Polygons

For the rest of this book, a (generalized) polygon Γ is defined by a collection of vertices w1, …, wn and real interior angles α1π, …, αnπ. It is convenient for indexing purposes to define wn+1 = w1 and w0 = wn. The vertices, which lie in the extended complex plane C ∪ {∞}, are given in counterclockwise order with respect to the interior of the polygon (i. e., locally the polygon is “to the left” as one traverses the side from wk to wk+1).

The interior angle at vertex k is defined as the angle swept from the outgoing side at wk to the incoming side. If |wk| < ∞, we have αk ∈ (0, 2]. If αk = 2, the sides incident on wk are collinear, and wk is the tip of a slit. The definition of the interior angle is applied on the Riemann sphere if wk = ∞. In this case, αk ∈ [–2, 0]. See Figure 2.1. Specifying αk is redundant if wk and its neighbors are finite, but otherwise αk is needed to determine the polygon uniquely.

In addition to the preceding restrictions on the angles αk, we require that the polygon make a total turn of 2π.