Lissajous quadratic

The curve is an extension of the Lissajous curve, and is characterised by his elegant handwriting-like curves. I haven't found the curves at other mathematicians, but I guess they exist, please let me know. For b = 1 and d = 1, the curve is a quartic1. Interchanging the values for a and c gets the same picture, so when can confine ourselves to the situation that c <= a. Some examples: Some examples for other values of b and d:

notes

1) with Cartesian equation, thanks to Piers Barber: a⁴ y⁴ - 4 a³ c x y³ + 2 a² c y³ + 6 a² c² x² y² - 4 a c² x y² - 2 a³ x y² - 2 a² c² y² + c² y² + a² y² - 4 a c³ x³ y + 2 c³ x² y + 4 a² c x² y + 4 a c³ x y -2 c³ y - 4 a² c y + c⁴ x⁴ - 2 a c² x³ - 2 c⁴ x² + c² x² + a² x² + 2 a c² x = a² + c² - c⁴