I'm looking for a equation that describes the shape of a "Looping" in the best way. I really don't know how to start here, as it isn't even a function (if it were, I could just use spline interpolation), so do you have any idea on how to best approximate that shape?
Just to make clear what shape I'm talking about (sorry for my really bad paint skills): I'm not looking for a circle and a line as in (2), but rather for a "rollercoaster like" looping as in (1).
On
Use a parametric spline, in which $x$ and $y$ are spline functions (or even just polynomial functions) of some independent parameter $t$.
Here is a parametric cubic spline with 4 segments created in Powerpoint. Or, looking at it another way, this is just a string of four cubic Bezier curves that join smoothly.
And here's a nicer curve. It's a Bezier curve of degree 6, and its control points are $(2,0)$, $(6,0)$, $(7,2)$, $(4,5)$, $(1,2)$, $(2,0)$, $(6,0)$.
$$x^2+y(y+1)^2=0$$ gives something like what you want.
See also Graph of an infinitely extending rollercoaster loop