Parametric Equation for A Wiggly Tube

Question

I need to form a shape where the side view in the $xz$-plane is parallel inverse sines, and the surface is a pipe with circular cross-sections. Is there a name for this shape?

I tried messing around with ParametricPlot3D in Mathematica, but couldn't figure it out. I tried messing around with the equation of circles for two slots and inverse sine for the other. Fig. 1:

Fig. 2:

Answer 1

Since the cross section is a circle and the cross section from the side are parallel arc sine functions, we can form the shape as a continuous sequence of circles whose height changes as we go along the x-axis. With circles lying in the xz-plane and the arcsines in flat in the xz-plane we get this parametric function:

$x=t, y = r_0cos(\theta),z=r_0sin(\theta)-sin^{-1}(t)$ with $0 \leq \theta \leq 2\pi$ and $a \leq t \leq b$

So with $r_0 = 1, a = -10$, and $b = 10$ we have the graph:

Answer 2

This is a circle with a hight dependent centre.

One possible parametrization is

$$x(t)=f_x(z(t)) + r\cos(\omega t)$$ $$y(t)=f_y(z(t)) + r\sin(\omega t)$$ $$z(t) = K t .$$

If you only want the whigglyness in one direction then you can set $f_y(z(t))=0$ and $f_x(z(t))=\sin(z(t))=\sin(Kt)$.