  In this paper, we systematically study non-crossing chords of simple polygons
in the plane. We first introduce the reduced Euler characteristic of a family
of line-segments, and subsequently investigate the structure of the diagonals
and epigonals of a polygon. Interestingly enough, the reduced Euler
characteristic of a subfamily of diagonals and epigonals characterizes the
geometric convexity of polygons. In particular, an alternative and complete
answer is given for a problem proposed by G. C. Shephard. Meanwhile, we extend
such research to non-crossing diagonals and epigonals with forbidden positions
in some appropriate sense. We prove that the reduced Euler characteristic of
diagonals with forbidden positions only depends on the information involving
convex partitions by those forbidden diagonals, and it determines the shapes of
polygons in a surprising way. Incidentally, some kinds of generalized Catalan's
numbers naturally arise.
