In This Chapter, We Have Looked At The Point Location Proble ✓ Solved

In this chapter we have looked at the point location problem

1. In this chapter we have looked at the point location problem with preprocessing. We have not looked at the single shot problem, where the subdivision and the query point are given at the same time, and we have no special preprocessing to speed up the searches. In this exercise and some of the following ones, we have a look at such problems. Given a simple polygon P with n vertices and a query point q, here is an algorithm to determine whether q lies in P. Consider the ray Ï := {(qx + λ, qy) : λ > 0} (this is the horizontal ray starting in q and going rightwards). Determine for every edge e of P whether it intersects Ï. If the number of intersecting edges is odd, then q ∈ P, otherwise q ∈ P. Prove that this algorithm is correct, and explain how to deal with degenerate cases. (One degenerate case is when Ï intersects an endpoint of an edge. Are there other special cases?) What is the running time of the algorithm?

2. Suppose you are given an n-vertex simple polygon, P. Describe how to build an efficient data structure for determining in O(log n) time whether a query point, q, is inside of P or not. What is the space and preprocessing time for your data structure? 3. The ray shooting problem occurs in computer graphics (see Chapter 8). A 2-dimensional version can be given as follows: Store a set S of n non-crossing line segments such that one can quickly answer queries of the type: “Given a query ray Ï—a ray is a half-line starting at some point—find the first segment in S intersected by Ï." (We leave it to you to define the behavior for degenerate cases.) In this exercise, we look at vertical ray shooting, where the query ray must be a vertical ray pointing upwards. Only the starting point need be specified in such a query. Give a data structure for the vertical ray shooting problem for a set S of n non-crossing line segments in general position. Bound the query time and storage requirement of your data structure. What is the preprocessing time?

Paper For Above Instructions

The challenge of the point location problem and its related algorithms have been widely studied in computational geometry. The task at hand involves determining whether a query point q lies within a simple polygon P defined by n vertices, employing a horizontal ray shooting algorithm. The algorithm relies on casting a horizontal ray from the query point and counting the number of times it intersects with the edges of the polygon. This method, known as the Ray-Casting Algorithm, is a common approach for point-in-polygon testing.

The fundamental idea behind the Ray-Casting Algorithm is straightforward: if the number of times the ray intersects polygon edges is odd, the point lies within the polygon; if even, it lies outside. Mathematically, this is represented as follows. Given any edge e of the polygon defined by its endpoints (x1, y1) and (x2, y2), the intersection is computed by checking if the y-coordinate of the query point falls between the y-coordinates of the edge's endpoints, and if so, finding the corresponding x-coordinate of the intersection. This can be expressed as:

1. Check if min(y1, y2)

2. Compute the intersection point x of the ray with the edge using linear interpolation:

x = x1 + (q.y - y1) * (x2 - x1) / (y2 - y1).

If qx

Regarding degeneracies, one particularly troublesome case arises when the ray intersects at a vertex or endpoint of an edge. To handle this, it is crucial to establish a consistent rule, such as treating intersections at vertices as belonging to the polygon, unless it is determined that the edge connected to the vertex does not lead into the polygon. Other cases include rays that might be parallel to an edge or those that intersect multiple edges at the same point. These can be resolved by employing a systematic approach that prioritizes how edges are traversed and counted, perhaps by always ordering edges from left to right based on their x-coordinates.

The complexity of the Ray-Casting Algorithm is O(n) in the worst case, as one might need to check each edge of the polygon. However, when dealing with a larger dataset or scenarios requiring rapid subsequent queries, constructing a data structure for efficient point location becomes necessary. One efficient structure often employed is the trapezoidal map.

The trapezoidal map is constructed by dividing the space with horizontal lines determined by the edges of the polygon, effectively creating trapezoidal regions that allow rapid query point location. Each query point can be located using a binary search within these trapezoids, allowing for a time complexity of O(log n). The preprocessing for the trapezoidal map involves decomposing the polygon's edges and building the structure, which can be accomplished in O(n log n) time. The space requirement for the trapezoidal map is O(n), representing the number of edges and their active subdivisions.

Next, we move on to the vertical ray shooting problem, particularly in the context of computer graphics. Here, the objective is to efficiently determine which of a set S of n non-crossing line segments is first intersected by a vertical ray starting from a given point. This requires a suitable data structure to facilitate rapid queries.

A common approach is to utilize an interval tree or a segment tree, which allows storage of the segments in a manner that is conducive to querying. Each segment is stored in nodes divided by their corresponding vertical ranges, permitting logarithmic time complexity for query operations. The overall preprocessing time for organizing the segments into an interval tree is O(n log n), which encompasses sorting the segments and constructing the tree itself. The space complexity matches this requirement at O(n).

When a vertical ray is queried, it is directed upwards from its starting point. The data structure checks which segments lie in the vertical range of the query ray and returns the lowest (highest y-coordinate) endpoint. This way, the first segment that the ray encounters is efficiently determined.

In conclusion, the point location problem and ray shooting problem present significant challenges in computational geometry and computer graphics. Using structured algorithms and data representation techniques, one can achieve efficient query responses, facilitating a range of applications from robotics navigation to computer-aided design.

References

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