Rescaling a section of a ContourPlot3D

Question

So I have the following 3D contour plot:

first = ContourPlot3D[
  z Sin[Pi x] == -(( Sqrt[44 + y^2 ] - y) x^2 + y) Cos[Pi x], {x, 0, 
   1}, {y, -10, 10}, {z, -150, 150}, Axes -> False, 
  Contours -> Automatic, Boxed -> False, 
  Method -> {"Extrusion" -> .05}]

The goal is to generate a surface which I will 3D print. As you can see from the axis bounds it is extremely long in the z axis and short in the other two. When I load the surface into my 3d printing software I rescale the axis's so the surface fits nicely within a square. The problem that arises is that when I extrude the plot, it is very thin in the x-axis, see the attached photo. What I am looking to do is extrude evenly the surface relative to the image I attached and not the bounds I used in the contourplot3d code.

Answer 1

Rescale your function before applying ContourPlot3D

cont = z Sin[Pi x] + ((Sqrt[44 + y^2] - y) x^2 + y) Cos[Pi x] /. 
{z ->150 dz, x -> (dx + 1)/2, y -> dy 10}
ContourPlot3D[ cont == 0, {dx, -1, 1}, {dy, -1 , 1 }, {dz, -1 , 1 }, 
Contours -> Automatic, Boxed -> False,Method -> {"Extrusion" -> .05} , Axes -> False]

Answer 2

As noted by Ulrich rescaling the coordinates so that the ranges are similar in three directions fixes the issue.

It turns out re-scaling only the third coordinate (using ScalingFunctions) is sufficient:

ContourPlot3D[z Sin[Pi x] == -((Sqrt[44 + y^2] - y) x^2 + y) Cos[Pi x], 
  {x, 0, 1}, {y, -10, 10}, {z, -150, 150}, 
 Axes -> False, 
 Contours -> Automatic, Boxed -> False, 
 Method -> {"Extrusion" -> .05}, 
 ScalingFunctions -> {"Linear", "Linear", 
  {Rescale[#, {-150, 150}] &, Rescale[#, {0, 1}, {-150, 150}] &}}]