Polygon Merging in Adjacent Tiles for Implicit Relation Plotting
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current methods for plotting implicit relations in mathematical software packages face challenges in efficiently merging polygons in adjacent tiles, leading to high computational complexity and memory requirements, especially for devices with limited resources.
Innovation Solution
A method for merging polygons in adjacent tiles by identifying overlapping invalid edges and generating a new edge linked list for a merged polygon, which includes determining matching invalid edges, checking for overlap, and merging or splitting edges as necessary to create a cohesive polygon representation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If recursive mesh division is used to detect contours of implicit relations, then plotting accuracy is improved, but computational complexity and memory requirements increase
Solution Approach 1:
The graphing region is divided into a rectangular mesh of tiles, with each tile independently processed to determine contour segments. This segmentation allows the complex global problem to be broken into manageable local problems, reducing computational complexity while maintaining plotting accuracy through systematic exploration of each tile.
Solution Approach 2:
The method performs preliminary evaluation of function values at tile corners before recursively dividing tiles. This preliminary action identifies tiles that definitely contain or exclude contours, allowing early termination of recursion for many tiles and significantly reducing the number of recursive divisions needed, thus lowering computational complexity while preserving accuracy.
2Measurement precision
If recursive mesh division is used to detect contours of implicit relations, then plotting accuracy is improved, but memory requirements increase
Solution Approach 1:
The method extracts and processes only the necessary information from each tile - specifically, whether contour segments exist and their locations - rather than storing complete polygon data for all tiles simultaneously. This extraction approach minimizes memory requirements by keeping only essential contour information while maintaining plotting accuracy.
Solution Approach 2:
The method performs preliminary determination of contour presence in each tile before full processing. Tiles that are determined to not contain contours are excluded from further processing and memory storage, significantly reducing memory requirements while preserving accuracy by ensuring all actual contours are still captured.
3Measurement precision
If contour detection is performed on fine-grained rectangular meshes, then plotting precision is improved, but processing time increases
Solution Approach 1:
The method performs preliminary evaluation of function values at tile corners to quickly determine which tiles contain contours. This preliminary action filters out the majority of tiles that do not require further processing, allowing the system to achieve high plotting precision on fine-grained meshes without the proportional increase in processing time that would result from uniformly processing all tiles.
Solution Approach 2:
The method applies full recursive contour detection only to tiles that are determined to contain contours, while using simpler evaluation for tiles that clearly do not contain contours. This partial application of the computationally intensive algorithm reduces processing time while maintaining plotting precision by ensuring all actual contours are detected with full accuracy.
Data Source
AI summary
A method for merging polygons in adjacent tiles is provided that includes determining that a first polygon in a tile having a first edge linked list and a second polygon in an adjacent tile having a second edge linked list have overlapping invalid edges, wherein an edge includes an ordered set of point coordinates in which an initial two point coordinates lie on a tile boundary and last point coordinates of the set also lie on a tile boundary, and an edge linked list is an ordered list of edges of a polygon, and merging the first edge linked list and the second edge linked list to generate a third edge linked list for a merged polygon comprising the first polygon and the second polygon.