Bivariate Polynomial Visualization via Vector Contour Extraction
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Solution Overview
Problem
Existing methods for visualizing bivariate polynomials result in large datafiles that hinder server-client configurations due to bandwidth limitations, and the additional step of determining contours increases processing time, delaying visualization and increasing energy consumption.
Innovation Solution
The method involves determining distinct z-values and shape-characteristic parameters to represent polynomial contours as vectors, eliminating the need for bitmap datafiles and allowing faster processing, enabling immediate visualization adjustments and reduced energy consumption.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the processing step calculates z-values on every x-y coordinate of a predefined x-y grid, then the visualization data is complete and accurate, but the resulting datafile size becomes relatively large
Solution Approach 1:
The patent extracts only the essential information needed for visualization by determining contour lines at specific z-values rather than calculating and storing all z-values for every grid point. This extraction approach retains the visual representation capability while significantly reducing data volume.
Solution Approach 2:
Instead of the conventional approach of calculating all z-values first and then generating contours, the patent inverts the process by directly determining contour lines through mathematical analysis of the polynomial equation, bypassing the need to compute the complete z-value matrix.
2Quantity of substance
If the processing step determines contours of the polynomial at certain z-values, then the datafile size is reduced, but the processing time increases
Solution Approach 1:
The patent replaces the mechanical/computational approach of grid-based z-value calculation with a mathematical analytical approach. By solving the polynomial equation directly for contour lines using algebraic methods, the system achieves faster processing than iterative numerical computation while maintaining accuracy.
Solution Approach 2:
The patent performs preliminary mathematical analysis of the polynomial structure to identify critical points and contour characteristics before generating the actual contour data. This preliminary characterization enables more efficient contour determination and reduces the computational workload during the main processing step.
3Measurement precision
If the processing step determines contours by calculating z-values on every x-y coordinate, then the contour accuracy is maintained, but the energy consumption increases
Solution Approach 1:
The patent extracts only the essential contour information needed for accurate visualization by directly solving for contour lines at specific z-values rather than computing the complete z-value matrix. This extraction maintains contour accuracy while significantly reducing the computational energy required.
Solution Approach 2:
The patent changes the computational parameters from calculating all grid point z-values to directly determining contour line equations through mathematical analysis. This parameter change in the computational approach maintains accuracy while reducing energy consumption by avoiding redundant calculations.
Data Source
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Figure 3B
AI summary
A method for the visualization of bivariate polynomials, comprises a step of loading a polynomial and boundary conditions into computer processing means, a step of processing (11-16), by the computer processing means, of the polynomial within the boundary conditions for visualization and a step of visualizing (17) the polynomial on visualization means. The processing step comprises a step of determining (11) a plurality of distinct z-values, which may overlap z-values spanning the boundary conditions. The processing step further comprises a step of determining (12) a combination of shape-characteristic parameters of the polynomial. Additionally, the processing step comprises determining (15) at least one or a combination of vector-characteristic parameters based on the corresponding combination of shape-characteristic parameters. The at least one vector-characteristic parameter, in combination with the combination of shape-characteristic parameters, is representative for a contour of the polynomial at each distinct z-value. Drawing directives, which may be visualization system-specific, can be set up (16) on the basis of the combination of vector-characteristic parameters.