Strophoid
The right strophoid y²(a − x) = x²(a + x): a loop through the origin and two arms reaching for the asymptote x = a.
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Formulas and key values
- Cartesian equation
- Polar equation
- Asymptote
- Area · inside the loopIn the graph ≈ 0.4292
a sets the size: the loop passes through (−a, 0) and the asymptote is x = a.
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Start the video01 · Definition
What is the Strophoid?
The right strophoid is a cubic curve with a node: it crosses itself at the origin, where its two tangents are the lines y = ±x. On the left it closes into a loop through (−a, 0); on the right its two arms climb towards the vertical asymptote x = a. Its name comes from the Greek for a twisted band.
The polar form r = −a·cos 2θ / cos θ shows the whole shape at once: r is zero at θ = ±45°, which is the node, and grows without bound as θ approaches ±90°, which is the asymptote.
02 · Properties
Key properties
- Node at the origin with tangents y = ±x.
- Vertical asymptote x = a.
- The loop passes through (−a, 0) and encloses area (2 − π/2)a².
- Polar form: r = −a·cos 2θ / cos θ.
- Symmetric about the x-axis.
03 · Questions
Strophoid: common questions
- What is the equation of the Strophoid?
- Cartesian equation: y^2(a - x) = x^2(a + x); Polar equation: r = -a (cos 2θ)/(cos θ). a sets the size: the loop passes through (−a, 0) and the asymptote is x = a.
- What is the area of the Strophoid?
- Area (inside the loop): A = (2 - π/2)a^2. a sets the size: the loop passes through (−a, 0) and the asymptote is x = a.
04 · Keep exploring