Finding a nonredundant component in a polygon

Let v_i be a reflex vertex (internal angle greater than pi) of a polygon P with n vertices. Extend the clockwise edge of v_i as a ray until it hits P, and then walk clockwise from v_i to the hitpoint. The chain we walked defines the clockwise component of v_i (it also has a counterclockwise component). In O(n) time we find some component of P that does not entirely contain another component, without using a general O(n) time triangulation algorithm. This time bound has already been achieved using such a triangulation algorithm, but we show it is possible without it. Our central algorithm simultaneously walks a component in the clockwise and counterclockwise directions. In these walks, it shoots rays and finds acceptable hitpoints that are not necessarily correct. For a particular hitpoint, the algorithm either validates it, disqualifies it and finds another, or shoots a new ray and finds a finds a new hitpoint.