I'm confused in regards of an explanation regarding infinite faces of a graph when explained through the mapping of the graph through stereographic projection onto a sphere. This is from "Introduction to graph theory" by Robert J. Wilson:
"There is nothing special about the infinite face - in fact, any face can be chosen as the infinite face. To see this, we map the graph onto the surface of a sphere by stereo-graphic projection . We now rotate the sphere so that the point of projection (the north pole) lies inside the face we want as the infinite face, and then project the graph down onto the plane tangent to the sphere at the south pole. The chosen face is now the infinite face."
When we mean rotating the sphere in this context do we mean that the graph gets turned "upside down" such that the two poles shift and the graph is then located where the south pole was previously? Is the meaning then that the graph gets turned upside down and we somehow project infinite faces. This seems very confusing to me. Would love to hear an explanation perhaps even with a diagram.
Rotating here means simply moving the graph on the sphere. If you think of your sphere as the earth then a vertex that is at the location of New York gets moved to San Francisco. The other vertices and the connecting edges move similarly.
To wrap your head around the infinite face statement I would recommend getting an orange or something similar and then draw a graph on it and think about the different projections on a piece of paper.