Centroid of ParametricPlot3D

Question

Helo All

I have some Question.

i just have this shape with ParametricPlot3D.

how can I calculate this shape with RegionCentroid Function? here my code

r = 10;
scc = 1;
zx = 5;
ppp = {x[1], y[1], z[1]} = {r*Sin[u], scc*v, r*Cos[u]};
sp = 116*Cot[\[Pi]/180*72];
pcg1 = {};
For[i = 1, i <= zx, i++,
 \[CurlyPhi] = (\[Pi]/2)/zx*i;
 {xi, yi, zi, dummy} = ( {
     {Cos[\[CurlyPhi]], -Sin[\[CurlyPhi]], 0, 0},
     {Sin[\[CurlyPhi]], Cos[\[CurlyPhi]], 0, 0},
     {0, 0, 1, sp*\[CurlyPhi]},
     {0, 0, 0, 1}
    } ).( {
     {1, 0, 0, 0},
     {0, 1, 0, 6 r},
     {0, 0, 1, 0},
     {0, 0, 0, 1}
    } ).Insert[ppp, 1, 4];
 AppendTo[pcg1, {xi, yi, zi}]
 ]
ParametricPlot3D[pcg1, {u, 0, 2 \[Pi]}, {v, 0, 10}, 
 AxesLabel -> {x, y, z}, PlotRange -> Full]

Thank you

Answer 1

Method-1

Using the code by author.

RegionCentroid@*DiscretizeGraphics@
 ParametricPlot3D[pcg1, {u, 0, 2 π}, {v, 0, 10}, 
  AxesLabel -> {x, y, z}, PlotRange -> Full]

{-47.5395, 34.5392, 35.5237}

Method-2

We rewrite the code,replace For by Table etc.

Clear["Global`*"];
r = 10;
scc = 1;
zx = 5;
ppp = {r*Sin[u], scc*v, r*Cos[u]};
sp = 116*Cot[π/180*72];
pcg1 = Table[Block[{φ = (π/2)/zx*i},
    ({{Cos[φ], -Sin[φ], 0, 0}, {Sin[φ], 
        Cos[φ], 0, 0}, {0, 0, 1, sp*φ}, {0, 0, 0, 
        1}}) . ({{1, 0, 0, 0}, {0, 1, 0, 6 r}, {0, 0, 1, 0}, {0, 0, 0,
         1}}) . PadRight[ppp, 4, 1]], {i, 1, zx}];
pcg1 = pcg1[[;; , 1 ;; 3]];
ParametricPlot3D[pcg1, {u, 0, 2 π}, {v, 0, 10}, 
   AxesLabel -> {x, y, z}, PlotRange -> All] // 
  DiscretizeGraphics // RegionCentroid

{-47.5395, 34.5392, 35.5237}

Method-3

Rewrite all the code，replace the matrix product by geometric transformation and using TransformedRegion.

Clear["Global`*"];
r = 10;
scc = 1;
zx = 5;
ppp = {r*Sin[u], scc*v, r*Cos[u]};
sp = 116*Cot[π/180*72];
reg = ParametricRegion[ppp, {{u, 0, 2 π}, {v, 0, 10}}];
Table[Block[{φ = (π/2)/zx*i}, 
     TransformedRegion[reg, 
      RotationTransform[φ, {0, 0, 1}]@*
       TranslationTransform[sp*φ {0, 0, 1}]@*
       TranslationTransform[6 r {0, 1, 0}]]], {i, 1, zx}] // 
   Map@DiscretizeRegion // RegionUnion
% // RegionCentroid

{-47.538, 34.543, 35.5203}

Answer 2

$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

ClearAll["Global`*"]

r = 10;
scc = 1;
zx = 5;
ppp = {x[1], y[1], z[1]} = {r*Sin[u], scc*v, r*Cos[u]};
sp = 116*Cot[π/180*72];
pcg1 = {};
For[i = 1, i <= zx, i++, φ = (π/2)/zx*i;
 {xi, yi, zi, 
   dummy} = ({{Cos[φ], -Sin[φ], 0, 0}, {Sin[φ], 
      Cos[φ], 0, 0}, {0, 0, 1, sp*φ}, {0, 0, 0, 
      1}}) . ({{1, 0, 0, 0}, {0, 1, 0, 6 r}, {0, 0, 1, 0}, {0, 0, 0, 1}}) . 
   Insert[ppp, 1, 4];
 AppendTo[pcg1, {xi, yi, zi}]]

pcg1 = pcg1 // Simplify;

The individual regions are

paraRgns = ParametricRegion[#, {{u, 0, 2 π}, {v, 0, 10}}] & /@ pcg1;

All of the regions are the same size

SameQ @@ (RegionMeasure /@ paraRgns)

(* True *)

Average the individual centroids

centroid = Mean[RegionCentroid /@ paraRgns] //
  FullSimplify

(* {-(13/2) (2 + Sqrt[5] + Sqrt[5 + 2 Sqrt[5]]), 
 13/2 (Sqrt[5] + Sqrt[5 + 2 Sqrt[5]]), 174/5 Sqrt[1 - 2/Sqrt[5]] π} *)

centroid // N

(* {-47.5394, 34.5394, 35.5226} *)

Show[
 ParametricPlot3D[pcg1, {u, 0, 2 π}, {v, 0, 10},
  AxesLabel -> {x, y, z}, PlotRange -> Full],
 Graphics3D[{Red, AbsolutePointSize[8], Point[centroid]}]]