Calculate Centroid three-dimensional planar polygon

Question

This seems like a question here .

Given a list of three-dimensional coordinates defining surface ( Point3D1, Point3D2, Point3D3etc.), how to calculate the surface center of gravity?

In 2D, the calculation is given by the following formula :

What about the 3D counterpart?

Answer 1

If it is a flat surface, you can convert it to a coordinate system local to the plane, calculate the centroid using the formulas that you presented, and then convert it back to get your coordinates in three-dimensional space.

Answer 2

, , z y.

, x-y, - x-z. , x, y z, .

: (x1, y1, z1), (x2, y2, z2),..., xy, (Cx, Cy), , (x1, y1), (x2, y2),... xz (Cx, Cz) (x1, z1), (x2, z2),.... - 2D-, z y . 3D- (Cx, Cy, Cz). , x-y, x-z y-z ( ).

Answer 3

v ₀, v ₁,..., v _N , v _i= (x _i, y _i, z _i).

(v ₀, v ₁, v ₂), (v ₀, v ₂, v ₃),..., (v ₀, v _i, v ₊₁),..., (v ₀, v _N-1, v _N) N-1 .

| (v _i & minus; v ₀) & times; (v _{+ 1} & minus; v ₀) | &; 2, & times; | & ; | .

, . (v _i & minus; v ₀) & times; (v _{+ 1} & minus; v ₀) & middot; (v ₁ & minus; v ₀) & times; (v ₂ & minus; v ₀). , .

2D- , (, z), , (v _i & minus; v ₀) & times; (v _{+ 1} & minus; v ₀) & middot; z . , .

(v ₀ + v _i + v _{+ 1}) & divide; 3.

, , ,

                1       N-1
centroid = ——————————    ∑  ( centroid-of-triangle-i × area-of-triangle-i )
           total-area   i=1

(For sizes ≥ 4D, the area should be calculated using A _i =? Frac12; | v _i & minus; v ₀ | | v _{i + 1} ? Min ₀ v sin | theta; _i , where cos? Theta; _i = (v _i & minus; v ₀ )? (v _{i + 1} & minus; v ₀ ).)