I consider Point Location Problem in Polygon in repetitive mode in the case of simple polygon.
In computational geometry,Point Location Problem in Polygon problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon.
There are few method that work in Single-Shot approach, where the input is a polygon $P$ and a single point $q$ (no preprocessing time). Ray casting algorithm is the famous algorithm for single-shot, it takes $O(n)$ to determine whether a point $q$ belongs to polygon $P$.
In addition, there is a repetitive approach, where instead of single point $q$ we should check the sequence of points, therefore the preprocessing is required. Division wedge is a algorithm that works in repetitive mode. Query time of division wedge is $O(\log n)$ and preprocessing time is $O(n)$. Division wedge assumes that there is a central point in polygon, visible from every vertex of polygon (part of the kernel of the polygon). The problem is a central point can be easily determined in convex polygon as well as in star-shaped polygon, but what to do in the case of simple polygon.
If division wedge is applied in the case of simple polygon how we can determine a central point in simple polygon? If division edge in not applied if there is the more efficient way to solve a problem in simple polygon than in arbitrary planar subdivision.