How to compute the centroid of a mesh with triangular faces?

Question

I want to compute a new centroid for my meshes given the following description. But I do not want to use Blender's built-in functions for computing centroids as explained here as it seems that they do not give me the kind of centroid I expect to get. First, I want to compute the centers of faces (triangle) of a mesh centroid of a mesh. Then I need to compute the faces area. The new centroid is the average of the mesh faces' centers, weighted by their area. How can I do this in Python (but not necessarily using Blender's Python API)?

Your description is exactly how I'd go about it, even though I've never had to do this before. Which part is giving you trouble? — Mark Ransom, Feb 22 '18 at 03:11
@MarkRansom I would say computing the area of the triangles and the last step to get the centroid point. I'm not entirely sure about the final normalization step. — Amir, Feb 22 '18 at 03:51
Questions on this site get a lot more attention when you demonstrate at least a *little* understanding and effort. Nobody wants to do all your work for you. Edit the question to show the parts you've figured out. — Mark Ransom, Feb 22 '18 at 04:07

score 4 · Answer 1 · answered Apr 13 '21 at 15:58

Note that Spektre's answer gives the centroid of the surface area of the mesh, which might be what you want.

If you want the center of the mesh volume instead (like a center of mass assuming constant density), you would need to do the following:

Create a tetrahedron from each triangle using the 3 vertices of the triangle plus the origin point.
Calculate the signed volume and center for each tetrahedron
Total up the volumes and the volume-weighted centers
Get the mesh center by dividing the sum of the volume-weighted centers by the total volume

Pseudocode:

meshVolume = 0
temp = (0,0,0)

for each triangle in mesh (with vertices v1, v2, v3)
  center = (v1 + v2 + v3) / 4          // center of tetrahedron
  volume = dot(v1, cross(v2, v3)) / 6  // signed volume of tetrahedron
  meshVolume += volume
  temp = center * volume

meshCenter = temp / totalVolume

Is it supposed be `temp += center * volume`? I can't figure out why it would be `temp = center * volume`. — gfaster, May 31 '23 at 00:06

score 3 · Accepted Answer · edited Feb 28 '18 at 02:17

let define each triangle with 3 vertexes p0,p1,p2 the center is easy

center = (p0+p1+p2) /3

it is just the average of all vertices which forms it. The area can be computed by cross product as:

area = 0.5 * | (p1-p0) x (p2-p0) |
area = 0.5 * length(cross( p1-p0, p2-p0 ))

Both are the same just in different notation... So the centroid you are describing should be computed like this (in C++):

float area_sum=0.0;
vec3 centroid=vec3(0.0,0.0,0.0);
for (int i=0;i<triangles;i++)
 {
 t = triangle[i];
 vec3  center = (t.p0+t.p1+t.p2) /3; 
 float area = 0.5 * length(cross(t.p1-t.p0, t.p2-t.p0));
 centroid += area*center;
 area_sum += area;
 }
centroid /= area_sum;

where triangle[trianges] is array of your faces where each face has p0,p1,p2 as 3D vectors. Sorry I do not use Python so you need to adapt the vector math to your style/environment. If you do not know how to compute cross product look here:

Understanding 4x4 homogenous transform matrices

vector math is at the bottom ...

How to compute the centroid of a mesh with triangular faces?

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