Adaptive Sampling for Web Graph Plotting
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Solution Overview
Problem
Existing computer software packages and devices for graphing mathematical functions are difficult to learn and time-consuming to use, making it challenging for students to visualize and understand abstract mathematical concepts without prior knowledge.
Innovation Solution
A method for displaying a graph of a mathematical function on a web page by dividing the range of variable values into sub-ranges, selecting initial sample values, calculating function values, and modifying the sub-range based on differences until a threshold is met, allowing for plotting without requiring user knowledge of complex software or devices.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If traditional graphing software packages or devices are used, then graphing capability is provided, but the system becomes difficult to learn and time-consuming to use
Solution Approach 1:
The system automatically determines sampling points and graph ranges without requiring user input or knowledge. The computer executes algorithms that self-adjust the graphing parameters based on the mathematical function provided, eliminating the need for users to learn complex software operations or manually configure graphing settings.
Solution Approach 2:
The system pre-calculates optimal sampling points and graph ranges before displaying the graph. By performing preliminary analysis of the mathematical function to identify critical points, asymptotes, and appropriate viewing ranges, the system prepares all necessary graphing parameters in advance, eliminating time-consuming manual configuration steps.
2Ease of operation
If traditional graphing software is used, then graphing functionality is achieved, but prior knowledge is required to correctly graph functions
Solution Approach 1:
The graphing system performs self-analysis of the mathematical function to automatically determine appropriate graphing parameters, sampling density, and display ranges. This eliminates the need for users to understand complex graphing concepts or software settings, as the system independently handles all technical decisions required for correct function visualization.
Solution Approach 2:
The system introduces an intelligent intermediary layer between the user and the graphing process. This intermediary automatically translates the mathematical function into appropriate graphing parameters and visual representations, shielding users from the complexity of graphing software while ensuring accurate and meaningful visualizations.
3Measurement precision
If uniform sampling is used across the entire range, then graph coverage is complete, but accuracy decreases in regions of rapid change
Solution Approach 1:
The system applies different sampling densities to different regions of the graph based on local characteristics. Regions with rapid function changes receive higher sampling density to capture critical behavior, while regions with slow changes use lower density. This local adaptation maintains high accuracy where needed while preserving overall graphing efficiency.
Solution Approach 2:
The sampling strategy dynamically adjusts based on the function's behavior at each point. The system continuously evaluates the derivative or rate of change and modifies sampling density in real-time during graph generation, concentrating computational resources on regions requiring higher precision while maintaining productivity in less critical areas.
4Measurement precision
If more sampling points are used, then graph accuracy improves, but computation time increases
Solution Approach 1:
The system concentrates computational effort on regions where high accuracy is most valuable, such as areas with rapid function changes, critical points, or user-specified regions of interest. By applying high sampling density only where necessary rather than uniformly across the entire range, the system achieves high graph accuracy while minimizing unnecessary computation time.
Solution Approach 2:
The system uses adaptive sampling that applies excessive precision only where required by the function's behavior. Rather than uniformly over-sampling the entire range, the system performs partial high-precision sampling in critical regions and lower-precision sampling elsewhere, optimizing the balance between accuracy and computation time.
Data Source
AI summary
Technology is described herein for quickly, efficiently and accurately rendering a plot of the mathematical function within an interactive graph on a web page, such as a search results web page. The technology includes determining a sampling resolution indicating the points of the mathematical function to be plotted within a display region of the plot. The sampling resolution includes a higher number of sample points in sections of the mathematical function with function values that change considerably, and includes a lower number of sample points in sections with function values that change slightly. The interactive graph provides the user with a rich interactive experience that facilitates and encourages the user's understanding of mathematical concepts and underlying principles.