Finding the coordinates x y z of the center of a surface - ruby

Question

I have a surface in a 3D space, and I need to calculate the coordinates of the center of that surface. The surface is a polygon.
I found this formula:
X1 += SUM[(xi + xi+1 ) * (xi* yi+1 - xi+1 * yi )]/6/area
Y1 += SUM[(yi + yi+1) * (xi* yi+1 - xi+1 * yi )]/6/area
But it only works for 2D. It gives x and y values. In my case I need 3 coordinates, x y z.
How can I do that ?
I need the center of the surfaces selected, but they may have any shape as they are a polygon.
Thanks a lot.

Answer 1

It's easy to prove that the centroid of the projection of a polygon on any plane is the projection of the centroid of that polygon on that plane.
So just calculate the centroid of the projections of the polygon on x-y plane and y-z plane, and you'll get the coordinate of the centroid of that polygon.
#!/usr/bin/env ruby
Point = Struct.new(:x, :y, :z)
def centroid(vertices, dimensions: [:x, :y])
area = 0
centroid_x, centroid_y = 0, 0
vertices.count.times do |i|
v1, v2 = vertices[i, 2]
v2 ||= vertices[0]
x1, y1 = dimensions.map{|d| v1.send(d)}
x2, y2 = dimensions.map{|d| v2.send(d)}
a = x1 * y2 - x2 * y1
area += a
centroid_x += (x1 + x2) * a
centroid_y += (y1 + y2) * a
end
area *= 0.5
centroid_x /= (6.0 * area)
centroid_y /= (6.0 * area)
[centroid_x, centroid_y]
end
vertices = [
Point.new(1, 0, 0),
Point.new(0, 2, 0),
Point.new(0, 0, 3)
]
p centroid(vertices, dimensions: [:x, :y])
p centroid(vertices, dimensions: [:y, :z])
p centroid(vertices, dimensions: [:z, :x])
prints
[0.3333333333333333, 0.6666666666666666]
[0.6666666666666666, 1.0]
[1.0, 0.3333333333333333]

Answer 2

#Aetherus it didn't work for my example:
X1 = 0
Y1 = 0
Y11 = 0
Z1 = 0
for i in 0..d.vertices.size-2
X1 += (d.vertices[i].position[0] + d.vertices[i+1].position[0]) * (d.vertices[i].position[0] * d.vertices[i+1].position[1] - d.vertices[i+1].position[0] * d.vertices[i].position[1])
Y1 += (d.vertices[i].position[1] + d.vertices[i+1].position[1]) * (d.vertices[i].position[0] * d.vertices[i+1].position[1] - d.vertices[i+1].position[0] * d.vertices[i].position[1])
Y11 += (d.vertices[i].position[1] + d.vertices[i+1].position[1]) * (d.vertices[i].position[1] * d.vertices[i+1].position[2] - d.vertices[i+1].position[1] * d.vertices[i].position[2])
Z1 += (d.vertices[i].position[2] + d.vertices[i+1].position[2]) * (d.vertices[i].position[1] * d.vertices[i+1].position[2] - d.vertices[i+1].position[1] * d.vertices[i].position[2])
end
x=X1/(6.0*(d.area))
y=Y1/(6.0*(d.area))
y1=Y11/(6.0*(d.area))
z=Z1/(6.0*(d.area))
UI.messagebox("x1 #{x} Y1 #{y} \n y11 #{y1} z1 #{z}")
With "d" is my polygon the xy is for x-y projection and y1z for y-z projection.