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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2022.tde-30032022-172201
Document
Author
Full name
Arthur Gabriel de Santana
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2022
Supervisor
Committee
Birgin, Ernesto Julian Goldberg (President)
Gardenghi, John Lenon Cardoso
Laurain, Antoine
Title in Portuguese
Cobertura com círculos de raio mínimo
Keywords in Portuguese
Algoritmo de Fortune
Algoritmo de Sutherland-Hodgman
Cobertura com círculos
Diagramas de Voronoi
Abstract in Portuguese
Neste trabalho, investigamos o problema de cobrir conjuntos de polígonos convexos usando círculos de mesmo raio mínimo. Utilizamos uma abordagem de otimização não-linear, definindo as restrições de viabilidade como diferenças entre áreas de polígonos curvilineares. Utilizando um particionamento baseado em Diagramas de Voronoi, apresentamos algoritmos para o cálculo exato das funções de restrição, além de suas primeiras derivadas. São expostos também os métodos usados nesse processo para o cálculo de Diagramas de Voronoi, interseções entre poliedros, polígonos e polígonos curvilineares, além do cálculo de áreas e comprimentos de interesse.
Title in English
Covering with circles of minimum radius
Keywords in English
Covering with circles
Fortunes Algorithm
Sutherland- Hodgman Algorithm
Voronoi Diagrams
Abstract in English
In this work, we investigate the problem of covering sets of convex polygons using circles of the same, minimal, radius. We utilize a nonlinear optimization approach, defining the feasibility constraints as differences between areas of curvilinear polygons. By utilizing a partition based on Voronoi Diagrams, we present algorithms for the exact computation of the constraint functions and its first derivatives. Methods used in the process are also shown, for the computation of the Voronoi Diagrams, intersections between polyhedra, polygons and curvilinear polygons, and calculation of areas and lengths of interest.
 
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Publishing Date
2022-03-30
 
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