Abstract
This article presents a new algorithm for interval plotting of the function y = f(x) based on combined sampling. The proposed method synthesizes the uniform and adaptive sampling approaches and provides a more compact and efficient function representation. During the combined sampling, the polygonal approximation with a given threshold α between the adjacent segments is constructed. The automated detection and treatment of the discontinuities based on the LR criterion are involved. Two implementations, the recursive-based and stack-based, are introduced. Finally, several tests of the proposed algorithms for the different functions involving the discontinuities and several map projection graticules are presented. The proposed method may be used for more efficient sampling the curves (map projection graticules, contour lines, or buffers) in geoinformatics.
Highlights
A function y = f (x) on interval Ω = [a, b] may have different form
The combined sampling algorithm is based on the idea of the hierarchical reconstruction of the curve shape, which follows the recursive approach with the multiple calls mixing the uniform and adaptive techniques
This article presented a new algorithm combining the uniform and adaptive sampling techniques applicable to the functions involving the discontinuities, which are detected by the LR criterion
Summary
Its polygonal approximation needs to be constructed. Adaptive sampling brings several benefits, it adapts to a different curvature of the function, reduces the amount of redundant data and provides a natural and smooth plot of the function without the jumps and breaks. This technique is popular in computer graphics; recall the well-known deCasteljau or Chaikin’s algorithms for the curve approximation. For each interval Ωkg, a polygonal approximation of f (x) is constructed using combined sampling.
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