Anti-podal pairs between two convex polygons
Directed lines of support
A directed line of support is exactly what its name indicates. The direction is necessary when it is required to differentiate between parallel lines of support that have the same or opposite direction.
It is further assumed that polygons are given in clockwise order (i.e. the direction if one follows the vertices in order is clockwise) and that a polygon's line of support is directed such that the polygon lies to the right of the line. Alternatively, polygons can be given in counterclockwise order if the lines of support are such that the polygons lie to their left.
Although this is just by convention, it is necessary to define some standard, in order to avoid cluttering otherwise simple definitions and results. Since this convention in no way alters results or puts any further constraints, it is better to adopt it.
Note: Lines of support are defined along with anti-podal pairs.
Anti-podal pairs between two convex polygons
Given two polygons P and Q, a pair of points (p, q) (belonging to P and Q respectively) form an anti-podal pair between P and Q if the polygons admit (directed) parallel lines of support in different directions at p and q.
Two such lines of support always determine at least one anti-podal pair. Depending on how the lines intersect the polygon, three cases arise:
- Vertex-vertex anti-podal pair
- Vertex-edge anti-podal pair
- Edge-edge anti-podal pair
Case 1 occurs when the lines of support intersects the polygon at two vertices only, as illutrated. The vertices shown as black dots form an anti-podal pair.
Case 2 occurs when one line of support intersects its polygon at an edge while the other line of support intersects its polygon at a vertex only. Note that the existence of such lines of support automatically implies the existence of two distinct vertex-vertex anti-podal pairs.
Case 3 occurs only when the two polygons have parallel edges. The lines of support intersect the polygons at these edges. In this case, the lines of support also determine four distinct vertex-vertex anti-podal pairs.
December 17th, 1998