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Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains
Department of Mechanical Engineering, Institute for Electronics Engineering and Informatics of Aveiro, University of Aveiro, 3810-193 Aveiro, Portugal
Algorithms 2024, 17(10), 444; https://doi.org/10.3390/a17100444 (registering DOI)
Submission received: 27 July 2024
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Revised: 19 September 2024
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Accepted: 3 October 2024
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Published: 5 October 2024
(This article belongs to the Special Issue Numerical Optimization and Algorithms: 2nd Edition)
Abstract
This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve.
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MDPI and ACS Style
Santos, V. Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains. Algorithms 2024, 17, 444. https://doi.org/10.3390/a17100444
AMA Style
Santos V. Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains. Algorithms. 2024; 17(10):444. https://doi.org/10.3390/a17100444
Chicago/Turabian StyleSantos, Vitor. 2024. "Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains" Algorithms 17, no. 10: 444. https://doi.org/10.3390/a17100444
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.
