Parametric Curve Intersection via Error-Controlled Discretization
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Solution Overview
Problem
Current methods for finding intersection points of arbitrary parametric curves in integrated circuit design are computationally intensive and struggle with precision, especially when dealing with complex shapes and self-intersections, leading to inefficiencies in mask fabrication.
Innovation Solution
A computer-implemented method using error-controlled discretization based on Taylor series approximation and symbolic algebra to approximate curves as linear segments, allowing for efficient computation of intersection points by recursively decreasing discretization tolerance, thereby reducing computational burden and improving precision.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If current methods are used to find intersection points of arbitrary parametric curves, then computational precision can be maintained, but computational resources and processing time increase significantly
Solution Approach 1:
The patent divides the parametric curve into multiple discrete segments based on parameter intervals. By segmenting the continuous curve into manageable pieces, the method enables efficient intersection detection between segments while maintaining overall curve accuracy, thus resolving the contradiction between precision and computational efficiency
Solution Approach 2:
The patent applies partial action by only computing intersections where segments are closer than a threshold distance, rather than checking all possible segment pairs. This selective approach maintains precision for critical regions while reducing unnecessary computations in distant regions, balancing precision and efficiency
2Manufacturing precision
If discretization tolerance is decreased to improve precision, then intersection point accuracy increases, but computational burden increases
Solution Approach 1:
The patent employs adaptive discretization where the tolerance value dynamically adjusts based on local curve characteristics and intersection proximity. In regions requiring high precision (near potential intersections), tolerance decreases; in other regions, tolerance increases. This dynamic approach maintains manufacturing precision while avoiding uniform high-resolution discretization that would increase overall computational complexity
Solution Approach 2:
The patent applies different discretization tolerances to different segments of the curve based on their local geometric properties and their proximity to other segments. This local quality approach ensures high precision where needed (near intersections) while using coarser discretization elsewhere, resolving the contradiction between overall precision and computational complexity
3Reliability
If all segments are checked for intersections, then no intersection points are missed, but processing time increases
Solution Approach 1:
The patent performs preliminary filtering by first identifying segments that are closer than a threshold distance before conducting detailed intersection analysis. This preliminary action eliminates obviously non-intersecting segment pairs from further processing, ensuring no true intersections are missed while significantly reducing the number of computations required
Solution Approach 2:
The patent replaces the brute-force mechanical checking of all segment pairs with a more efficient computational approach using distance thresholds and spatial filtering. This substitution maintains the reliability of complete intersection detection while reducing processing time by avoiding unnecessary distance calculations for distant segment pairs
Data Source
AI summary
A method includes finding, for a discretized curve comprising multiple segments, first and second segments that are closer than a threshold. The method includes determining an intersection point of the first segment and the second segment, the intersection point associated with a first parameter value for the first segment and a second parameter value for the second segment, determining an error value as a distance between a first point in the parametric curve corresponding to the first parameter value and a second point in the parametric curve corresponding to the second parameter value, and selecting the intersection point when the error value is smaller than a precision tolerance. The method includes transforming the discretized curve by removing at least a segment between the first segment and the second segment, and providing the discretized curve to manufacture a mask for reproducing the feature in the integrated circuit.