Crooked egg curve

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In algebraic geometry, a curve which has been called the crooked egg curve, and also the bean curve, is a plane algebraic curve of degree four and genus zero, with equation

(x² + y²)² = x³ + y³.

The equation can also be given in polar form, as

r = sin³θ + cos³θ.

It is not the same as another curve called the bean curve, but has some similar properties in terms of degree and the nature of its single singularity, as well as in appearance. Both curves belong to a family of bean-shaped curves of genus zero with a single singularity with an ordinary triple point at the origin, with equation

(x² + y²)² = x³ + y³ + ay²(x² + x − y),

which gives the crooked egg curve when a is zero, and the bean curve when a is one.