coordinate system transformation

Question

I am trying to define a radial deformation (same deformation radially) regarding Cartesian coordinates. The problem is that the cylinder is curved, i.e., the center of the cylinder doesn't follow the axis, i.e.:

I know how to do the coordinate transformation for the straight section, just not sure how to do it for the curved section. The center of the pipe in this region can be defined as: $$y = \sqrt{(0.03)^2 - (x-0.03)^2} + 0.15, y > 0.15$$

Thanks

Answer 1

You can do that with the blending of a cylinder and a torus as shown in the attached graphics.

The torus parametric equation is

$$ \left\{ \begin{array}{rcl} x & = & (a\cos (u)+c) \cos (v) \\ y & = & (a\cos (u)+c) \sin (v) \\ z & = & a\sin (u) \\ \end{array} \right. $$

such that

$$ c = \;\;\mbox{torus main radius}\\ a = \;\;\mbox{tube radius} $$