nodal curve

When a is a rational number, the curve is an algebraic curve.
Each curve consists of equal branches, that merge in a rotation of p/a.

La Gournerie studied the curve in 1851.

Some specific specimen are the following:

For a=1, the kappa curve, a quartic, results.
For a=2 we see the windmill curve, a sextic ¹⁾.
For a being integer, the power of the curve is 2*a +2.

For a=1/2, you find the straight strophoid.

notes

1) With formula: x²y² (x² + y²) = (x² - y²)²