Geometric Distance Calculation Using Quadtree Spatial Partitioning
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Calculating distances between complex geographical entities modeled as polygons or non-areal shapes is computationally expensive and time-consuming, especially when dealing with large numbers of vertices or holes, requiring efficient methods to optimize processing.
Innovation Solution
The distance calculation system employs PM quadtree rules to subdivide the coordinate space into quadrants, recursively searching for optimal edges and pruning non-essential regions, using an iterative method to determine minimum or maximum distances by focusing on the most proximal quadrants and updating values as closer edge pairs are discovered.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the naive approach of computing distance for every vertex of one polygon against every edge of another polygon is used, then complete accuracy is achieved, but computational expense becomes very high
Solution Approach 1:
The patent segments the coordinate space into quadrants using a quadtree structure, dividing the computational domain into manageable regions. This segmentation allows the algorithm to focus computations only on relevant quadrants containing vertices or edges, rather than computing distances for all vertex-edge pairs across the entire polygon set, thereby reducing computational expense while maintaining accuracy.
Solution Approach 2:
The patent introduces a spatial dimension by organizing vertices and edges into a quadtree data structure based on their coordinate positions. This dimensional organization allows for efficient pruning of quadrants that cannot contain the minimum distance, transforming the problem from a brute-force O(n²) approach to a more efficient spatial partitioning approach that reduces computational complexity.
2Productivity
If recursive subdivision of coordinate space into quadrants is performed, then computational expense is reduced, but algorithm complexity increases
Solution Approach 1:
The coordinate space is recursively segmented into quadrants, creating a hierarchical quadtree structure. Each quadrant is further subdivided until base cases are reached (quadrants containing single vertices or edges). This systematic segmentation reduces the search space for minimum distance calculations while providing a structured approach to managing algorithmic complexity through clear recursive termination conditions.
Solution Approach 2:
The algorithm performs preliminary actions by pre-organizing all vertices and edges into the quadtree structure before executing the distance calculation. This preliminary spatial organization allows subsequent distance computations to efficiently prune entire quadrants that cannot contain the minimum distance, reducing the overall computational workload while the recursive nature manages complexity through well-defined base cases.
3Measurement precision
If all quadrants are processed to ensure complete search, then accuracy is maintained, but processing time increases
Solution Approach 1:
The algorithm extracts and processes only the essential quadrants that can potentially contain the minimum distance between polygons. By using the quadtree structure to identify and isolate relevant quadrants (those containing vertices or edges that could form the minimum distance pair), the algorithm excludes non-essential quadrants from processing, thereby maintaining accuracy while significantly reducing processing time.
Solution Approach 2:
The algorithm performs partial action by processing only the necessary quadrants rather than all quadrants in the coordinate space. The recursive quadtree approach allows the algorithm to stop subdividing and processing quadrants once certain conditions are met (such as when a quadrant contains only one vertex or edge, or when the quadrant is proven not to contain the minimum distance), thus avoiding excessive computation while maintaining complete accuracy for the relevant search space.
Data Source
AI summary
Distances between geographic real-world entities modeled as geometric shapes are measured. At least two shapes are positioned in a coordinate space based on their respective sets of points. A non-empty quadrant set from a plurality of non-empty quadrant sets formed of non-empty quadrants in the coordinate space is identified such that the non-empty quadrants of the identified quadrant set associated with each of the at least two shapes satisfy a predetermined condition and a distance between portions of the at least two shapes in the non-empty quadrants of the identified non-empty quadrant set is an optimal value for the requested distance. The distance between the portions of the shapes in the identified quadrant set is determined to be the requested distance.