Lissajous quadratic

higher

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