In this paper we investigate non-crossing chords of simple polygons in the plane systematically. We first develop the Euler characteristic of a family of line-segments, and subsequently study the structure of the diagonals and epigonals of a polygon. A special phenomenon is that the Euler characteristic of a set of diagonals (or epigonals) characterizes the geometric property of polygons, such as convexity. In particular, a positive answer is given to an open problem proposed by Shephard. The main contributions of the present paper extend such research to non-crossing diagonals and epigonals with forbidden positions. We find that the Euler characteristic of diagonals (or epigonals) with forbidden positions determine the types of polygon in surprising ways. Incidentally, some kinds of generalized Catalan's number naturally arise.