Polygon self-intersection, 1
- Ryan Hirst
(Tool-- intersect ray and polygon: http://www.geogebratube.org/material/show/id/96426)
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....