Cassini oval
Points whose distances to two foci multiply to b²: one oval, a figure eight, or two separate loops.
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Formulas and key values
- Cartesian equation
- Polar equation
- Foci
- Product of focal distances
The foci are (±c, 0) with c = 1 in the graph; b² is the product of the two focal distances.
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Start the video01 · Definition
What is the Cassini oval?
An ellipse fixes the sum of the distances to two foci; a Cassini oval fixes their product. Giovanni Domenico Cassini proposed these curves in 1680 as a model for planetary orbits, before Newton’s gravity settled the matter in favour of Kepler’s ellipses.
With the foci at (±c, 0) and the product b², one ratio decides the shape. For b < c the curve splits into two loops around the foci; at b = c it becomes the lemniscate of Bernoulli; for b > c it is a single closed curve, pinched like a peanut until b reaches √2·c, after which it is convex.
02 · Properties
Key properties
- Defining property: the product of the distances to (±c, 0) equals b².
- b < c: two separate loops; b = c: the lemniscate of Bernoulli; b > c: one closed curve.
- The single curve is convex exactly when b ≥ √2·c.
- Polar form: r⁴ − 2c²r²·cos 2θ = b⁴ − c⁴.
- Symmetric about both axes for every b.
03 · Questions
Cassini oval: common questions
- What is the equation of the Cassini oval?
- Cartesian equation: ((x-c)^2+y^2)((x+c)^2+y^2) = b^4; Polar equation: r^4 - 2c^2r^2 cos 2θ = b^4 - c^4. The foci are (±c, 0) with c = 1 in the graph; b² is the product of the two focal distances.
04 · Keep exploring