Donghwan, I do not know how to control thickness value in plot option PlotTheme->"ThickSurface".

There may be other approaches if one wants to use Mathematica. See for example ImplicitRegion and ParametricRegion.

However if I myself wanted to construct a tightly controlled model to be 3D printed I would use a dedicated 3D CAD program. My own personal preference would be SolidWorks. But there are many other programs that could be used. In an earlier post you mention three of them.

Mathematica is not a 3D CAD program. It is something else.

Increasing the number of PlotPoints gives a smoother model. In the attached model the number of PlotPoints is 200. It has also been scaled by a factor of 50.

r[s_, a_, b_] := 1 + a Exp[-b s^2]
example =
    ParametricPlot3D[
    50 {r[s, .3, 10] Cos[t], r[s, .3, 10] Sin[t], s},
    {t, 0, 2 Pi}, {s, 0, 1 + 1/2 Cos[6 t]},
    PlotPoints -> 200, Mesh -> False, PlotTheme -> "ThickSurface"]

When opened in a CAD program:

Attachments:

Hello Donghwan,

However, no 3D program(Sketchup, Rhinocerus, Autodesk fusion 360, etc.) can recognize the mathematical plane into 3D printable object!

Whatever that means? I don't know Sketchup, Rhinocerus and so on and I have not idea how these programs work.

So this model need to have full thickness to be printed into real world

And what do you mean by "full thickness"? And what do you mean by "model"?

Regards, Hans

I don't understand what you mean by thickness of the base.

This perhaps?

r[s_, a_, b_] := 1 + a Exp[-b s^2]

ParametricPlot3D[{r[s, .3, 10] Cos[t], r[s, .3, 10] Sin[t], s}, 
{t, 0, 2 Pi}, {s, 0, 1 + 1/2 Cos[6 t]}, PlotPoints -> 50, 
PlotRange -> All, Mesh -> False]

Here is your cycloid, albeit it looks somewhat peculiar

cycl[a_] := Module[{},
  t = x /. FindRoot[a == x - Sin[x], {x, .1}];
  1 - Cos[t]
  ]

ParametricPlot3D[{r[s, .3, 10] Cos[t], r[s, .3, 10] Sin[t], 
  If[s < 1, s, 2 + cycl[5 t]]}, {t, 0, 2 Pi}, {s, 0, 2}, 
 PlotPoints -> 50, PlotRange -> All, Mesh -> False]

Another one. The cycloid is removend from the iterator and put in the vector- function

ParametricPlot3D[{Cos[t], Sin[t], If[s < 1, s, 2 + cycl[5 t]]}, {t, 0,
   2 Pi}, {s, 0, 2}, PlotPoints -> 50, PlotRange -> All, 
 Mesh -> False]

And some (simpler )thickness

rr[x_] := If[x < .5, 1.3, 1]

ParametricPlot3D[{rr[s] Cos[t], rr[s] Sin[t], 
If[s < 1, s, 2 + cycl[5 t]]}, {t, 0, 2 Pi}, {s, 0, 2}, 
PlotPoints -> 50, PlotRange -> All, Mesh -> False]

And last not least the "self-made one. Add a function of h in xx[ a, h ] to Sin and Cos to get some thickness

xx[a_, h_] := {Cos[a], Sin[a], Min[h, 1 + cycl[5 a]]}

pol[a_, h_, da_, dh_] := 
 Polygon[{xx[a, h], xx[a + da, h], xx[a + da, h + dh], xx[a, h + dh]}]

nda = 60; ndh = 20; ffh = .48;
da = 2 Pi/nda; dh = 5/ndh;
zylo = Table[{Hue[ffh n dh], 
    Table[pol[m da, n dh, da, dh], {m, 0, nda - 1}]}, {n, 0, ndh - 1}];
Graphics3D[zylo
 ]

[?] Plot in 3D Cycloid curves on Zylo ??

Or we can do it for our own:

xx[a_] := {Cos[a], Sin[a], 0}
zz[a_] := {Cos[a], Sin[a], 1 + Cos[5 a]/2}
pol[a_, da_] := Polygon[{xx[a], xx[a + da], zz[a + da], zz[a]}]

da = 2 Pi/60;
Graphics3D[
 Table[{Hue[zz[a][[3]]], pol[a, da]}, {a, 0, 2 Pi - da, da}]]

or

pol1[a_, da_] := 
 Polygon[{xx[a], xx[a + da], zz[a + da], zz[a]}, 
  VertexColors -> {Red, Green, Blue}]

da = 2 Pi/60;
Graphics3D[Table[pol1[a, da], {a, 0, 2 Pi - da, da}]]

Both of these are nice als well

ParametricPlot3D[{Cos[t], Sin[t], s}, {t, 0, 2 Pi}, {s, 0,1 + 1/2 Cos[5 t]}, PlotPoints -> 50, PlotRange -> All, 
Mesh -> False, ColorFunction -> Function[{x, y, z}, Hue[.75 z]]]

and

Animate[
ParametricPlot3D[{Cos[t], Sin[t], s}, {t, 0, 2 Pi}, {s, 0, 1 + 1/2 Cos[5 t + u]}, PlotPoints -> 50, 
PlotRange -> All, Mesh -> False], {u, 0, 2 Pi, 0.1}]

But when I try to rotate the colored object my system crashes. What is going wrong?

Animate[
 ParametricPlot3D[{Cos[t], Sin[t], s}, {t, 0, 2 Pi}, {s, 0,  + 1/2 Cos[5 t + u]}, 
PlotPoints -> 50, PlotRange -> All, 
  Mesh -> False, ColorFunction -> Function[{x, y, z}, Hue[.75 z]]],
 {u, 0, 2 Pi, 0.1}]

ParametricPlot3D[{Cos[t],Sin[t],Sin[5t],{t,0,2Pi}]

Then How can I make a cylinder below to make a full 3D printable X-zylo? Please give some Ideas!