10 research outputs found
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
Geometric and statistical techniques for projective mapping of chocolate chip cookies with a large number of consumers
The so-called rapid sensory methods have proved to be useful for the sensory
study of foods by different types of panels, from trained assessors to
unexperienced consumers. Data from these methods have been traditionally
analyzed using statistical techniques, with some recent works proposing the use
of geometric techniques and graph theory. The present work aims to deepen this
line of research introducing a new method, mixing tools from statistics and
graph theory, for the analysis of data from Projective Mapping. In addition, a
large number of n=349 unexperienced consumers is considered for the first time
in Projective Mapping, evaluating nine commercial chocolate chips cookies which
include a blind duplicate of a multinational best-selling brand and seven
private labels. The data obtained are processed using the standard statistical
technique Multiple Factor Analysis (MFA), the recently appeared geometric
method SensoGraph using Gabriel clustering, and the novel variant introduced
here which is based on the pairwise distances between samples. All methods
provide the same groups of samples, with the blind duplicates appearing close
together. Finally, the stability of the results is studied using bootstrapping
and the RV and Mantel coefficients. The results suggest that, even for
unexperienced consumers, highly stable results can be achieved for MFA and
SensoGraph when considering a large enough number of assessors, around 200 for
the consensus map of MFA or the global similarity matrix of SensoGraph.Comment: 21 pages, 16 figures, 1 tabl
Generalized kernels of polygons under rotation
Given a set of orientations in the plane, two points inside
a simple polygon -see each other if there is an
-staircase contained in that connects them. The
-kernel of is the subset of points which -see all
the other points in . This work initiates the study of the computation and
maintenance of the - of a polygon as we rotate
the set by an angle , denoted -. In particular, we design efficient algorithms for (i)
computing and maintaining - while
varies in , obtaining the angular intervals
where the - is not empty and (ii) for
orthogonal polygons , computing the orientation such that the area and/or the perimeter of the
- are maximum or minimum. These
results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.Comment: 12 pages, 4 figures, a version omitting some proofs appeared at the
34th European Workshop on Computational Geometry (EuroCG 2018
A connection between Inverse Problems and Nonstandard Analysis
Nonstandard analysis was born in the decade of 1960 an attempt to give a formal context to the Leibniz approach to differential calculus, in particular in order to provide a rigorous foundation to the notions of infinitesimal and infinite quantities in analysis. The theory soon found further nontrivial applications in several areas of mathematics including Analysis, Ergodic Theory, Geometric Group Theory, Probability Theory, Number Theory or Combinatorics. In this context, we find a remarkable proof by Renling Jin in 2007 of an extension of a result from Additive Number Theory, the so-called Freiman's Little Theorem, proving a longstanding conjecture for which the only known proof uses tools of nonstandard analysis. The objective of this master thesis is to give an account of this highly nontrivial proof by providing the necessary background on nonstandard analysis, with the aim of making the proof accessible to a wider readership outside the nonstandard analysis community
A connection between Inverse Problems and Nonstandard Analysis
Nonstandard analysis was born in the decade of 1960 an attempt to give a formal context to the Leibniz approach to differential calculus, in particular in order to provide a rigorous foundation to the notions of infinitesimal and infinite quantities in analysis. The theory soon found further nontrivial applications in several areas of mathematics including Analysis, Ergodic Theory, Geometric Group Theory, Probability Theory, Number Theory or Combinatorics. In this context, we find a remarkable proof by Renling Jin in 2007 of an extension of a result from Additive Number Theory, the so-called Freiman's Little Theorem, proving a longstanding conjecture for which the only known proof uses tools of nonstandard analysis. The objective of this master thesis is to give an account of this highly nontrivial proof by providing the necessary background on nonstandard analysis, with the aim of making the proof accessible to a wider readership outside the nonstandard analysis community
On circles enclosing many points
We prove that every set of n red and n blue points in the plane contains a red and a blue point such that every circle through them encloses at least points of the set. This is a two-colored version of a problem posed by Neumann-Lara and Urrutia. We also show that every set S of n points contains two points such that every circle passing through them encloses at most points of S. The proofs make use of properties of higher order Voronoi diagrams, in the spirit of the work of Edelsbrunner, Hasan, Seidel and Shen on this topic. Closely related, we also study the number of collinear edges in higher order Voronoi diagrams and present several constructions.Peer ReviewedPostprint (author's final draft
The edge labeling of higher order Voronoi diagrams
We present an edge labeling of order-k Voronoi diagrams, Vk(S), of point sets S in the plane, and study properties of the regions defined by them. Among them, we show that Vk(S) has a small orientable cycle and path double cover, and we identify configurations that cannot appear in V3(S).Postprint (published version
The edge labeling of higher order Voronoi diagrams
We present an edge labeling of order-k Voronoi diagrams, Vk(S), of point sets S in the plane, and study properties of the regions defined by them. Among them, we show that Vk(S) has a small orientable cycle and path double cover, and we identify configurations that cannot appear in V3(S).Postprint (published version
Flight level assignment using graph coloring
This paper models an air traffic optimization problem where, on the one hand, flight operators seek to minimize fuel consumption flying at optimal cruise levels and, on the other hand, air traffic managers aim to keep intersecting airways at as distant as possible flight levels. We study such a problem as a factorized optimization, which is addressed through a spectrum graph coloring model, evaluating the effect that safety constraints have on fuel consumption, and comparing different heuristic approaches for allocation