Diagonals: Part I
2. What is a diagonal?
Since we might want the blue line below, for a non-convex polygon (here a quadrilateral) to be thought of as diagonal, we should keep the definition of "diagonal" fairly general.
Polygons that have three consecutive vertices x, y, and z where the angle at y is 180 degrees (i.e. x, y, and z lie along a straight line) are a bit of a nuisance, and we will not allow them. (Sometimes it is even insisted that no three points of the polygon lie on a straight line.) By the Jordan Curve Theorem, a non-self-intersecting polygon, usually called a simple polygon, divides the points of the plane not on the polygon itself into two sets: the inside of the polygon and the outside. If we restrict ourselves to drawing diagonals that lie totally in the interior of the polygon, it is clear that we can find such diagonals for this polygon. Such a diagonal of a simple polygon will be called an interior diagonal. Is it possible that a polygon be so convoluted that it would be impossible to find a single diagonal which lies totally in its interior?
For a very convoluted polygon such as the one above, it is perhaps not entirely clear that there is always an interior diagonal. Finding an interior diagonal is the first step in showing that every plane simple polygon can be triangulated. This means that interior diagonals can be added which result in all the interior faces of the resulting structure having three sides. For a very convoluted polygon it may not even be clear which points are inside the polygon and which ones are outside. There is an easy way to determine this. Start at any point outside the polygon. Now draw a line segment from this point to the point in question. Count the number of times this line segment intersects the polygon. If this number is even, the point in question is outside the polygon. If this number is odd, the point is inside the polygon!
In the diagram above, the green line cuts the polygon in 5 places so the point whose interior vs. exterior status might be unclear must be inside the polygon.
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