Based on Finding the centroid of a polygon? (P. Bourke), I would like to compute a weighted centroid (i.e. each vertex has an associated weight). There is a formula given in Find the centroid of a polygon with weighted vertices but I think it is not valid (if weights are all the same, you don't get the centroid (see https://math.stackexchange.com/questions/3177/why-doesnt-a-simple-mean-give-the-position-of-a-centroid-in-a-polygon).
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You cannot compare centroid for polygon with masses in vertices only (imagine iron balls connected with cocktail tubes) and centroid for polygon with mass distributed over the area (imagine polygon carved from steel sheet).
In the first case use formula from Find the centroid of a polygon with weighted vertices, in the second- the last link given
