Piriform curve
The pear-shaped quartic a⁴y² = b²x³(2a − x): a teardrop with a cusp, enclosing exactly the area πab of an ellipse.
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Formulas and key values
- Cartesian equation
- Parametric equations
- AreaIn the graph ≈ 2.199
a is half the length (a = 1 in the graph: the curve runs from 0 to 2a) and b sets the width.
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Start the video01 · Definition
What is the Piriform curve?
The piriform (Latin pirum, pear) is a quartic shaped like a teardrop lying on its side: it runs from a cusp at the origin to its blunt end at (2a, 0). Near the cusp it behaves like the semicubical parabola, since y² is proportional to x³ there.
Its parametrisation x = a(1 + sin t), y = b·cos t·(1 + sin t) makes the area easy to compute, and the answer is a pleasant surprise: πab, exactly the area of the ellipse with semi-axes a and b.
02 · Properties
Key properties
- Enclosed area πab, equal to the ellipse with the same a and b.
- Cusp at the origin; the far end is (2a, 0).
- Parametrisation: x = a(1 + sin t), y = b·cos t·(1 + sin t).
- Symmetric about the x-axis.
- Near the cusp, y² ≈ (2b²/a³)·x³, like a semicubical parabola.
03 · Questions
Piriform curve: common questions
- What is the equation of the Piriform curve?
- Cartesian equation: a^4y^2 = b^2x^3(2a - x); Parametric equations: x = a(1 + sin t), y = b cos t (1 + sin t). a is half the length (a = 1 in the graph: the curve runs from 0 to 2a) and b sets the width.
- What is the area of the Piriform curve?
- Area: A = π ab. a is half the length (a = 1 in the graph: the curve runs from 0 to 2a) and b sets the width.
04 · Keep exploring