Approximating Point Sets via Polytope Intersection
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Solution Overview
Problem
Existing methods for approximating ordered sets of points with non-zero accuracy require high computational power and memory, and lack efficient mechanisms for data compression, as they directly analyze raw sets of points without effectively utilizing geometric elements for representation.
Innovation Solution
The method involves approximating an ordered set of physical points in physical space by generating bundles of geometric elements, such as straight line segments and circular arcs, and mapping these elements to polytopes in a parametric space, where the intersection of these polytopes determines the approximate best-fit geometric elements that represent the points with specified accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If raw sets of points are directly analyzed without approximation, then measurement precision is maintained, but computational complexity and memory requirements increase significantly
Solution Approach 1:
The ordered set of points is divided into multiple subgroups, with each subgroup approximated by a separate geometric element. This segmentation allows the complex problem of approximating all points to be broken down into simpler subproblems, reducing computational complexity while maintaining position accuracy within acceptable tolerances for each segment.
Solution Approach 2:
The method transforms the problem from operating on raw point coordinates to operating on geometric element parameters (such as line segment endpoints, circular arc center and radius). This parameter transformation reduces the data representation from N points to fewer geometric parameters, lowering computational complexity and memory requirements while preserving measurement precision through controlled approximation.
2Quantity of substance
If geometric elements are used to approximate points, then data compression is achieved, but manufacturing precision of the approximation decreases
Solution Approach 1:
The method generates multiple candidate geometric elements (excessive action) for each subgroup of points, then selects the best approximation by evaluating which candidate minimizes the deviation from the actual points. This approach ensures that even though geometric elements are used for compression, the approximation accuracy is optimized by choosing from multiple possibilities rather than using a single fixed element.
Solution Approach 2:
The patent replaces direct geometric approximation with a parametric space transformation approach. By mapping geometric elements to polytopes in parametric space and using intersection operations, the method achieves more precise control over approximation accuracy while maintaining data compression benefits. This substitution of the approximation mechanism allows for better precision control.
3Measurement precision
If high power processors and large memory are used for analyzing raw points, then analysis accuracy is improved, but loss of energy and processing time increase
Solution Approach 1:
The method extracts only the essential geometric characteristics from the raw point sets by fitting geometric elements (line segments, circular arcs) to the data. This extraction process removes redundant information while retaining the critical geometric features needed for analysis, thereby reducing the computational workload and energy consumption required for subsequent processing while maintaining analysis accuracy.
Data Source
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AI summary
An ordered set of physical points, each comprising a nominal point and an associated allowable deviation, is approximated by a sequence of geometric elements determined by a method of intersecting polytopes in a parametric space. A first bundle of geometric elements connecting a first subgroup of physical points is generated and mapped to a first polytope. A second bundle of geometric elements connecting a second subgroup of physical points is generated and mapped to a second polytope. If the intersection between the first polytope and the second polytope is not null, the points in the intersection region correspond to geometric elements which approximate the physical points in the combined two subgroups. The process is repeated iteratively for additional subgroups. The center point of the final intersection region corresponds to an approximate best-fit geometric element.