As part of my math project in school I am trying to derive a parametrization of a klein bottle. I am starting by creating a tubular/canal surface around the space curve p:
$$x(t)=5(1+\sin(t))$$
$$y(t)=3\cos(t)(1+\sin(t))$$
$$z(t)=0$$
which is essentially a piriform curve lying on the $x$-$y$ plane. I hope to achieve an end product that looks something like this. I have tried using the Frenet-serret formulas for the TNB frame using these formulas instead of differentiating them with respect to arc length, as I realized that arc length could not be expressed as an elementary function in this particular case.
This was all so I could find the orthogonal basis by which I could parameterize a tubular/canal surface using the equation: $$p(u)+R(\cos(v)N(u)+\sin(v)B(u))$$
As I expected, the computations were quite tedious and I ended up having really long equations by the time I found the normal and binormal vectors. I plugged in them into the equation to form my surface, I chose the radius R to be $2$. For example:
$$x(u,v)= 5(1+\sin(u)) + 2(\cos(v)N(u)+\sin(v)B(u))$$
I didn't write out the actual normal and binormal vectors above as they are quite long, but basically I swapped out the t for u. However, when I plugged my equations into the geogebra surface plotter, I did not end up with my desired surface. I have checked my calculations, although having done them by hand, it is still prone to error.
Is there an error in my process? I was thinking that perhaps this particular curve may be a special case since the surface ends up intersecting itself, like a klein bottle self-intersecting in R3, or perhaps because of the nature of the piriform curve? If my method is wrong, what went wrong and how can I change it to eventually develop it for the parametrization of a klein bottle?
Ps- what are some good differentiation/integration calculators online for absurdly long equations which I can copy in format?