·sin θ = a·sin θ/θ tends to a, so the curve settles onto the horizontal asymptote y = a.
02 · Properties
Key properties
- Horizontal asymptote y = a as θ → 0.
- Winds around the pole infinitely often as θ → ∞, never reaching it.
- Inversion in the unit circle maps it to the Archimedean spiral r = θ/a.
- Each turn shrinks the radius by a factor θ/(θ + 2π), not a constant ratio.
- Studied by Pierre Varignon in 1704.
03 · Questions
Hyperbolic spiral: common questions
- What is the equation of the Hyperbolic spiral?
- Polar equation: r = a/θ. a sets the scale (a = 1 in the graph) and is also the height of the asymptote.
04 · Keep exploring