algorithm - Center of gravity of a polygon

Question

I am trying to write a PHP function that will calculate the center of gravity of a polygon.

I've looked at the other similar questions but I can't seem to find a solution to this.

My problem is that I need to be able to calculate the center of gravity for both regular and irregular polygons and even self intersecting polygons.

Is that possible?

I've also read that: http://paulbourke.net/geometry/polyarea/ But this is restricted to non self intersecting polygons.

How can I do this? Can you point me to the right direction?

Answer 1

The center of gravity (also known as "center of mass" or "centroid" can be calculated with the following formula:

X = SUM[(Xi + Xi+1) * (Xi * Yi+1 - Xi+1 * Yi)] / 6 / A
Y = SUM[(Yi + Yi+1) * (Xi * Yi+1 - Xi+1 * Yi)] / 6 / A

Extracted from Wikipedia: The centroid of a non-self-intersecting closed polygon defined by n vertices (x0,y0), (x1,y1), ..., (xn?1,yn?1) is the point (Cx, Cy), where


and where A is the polygon's signed area,

Example using VBasic:

' Find the polygon's centroid.
Public Sub FindCentroid(ByRef X As Single, ByRef Y As _
    Single)
Dim pt As Integer
Dim second_factor As Single
Dim polygon_area As Single

    ' Add the first point at the end of the array.
    ReDim Preserve m_Points(1 To m_NumPoints + 1)
    m_Points(m_NumPoints + 1) = m_Points(1)

    ' Find the centroid.
    X = 0
    Y = 0
    For pt = 1 To m_NumPoints
        second_factor = _
            m_Points(pt).X * m_Points(pt + 1).Y - _
            m_Points(pt + 1).X * m_Points(pt).Y
        X = X + (m_Points(pt).X + m_Points(pt + 1).X) * _
            second_factor
        Y = Y + (m_Points(pt).Y + m_Points(pt + 1).Y) * _
            second_factor
    Next pt

    ' Divide by 6 times the polygon's area.
    polygon_area = PolygonArea
    X = X / 6 / polygon_area
    Y = Y / 6 / polygon_area

    ' If the values are negative, the polygon is
    ' oriented counterclockwise. Reverse the signs.
    If X < 0 Then
        X = -X
        Y = -Y
    End If
End Sub

For more info check this website or Wikipedia.

Hope it helps.

Regards!