As I have shown before (http://www.geogebratube.org/material/show/id/93239), GGB is in perfect agreement with my Centroid.
But my Centroid is sick.
I want to make it better, but I am in a dilemma:
Cases of self-intersection are ambiguous. Consider the figure above. The region containing P might be filled, doubly or triply overlapped, or empty. Say, for example, this is a projection onto a plane. Are A5-A7 the points of a closed star? Then where do the lines come from? Perhaps several overlapped triangles. Or perhaps the pentagon containing P is actually empty. The information which determines how the figures relate in space is lost.
I might agree to draw the shadow: the smallest wholly enclosed polygonal region whose edges are taken from the given edges.
Or I might agree that at any given point on an edge, only one side is shaded. Given only the list of vertices, there is no rule: my choice may be arbitrary (I can make up whatever I want). At the same time, different answers will give different results.
So I need a working definition of a polygon.
Frown. And let us agree right now that point C (the green diamond) is not the Centroid of Jerome, irrespective of how we choose to resolve the pentagon surrounding P.
For simplicity, I adopt a conventional approach for plane figures:
1. Ignore overlap: a point P in the plane is either inside or outside Jerome.
2. Draw any ray originating from P. Let k be the number of times the ray intersects Jerome.
If k is even, P is outside Jerome
If k is odd, P is inside Jerome
So, in the given figure, the pentagon containing P is outside the polygon.
And the IsInRegion[] command agrees with Slumberland!
.....but the Polygon tool disagrees (the area is shaded).
.....as does Centroid[]
But neither are Centroid[] and Polygon[] in agreement.
So let us fix the problem by defining Polygons for ourselves in a consistent way....