Question

Given a point p and a vertex v of a convex polygon P in the plane, give an algorithm to classify in constant time vertex v with respect to pv​ (as concave or supporting or reflex).

Answer 1

To classify vertex v of a convex polygon P concerning a point p and line segment pv, simply verify if p is inside or on P, as all vertices in a convex polygon will be supporting; there are no concave or reflex vertices in a convex polygon.

Step-by-step explanation:

To classify a vertex v of a convex polygon P in relation to a point p with respect to the line segment pv, you can follow a simple algorithm since P is convex. In a convex polygon, all vertices form angles that are less than or equal to 180 degrees with adjacent vertices. Therefore, we can classify the vertex v as supporting if the line segment pv lies entirely inside P, or overlaps with one of P's edges. There is no need to classify v as concave or reflex because convex polygons do not have concave or reflex vertices. Thus, in constant time, we only need to verify if the point p lies inside or outside P to determine the classification of v.