Vtyjkyuk

Given a list of coordinates of a coplanar plane $left(pt_1, pt_2, pt_3, cdots right)$, how to compute the centroid of the coplanar plane?

One way to do it is to project the plane onto $XY$ and $YZ$ plane, but I don't really favor this approach as you have to check the orientation of the coplanar plane first before doing the projection and computing the centroid.

More specifically, I'm looking for a natural extension of the 2D centroid plane algorithm in 3D:

beginalign
C_x&=frac16Asum_i=0^n-1(x_i+x_i+1)(x_iy_i+1-x_i+1y_i)\
C_y&=frac16Asum_i=0^n-1(y_i+y_i+1)(x_iy_i+1-x_i+1y_i)\
A&=frac12sum_i=0^n-1(x_iy_i+1-x_i+1y_i)
endalign

Any idea?

You can take any two orthogonal vectors $vece_1$ and $vece_2$ on the plane and use them as a basis. You also need some point $(x_0, y_0, z_0)$ on the plane as origin.

Given point with coordinates $(x_1, y_1, z_1)$ on your plane you calculate it's coordinates with respect to new basis:

$x = (x_1 - x_0) e_1x + (y_1 - y_0) e_1y + (z_1 - z_0) e_1z$

$y = (x_1 - x_0) e_2x + (y_1 - y_0) e_2y + (z_1 - z_0) e_2z$

And after that you can apply your formulae to get $C_x$ and $C_y$. Those coordinates are easyly transformed back into original 3d coordinates:

$x = x_0 + e_1x C_x + e_2x C_y$

$y = y_0 + e_1y C_x + e_2y C_y$

$z = z_0 + e_1z C_x + e_2z C_y$