Multidimensional Curve Point Alignment via Region Approximation
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Solution Overview
Problem
Existing methods for determining whether a point lies along a curve in multidimensional space are computationally intensive, particularly when tolerating displacement, as they involve measuring the shortest distance between the point and the curve.
Innovation Solution
The method involves approximating a complex curve with a simpler one by identifying regions encompassing portions of the original curve and using these regions to efficiently determine if a point lies within them, thereby indicating its alignment with the curve.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the shortest distance method is used to determine whether a point lies along a curve with tolerance, then measurement accuracy is improved, but computational complexity increases
Solution Approach 1:
The curve is divided into multiple line segments connecting consecutive vertices. Instead of computing distance to every point on the continuous curve, the method segments the curve into discrete linear portions, allowing efficient point-to-line segment distance calculations that satisfy tolerance requirements without exhaustive computation.
Solution Approach 2:
The method pre-identifies candidate line segments that could contain the closest point to the query point, using spatial relationships and tolerance thresholds to narrow down the search space before performing actual distance calculations. This preliminary filtering avoids computing distances to all curve points.
2Manufacturing precision
If a complex curve with many vertices is used to accurately represent data, then manufacturing precision is improved, but device complexity increases
Solution Approach 1:
The method uses a tolerance threshold to determine the sufficient level of curve detail needed. Rather than requiring exact representation with all vertices, the tolerance-based approach allows partial use of curve features - only those vertices and segments necessary to satisfy the tolerance requirement are considered, reducing complexity while maintaining adequate accuracy.
Data Source
AI summary
There is disclosed a computer-implemented method for determining whether a point lies along a first curve in a multidimensional space and program instructions executable for carrying out the method. The method comprises appreciating first curve coordinates defining each of a position and a form of the first curve in the multidimensional space; identifying regions of the multidimensional space each encompassing a respective portion of the first curve while generating a second curve approximating the first curve; and storing, in a non-transitory computer-readable medium, region coordinates of at least one of the regions; whereby an analysis of the region coordinates and point coordinates defining a position of the point in the multidimensional space is executable to provide to a user either an indication that the point lies along the first curve or an indication that the point does not lie along the first curve. Some implementations comprise executing said analysis and providing said indication.