189 research outputs found
Implementation of linear minimum area enclosing traingle algorithm
This article has been made available through the Brunel Open Access Publishing Fund.An algorithm which computes the minimum area triangle enclosing a convex polygon in linear time already exists in the literature. The paper describing the algorithm also proves that the provided solution is optimal and a lower complexity sequential algorithm cannot exist. However, only a high-level description of the algorithm was provided, making the implementation difficult to reproduce. The present note aims to contribute to the field by providing a detailed description of the algorithm which is easy to implement and reproduce, and a benchmark comprising 10,000 variable sized, randomly generated convex polygons for illustrating the linearity of the algorithm
Problems and applications of Discrete and Computational Geometry concerning graphs, polygons, and points in the plane
Esta tesistratasobreproblemasyaplicacionesdelageometríadiscretay
computacional enelplano,relacionadosconpolígonos,conjuntosdepuntos
y grafos.
Después deunprimercapítulointroductorio,enelcapítulo 2 estudiamos
una generalizacióndeunfamosoproblemadevisibilidadenelámbitodela
O-convexidad. Dadounconjuntodeorientaciones(ángulos) O, decimosque
una curvaes O-convexa si suintersecciónconcualquierrectaparalelaauna
orientaciónde O es conexa.Cuando O = {0◦, 90◦}, nosencontramosenel
caso delaortoconvexidad,consideradodeespecialrelevancia.El O-núcleo
de unpolígonoeselconjuntodepuntosdelmismoquepuedenserconectados
con cualquierotropuntodelpolígonomedianteunacurva O-convexa.En
este trabajoobtenemos,para O = {0◦} y O = {0◦, 90◦}, unalgoritmopara
calcular ymantenerel O-núcleodeunpolígonoconformeelconjuntode
orientaciones O rota. Dichoalgoritmoproporciona,además,losángulosde
rotación paralosqueel O-núcleotieneáreayperímetromáximos.
En elcapítulo 3 consideramos unaversiónbicromáticadeunproblema
combinatorioplanteadoporNeumann-LarayUrrutia.Enconcreto,de-
mostramos quetodoconjuntode n puntosazulesy n puntosrojosenel
plano contieneunparbicromáticodepuntostalquetodocírculoquelos
tenga ensufronteracontieneensuinterioralmenos n(1− 1 √2
)−o(n) puntos
del conjunto.Esteproblemaestáfuertementeligadoalcálculodelosdiagra-
mas deVoronoideordensuperiordelconjuntodepuntos,pueslasaristas
de estosdiagramascontienenprecisamentetodosloscentrosdeloscírculos
que pasanpordospuntosdelconjunto.Porello,nuestralíneadetrabajo
actual enesteproblemaconsisteenexplorarestaconexiónrealizandoun
estudio detalladodelaspropiedadesdelosdiagramasdeVoronoideorden
superior.
En loscapítulos 4 y 5, planteamosdosaplicacionesdelateoríadegrafos
6
7
al análisissensorialyalcontroldeltráficoaéreo,respectivamente.Enel
primer caso,presentamosunnuevométodoquecombinatécnicasestadísti-
cas ygeométricasparaanalizarlasopinionesdelosconsumidores,recogidas
a travésdemapeoproyectivo.Estemétodoesunavariacióndelmétodo
SensoGraph ypretendecapturarlaesenciadelmapeoproyectivomediante
el cálculodelasdistanciaseuclídeasentrelosparesdemuestrasysunor-
malización enelintervalo [0, 1]. Acontinuación,aplicamoselmétodoaun
ejemplo prácticoycomparamossusresultadosconlosobtenidosmediante
métodosclásicosdeanálisissensorialsobreelmismoconjuntodedatos.
En elsegundocaso,utilizamoslatécnicadelespectro-coloreadodegrafos
para plantearunmodelodecontroldeltráficoaéreoquepretendeoptimizar
el consumodecombustibledelosavionesalmismotiempoqueseevitan
colisiones entreellos.This thesisdealswithproblemsandapplicationsofdiscreteandcomputa-
tional geometryintheplane,concerningpolygons,pointsets,andgraphs.
After afirstintroductorychapter,inChapter 2 westudyageneraliza-
tion ofafamousvisibilityproblemintheframeworkof O-convexity. Given
a setoforientations(angles) O, wesaythatacurveis O-convex if itsin-
tersection withanylineparalleltoanorientationin O is connected.When
O = {0◦, 90◦}, wefindourselvesinthecaseoforthoconvexity,consideredof
specialrelevance.The O-kernel of apolygonisthesubsetofpointsofthe
polygonthatcanbeconnectedtoanyotherpointofthepolygonwithan
O-convexcurve.Inthisworkweobtain,for O = {0◦} and O = {0◦, 90◦}, an
algorithm tocomputeandmaintainthe O-kernelofapolygonasthesetof
orientations O rotates. Thisalgorithmalsoprovidestheanglesofrotation
that maximizetheareaandperimeterofthe O-kernel.
In Chapter 3, weconsiderabichromaticversionofacombinatorialprob-
lem posedbyNeumann-LaraandUrrutia.Specifically,weprovethatevery
set of n blue and n red pointsintheplanecontainsabichromaticpairof
pointssuchthateverycirclehavingthemonitsboundarycontainsatleast
n(1 − 1 √2
) − o(n) pointsofthesetinitsinterior.Thisproblemisclosely
related toobtainingthehigherorderVoronoidiagramsofthepointset.The
edges ofthesediagramscontain,precisely,allthecentersofthecirclesthat
pass throughtwopointsoftheset.Therefore,ourcurrentlineofresearch
on thisproblemconsistsonexploringthisconnectionbystudyingindetail
the propertiesofhigherorderVoronoidiagrams.
In Chapters 4 and 5, weconsidertwoapplicationsofgraphtheoryto
sensory analysisandairtrafficmanagement,respectively.Inthefirstcase,
weintroduceanewmethodwhichcombinesgeometricandstatisticaltech-
niques toanalyzeconsumeropinions,collectedthroughprojectivemapping.
This methodisavariationofthemethodSensoGraph.Itaimstocapture
4
5
the essenceofprojectivemappingbycomputingtheEcuclideandistances
betweenpairsofsamplesandnormalizingthemtotheinterval [0, 1]. Weap-
ply themethodtoareal-lifescenarioandcompareitsperformancewiththe
performanceofclassicmethodsofsensoryanalysisoverthesamedataset.
In thesecondcase,weusetheSpectrumGraphColoringtechniquetopro-
poseamodelforairtrafficmanagementthataimstooptimizetheamount
of fuelusedbytheairplanes,whileavoidingcollisionsbetweenthem
Reshaping Convex Polyhedra
Given a convex polyhedral surface P, we define a tailoring as excising from P
a simple polygonal domain that contains one vertex v, and whose boundary can be
sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In
particular, a digon-tailoring cuts off from P a digon containing v, a subset of
P bounded by two equal-length geodesic segments that share endpoints, and can
then zip closed.
In the first part of this monograph, we primarily study properties of the
tailoring operation on convex polyhedra. We show that P can be reshaped to any
polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings.
This investigation uncovered previously unexplored topics, including a notion
of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto
P.
In the second part of this monograph, we study vertex-merging processes on
convex polyhedra (each vertex-merge being in a sense the reverse of a
digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to
produce non-overlapping polyhedral and planar unfoldings, which led us to
develop an apparently new theory of convex sets, and of minimal length
enclosing polygons, on convex polyhedra.
All our theorem proofs are constructive, implying polynomial-time algorithms.Comment: Research monograph. 234 pages, 105 figures, 55 references. arXiv
admin note: text overlap with arXiv:2008.0175
A Boundary-Based Measure for Gerrymandering
This paper presents a new measure for quantifying legislative gerrymandering
based intuitive observation that in a non-gerrymandered district a randomly placed
observer should be able to walk in a straight line and only cross the boundary of the
district once, when the district is exited. We make this notion precise in terms of
the expected value of such crossings. The result is the Boundary Intersection
Number, or BIN. Properties of the BIN score are proven and its computational
properties discussed
Separating bichromatic point sets in the plane by restricted orientation convex hulls
The version of record is available online at: http://dx.doi.org/10.1007/s10898-022-01238-9We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and O be a set of k=2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of O for which the O-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlogn) time and O(n) space, n=|R|+|B|, we compute the set of rotation angles such that, after simultaneously rotating the lines of O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where O is formed by k=2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of O, let ai be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O(1/T·NlogN) time and O(1/T·N) space, where T=min{a1,…,ak} and N=max{k,|R|+|B|}. We finally consider the case in which O is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to p. We show that this last case can also be solved in optimal O(nlogn) time and O(n) space, where n=|R|+|B|.Carlos Alegría: Research supported by MIUR Proj. “AHeAD” no 20174LF3T8. David Orden:
Research supported by Project PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish
Ministry of Science and Innovation. Carlos Seara: Research supported by Project PID2019-104129GB-I00 /
AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation. Jorge Urrutia: Research
supported in part by SEP-CONACYThis project has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska–Curie Grant Agreement No 734922.Peer ReviewedPostprint (published version
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