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:
\begin{align} C_x&=\frac1{6A}\sum_{i=0}^{n-1}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)\\ C_y&=\frac1{6A}\sum_{i=0}^{n-1}(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)\\ A&=\frac12\sum_{i=0}^{n-1}(x_iy_{i+1}-x_{i+1}y_i) \end{align}
Any idea?