Anti-podal pairs
Lines of support
Given a convex polygons P, a line of support l is a line intersecting P and such that the interior of P lies to one side of l.
This concept is comparable to that of a tangent line.
Anti-podal pairs
If two points p and q (belonging to P) admit parallel lines of support, then they form an anti-podal pair.
Two distinct parallel 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 intersect 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 the polygon at an edge while the other line of support is tangent 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 lines of support intersect the polygon at parallel edges. In this case, the lines of support also determine four distinct vertex-vertex anti-podal pairs.
December 17th, 1998