Here is what I mean. I got the inner and outer curves in this picture by using a spline tool in CAD. I do not want to use a spline. I want to use an equation.

I've tried every single modification to the original equation that I can think of, and I've resorted to just embarrassing randomness. I've been at this for many hours and I'm wielding mathematics like a monkey with a machine gun at this point. Figured it was time to ask for help.