Plateau curve

These curves were studied by the Belgium Joseph Plateau.

As you can see on this page, nice pictures appear for different values for a.
For a = 1 the Plateau curve has no meaning.
The Plateau curve with parameter 1/a is equal to the Plateau curve with parameter a. So we can confine ourselves to values for the parameter a between -1 and 1.
For a = 1/2, the Plateau curve degenerates to an ellipse, for a = - 1/2 to a hyperbola.
For a = 0 a straight line results.

When the parameter 'a' is a positive rational number, the curve consists of a number of ellips-like forms. Give the parameter 'a' its simplest form, then the number of elliptic forms is equal to the numerator of 'a'.
The denominator of 'a' minus one gives the number of elliptic and asymptotic forms.

When the parameter 'a' is a negative rational number, the Plateau curve has the form of a number of hyperbolas. How many hyperbolas? Give the parameter 'a' its simplest form, then the number of hyperbolas is equal to the sum of the numerator and the denominator minus one. Remarkable.