Fault localization is the process of realizing the true source of a failure from a set of collected failure notifications. Isolating failure recovery within the network optical domain is necessary to resolve alarm storm problems. The introduction of the monitoring trail (m-trail) has been proven to deliver better performance by employing monitoring resources in a form of optical trails - a monitoring framework that generalizes all the previously reported counterparts. In this dissertation, the m-trail design is explored and a focus is given to the analysis on using m-trails with established lightpaths to achieve fault localization. This process saves network resources by reducing the number of the m-trails required for fault localization and therefore the number of wavelengths used in the network. A novel approach based on Geographic Midpoint Technique, an adapted version of the Chinese Postman’s Problem (CPP) solution and an adapted version of the Traveling Salesman’s Problem (TSP) solution algorithms is introduced. The desirable features of network architectures and the enabling of innovative technologies for delivering future millimeter-waveband (mm-WB) Radio-over-Fiber (RoF) systems for wireless services integrated in a Dense Wavelength Division Multiplexing (DWDM) is proposed in this dissertation. For the conceptual illustration, a DWDM RoF system with channel spacing of 12.5 GHz is considered. The mm-WB Radio Frequency (RF) signal is obtained at each Optical Network Unit (ONU) by simultaneously using optical heterodyning photo detection between two optical carriers. The generated RF modulated signal has a frequency of 12.5 GHz. This RoF system is easy, cost-effective, resistant to laser phase noise and also reduces maintenance needs, in principle. A revision of related RoF network proposals and experiments is also included. A number of models for Passive Optical Networks (PON)/ RoF-PON that combine both innovative and existing ideas along with a number of solutions for m-trail design problem of these models are proposed. The comparison between these models uses the expected survivability function which proved that these models are liable to be implemented in the new and existing PON/ RoF-PON systems. This dissertation is followed by recommendation of possible directions for future research in this area.

Fault localization is the process of realizing the true source of a failure from a set of collected failure notifications. Isolating failure recovery within the network optical domain is necessary to resolve alarm storm problems. The introduction of the monitoring trail (m-trail) has been proven to deliver better performance by employing monitoring resources in a form of optical trails - a monitoring framework that generalizes all the previously reported counterparts. In this dissertation, the m-trail design is explored and a focus is given to the analysis on using m-trails with established lightpaths to achieve fault localization. This process saves network resources by reducing the number of the m-trails required for fault localization and therefore the number of wavelengths used in the network. A novel approach based on Geographic Midpoint Technique, an adapted version of the Chinese Postman’s Problem (CPP) solution and an adapted version of the Traveling Salesman’s Problem (TSP) solution algorithms is introduced. The desirable features of network architectures and the enabling of innovative technologies for delivering future millimeter-waveband (mm-WB) Radio-over-Fiber (RoF) systems for wireless services integrated in a Dense Wavelength Division Multiplexing (DWDM) is proposed in this dissertation. For the conceptual illustration, a DWDM RoF system with channel spacing of 12.5 GHz is considered. The mm-WB Radio Frequency (RF) signal is obtained at each Optical Network Unit (ONU) by simultaneously using optical heterodyning photo detection between two optical carriers. The generated RF modulated signal has a frequency of 12.5 GHz. This RoF system is easy, cost-effective, resistant to laser phase noise and also reduces maintenance needs, in principle. A revision of related RoF network proposals and experiments is also included. A number of models for Passive Optical Networks (PON)/ RoF-PON that combine both innovative and existing ideas along with a number of solutions for m-trail design problem of these models are proposed. The comparison between these models uses the expected survivability function which proved that these models are liable to be implemented in the new and existing PON/ RoF-PON systems. This dissertation is followed by recommendation of possible directions for future research in this area.

This thesis investigated Stochastic Runge–Kutta Lawson (SRKL) schemes and their application to the Heston model. Two distinct SRKL discretization methods were used to simulate a single asset’s dynamics under the Heston model, notably the Euler–Maruyama and Midpoint schemes. Additionally, standard Monte Carlo and variance reduction techniques were implemented. European and Asian option prices were estimated and compared with a benchmark value regarding accuracy, effectiveness, and computational complexity. Findings showed that the SRKL Euler–Maruyama schemes exhibited promise in enhancing the price for simple and path-dependent options. Consequently, integrating SRKL numerical methods into option valuation provides notable advantages by addressing challenges posed by the Heston model’s SDEs. Given the limited scope of this research topic, it is imperative to conduct further studies to understand the use of SRKL schemes within other models.

Introdução: o ciclo de sono-vigília é o comportamento rítmico circadiano mais onipresente nos humanos. Ele é estabelecido por um sistema temporizador circadiano endógeno que é regulado pela presença ou ausência de luz no ambiente. O comportamento em relação a dormir e acordar varia entre os indivíduos e essa característica nomeia-se cronotipo. Efeitos negativos à saúde têm sido associados ao cronotipo que apresenta horários para iniciar e terminar o sono mais tarde. A hipótese considerada para explicar esses achados é de que demandas sociais, que não levam em conta essa variabilidade individual, sejam mediadoras dessa relação. Objetivos: avaliar a associação entre cronotipo e saúde mental (bem-estar psicológico e sintomas depressivos) levando-se em conta fatores demográficos (sexo e idade) e as rotinas de escola ou trabalho. Materiais e métodos: estudos transversais aninhados a um estudo epidemiológico em uma amostra de indivíduos do Vale do Taquari, no sul do Brasil. Na primeira avaliação, 6.506 participantes foram avaliados quanto a variáveis demográficas, dados de saúde, sintomas depressivos (Inventário de Depressão de Beck – BDI) e cronotipo (Questionário de Cronotipo de Munique – MCTQ). Na segunda etapa, 1.127 indivíduos entre 18 e 65 anos, selecionados a partir de seu cronotipo foram avaliados através do Índice de bem-estar de 5-itens da Organização Mundial da Saúde (OMS). O ponto médio do sono nos dias de rotinas de trabalho ou escolares foi utilizado como indicador do cronotipo, sendo “atrasado” aquele que tem o ponto médio mais tarde em relação ao início da noite e “avançado”, mais cedo. Resultados: no primeiro artigo, estudou-se a relação entre cronotipo e depressão em uma amostra de estudantes adolescentes. O modelo de regressão que melhor explicou a diferença entre os grupos com diferentes níveis de sintomas de depressão (BDI<10 X BDI!10) incluiu o sexo feminino e o cronotipo atrasado. O segundo artigo demonstrou que a escala de Bem-estar de 5-itens da OMS tem uma estrutura unidimensional, boa validade interna e externa e utilidade como instrumento de triagem para depressão quando comparada ao BDI. Assim, no terceiro artigo, foi avaliada a relação entre cronotipo e bem-estar, em indivíduos entre 18 e 65 anos. O modelo de regressão que incluiu o cronotipo atrasado, maior carga de trabalho, rotinas de trabalho mais cedo no dia e menor exposição à luz do sol, como variáveis preditoras, e piores escores na Escala de Bem-estar, como desfecho, foi significativo para o sexo feminino. Discussão: a presente tese explicitou a importância de considerar os fatores idade e sexo na expressão do cronotipo e na relação deste com saúde mental. Corroborou com a hipótese de que as demandas sociais mediam a relação entre cronotipo e os desfechos estudados. A expressão do cronotipo nos dias de rotinas escolares ou de trabalho foi identificada como a variável que estabeleceu a mais forte relação com piores escores de bemestar e com mais sintomas de depressão. Por fim, reforçou a necessidade de revermos as rotinas de trabalho e escolares que, ao não considerarem as diferenças fisiológicas individuais, têm-se associado, de forma consistente, a conseqüências negativas à saúde. / Introduction: the sleep-wake cycle is the most ubiquitous human circadian rhythmic behavior. It is established by an endogenous circadian timing system that is regulated by the presence or absence of light in the environment. Sleep and wake behavior varies among individuals and this feature has been termed chronotype. Negative health effects have been associated with the chronotype that presents later start and end sleep times. A hypothesis to explain these findings is that social demands, which do not take into account individual variability, are mediators of this relationship. Objectives: to evaluate the association between chronotype and mental health (psychological well-being and depressive symptoms) taking into account demographic factors (age and sex) and the routines of school or work. Materials and methods: the studies included here are cross-sectional nested to an epidemiological study in a sample of individuals from “Vale do Taquari”, in southern Brazil. In the first evaluation, 6,506 participants were assessed for demographic variables, health data, depressive symptoms (Beck Depression Inventory - BDI) and chronotype (Munich Chronotype Questionnaire - MCTQ). In the second stage, 1,127 individuals between 18 and 65 years, selected based on their chronotype, were evaluated through the Well-being 5 items Index from the World Health Organization (WHO). The midpoint of sleep on working or school days was used as an indicator of chronotype, and considered "delayed" or “late” those who have later midpoints in relation to environmental night and "advanced" or “early”, earlier. Results: in the first article, we studied the relationship between chronotype and depression in a sample of adolescent students. The regression model that best explained the difference between groups with different levels of depression symptoms (BDI<10 X BDI!10) included female sex and late chronotype. The second article demonstrated that the Well-being index has a unidimensional structure, good internal and external validities and might be usefulness as a screening tool for depression when compared to the BDI. Thus, in the third article, we evaluated the relationship between chronotype and well-being in individuals between 18 and 65 years. The regression model that included late chronotype, increased workload, earlier working routines in the day and less exposure to sunlight, as predictor variables, and worse scores on the well-being index, as the outcome was significant for females. Discussion: the present thesis content highlighted the importance of considering age and sex as factors influencing the expression of chronotype and the relationship with mental health outcomes. It corroborated the hypothesis that social demands mediate the relationship between chronotype and the studied outcomes. The expression of chronotype during the days of work or school routines was identified as the variable that established the strongest relationship with worse well-being scores and more depression symptoms. Finally, it reinforced the need to reconsider work and school routines that, by not taking into acount individual physiological differences, have been associated consistently with negative health consequences.

Introdução: o ciclo de sono-vigília é o comportamento rítmico circadiano mais onipresente nos humanos. Ele é estabelecido por um sistema temporizador circadiano endógeno que é regulado pela presença ou ausência de luz no ambiente. O comportamento em relação a dormir e acordar varia entre os indivíduos e essa característica nomeia-se cronotipo. Efeitos negativos à saúde têm sido associados ao cronotipo que apresenta horários para iniciar e terminar o sono mais tarde. A hipótese considerada para explicar esses achados é de que demandas sociais, que não levam em conta essa variabilidade individual, sejam mediadoras dessa relação. Objetivos: avaliar a associação entre cronotipo e saúde mental (bem-estar psicológico e sintomas depressivos) levando-se em conta fatores demográficos (sexo e idade) e as rotinas de escola ou trabalho. Materiais e métodos: estudos transversais aninhados a um estudo epidemiológico em uma amostra de indivíduos do Vale do Taquari, no sul do Brasil. Na primeira avaliação, 6.506 participantes foram avaliados quanto a variáveis demográficas, dados de saúde, sintomas depressivos (Inventário de Depressão de Beck – BDI) e cronotipo (Questionário de Cronotipo de Munique – MCTQ). Na segunda etapa, 1.127 indivíduos entre 18 e 65 anos, selecionados a partir de seu cronotipo foram avaliados através do Índice de bem-estar de 5-itens da Organização Mundial da Saúde (OMS). O ponto médio do sono nos dias de rotinas de trabalho ou escolares foi utilizado como indicador do cronotipo, sendo “atrasado” aquele que tem o ponto médio mais tarde em relação ao início da noite e “avançado”, mais cedo. Resultados: no primeiro artigo, estudou-se a relação entre cronotipo e depressão em uma amostra de estudantes adolescentes. O modelo de regressão que melhor explicou a diferença entre os grupos com diferentes níveis de sintomas de depressão (BDI<10 X BDI!10) incluiu o sexo feminino e o cronotipo atrasado. O segundo artigo demonstrou que a escala de Bem-estar de 5-itens da OMS tem uma estrutura unidimensional, boa validade interna e externa e utilidade como instrumento de triagem para depressão quando comparada ao BDI. Assim, no terceiro artigo, foi avaliada a relação entre cronotipo e bem-estar, em indivíduos entre 18 e 65 anos. O modelo de regressão que incluiu o cronotipo atrasado, maior carga de trabalho, rotinas de trabalho mais cedo no dia e menor exposição à luz do sol, como variáveis preditoras, e piores escores na Escala de Bem-estar, como desfecho, foi significativo para o sexo feminino. Discussão: a presente tese explicitou a importância de considerar os fatores idade e sexo na expressão do cronotipo e na relação deste com saúde mental. Corroborou com a hipótese de que as demandas sociais mediam a relação entre cronotipo e os desfechos estudados. A expressão do cronotipo nos dias de rotinas escolares ou de trabalho foi identificada como a variável que estabeleceu a mais forte relação com piores escores de bemestar e com mais sintomas de depressão. Por fim, reforçou a necessidade de revermos as rotinas de trabalho e escolares que, ao não considerarem as diferenças fisiológicas individuais, têm-se associado, de forma consistente, a conseqüências negativas à saúde. / Introduction: the sleep-wake cycle is the most ubiquitous human circadian rhythmic behavior. It is established by an endogenous circadian timing system that is regulated by the presence or absence of light in the environment. Sleep and wake behavior varies among individuals and this feature has been termed chronotype. Negative health effects have been associated with the chronotype that presents later start and end sleep times. A hypothesis to explain these findings is that social demands, which do not take into account individual variability, are mediators of this relationship. Objectives: to evaluate the association between chronotype and mental health (psychological well-being and depressive symptoms) taking into account demographic factors (age and sex) and the routines of school or work. Materials and methods: the studies included here are cross-sectional nested to an epidemiological study in a sample of individuals from “Vale do Taquari”, in southern Brazil. In the first evaluation, 6,506 participants were assessed for demographic variables, health data, depressive symptoms (Beck Depression Inventory - BDI) and chronotype (Munich Chronotype Questionnaire - MCTQ). In the second stage, 1,127 individuals between 18 and 65 years, selected based on their chronotype, were evaluated through the Well-being 5 items Index from the World Health Organization (WHO). The midpoint of sleep on working or school days was used as an indicator of chronotype, and considered "delayed" or “late” those who have later midpoints in relation to environmental night and "advanced" or “early”, earlier. Results: in the first article, we studied the relationship between chronotype and depression in a sample of adolescent students. The regression model that best explained the difference between groups with different levels of depression symptoms (BDI<10 X BDI!10) included female sex and late chronotype. The second article demonstrated that the Well-being index has a unidimensional structure, good internal and external validities and might be usefulness as a screening tool for depression when compared to the BDI. Thus, in the third article, we evaluated the relationship between chronotype and well-being in individuals between 18 and 65 years. The regression model that included late chronotype, increased workload, earlier working routines in the day and less exposure to sunlight, as predictor variables, and worse scores on the well-being index, as the outcome was significant for females. Discussion: the present thesis content highlighted the importance of considering age and sex as factors influencing the expression of chronotype and the relationship with mental health outcomes. It corroborated the hypothesis that social demands mediate the relationship between chronotype and the studied outcomes. The expression of chronotype during the days of work or school routines was identified as the variable that established the strongest relationship with worse well-being scores and more depression symptoms. Finally, it reinforced the need to reconsider work and school routines that, by not taking into acount individual physiological differences, have been associated consistently with negative health consequences.

Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.

Introdução: o ciclo de sono-vigília é o comportamento rítmico circadiano mais onipresente nos humanos. Ele é estabelecido por um sistema temporizador circadiano endógeno que é regulado pela presença ou ausência de luz no ambiente. O comportamento em relação a dormir e acordar varia entre os indivíduos e essa característica nomeia-se cronotipo. Efeitos negativos à saúde têm sido associados ao cronotipo que apresenta horários para iniciar e terminar o sono mais tarde. A hipótese considerada para explicar esses achados é de que demandas sociais, que não levam em conta essa variabilidade individual, sejam mediadoras dessa relação. Objetivos: avaliar a associação entre cronotipo e saúde mental (bem-estar psicológico e sintomas depressivos) levando-se em conta fatores demográficos (sexo e idade) e as rotinas de escola ou trabalho. Materiais e métodos: estudos transversais aninhados a um estudo epidemiológico em uma amostra de indivíduos do Vale do Taquari, no sul do Brasil. Na primeira avaliação, 6.506 participantes foram avaliados quanto a variáveis demográficas, dados de saúde, sintomas depressivos (Inventário de Depressão de Beck – BDI) e cronotipo (Questionário de Cronotipo de Munique – MCTQ). Na segunda etapa, 1.127 indivíduos entre 18 e 65 anos, selecionados a partir de seu cronotipo foram avaliados através do Índice de bem-estar de 5-itens da Organização Mundial da Saúde (OMS). O ponto médio do sono nos dias de rotinas de trabalho ou escolares foi utilizado como indicador do cronotipo, sendo “atrasado” aquele que tem o ponto médio mais tarde em relação ao início da noite e “avançado”, mais cedo. Resultados: no primeiro artigo, estudou-se a relação entre cronotipo e depressão em uma amostra de estudantes adolescentes. O modelo de regressão que melhor explicou a diferença entre os grupos com diferentes níveis de sintomas de depressão (BDI<10 X BDI!10) incluiu o sexo feminino e o cronotipo atrasado. O segundo artigo demonstrou que a escala de Bem-estar de 5-itens da OMS tem uma estrutura unidimensional, boa validade interna e externa e utilidade como instrumento de triagem para depressão quando comparada ao BDI. Assim, no terceiro artigo, foi avaliada a relação entre cronotipo e bem-estar, em indivíduos entre 18 e 65 anos. O modelo de regressão que incluiu o cronotipo atrasado, maior carga de trabalho, rotinas de trabalho mais cedo no dia e menor exposição à luz do sol, como variáveis preditoras, e piores escores na Escala de Bem-estar, como desfecho, foi significativo para o sexo feminino. Discussão: a presente tese explicitou a importância de considerar os fatores idade e sexo na expressão do cronotipo e na relação deste com saúde mental. Corroborou com a hipótese de que as demandas sociais mediam a relação entre cronotipo e os desfechos estudados. A expressão do cronotipo nos dias de rotinas escolares ou de trabalho foi identificada como a variável que estabeleceu a mais forte relação com piores escores de bemestar e com mais sintomas de depressão. Por fim, reforçou a necessidade de revermos as rotinas de trabalho e escolares que, ao não considerarem as diferenças fisiológicas individuais, têm-se associado, de forma consistente, a conseqüências negativas à saúde. / Introduction: the sleep-wake cycle is the most ubiquitous human circadian rhythmic behavior. It is established by an endogenous circadian timing system that is regulated by the presence or absence of light in the environment. Sleep and wake behavior varies among individuals and this feature has been termed chronotype. Negative health effects have been associated with the chronotype that presents later start and end sleep times. A hypothesis to explain these findings is that social demands, which do not take into account individual variability, are mediators of this relationship. Objectives: to evaluate the association between chronotype and mental health (psychological well-being and depressive symptoms) taking into account demographic factors (age and sex) and the routines of school or work. Materials and methods: the studies included here are cross-sectional nested to an epidemiological study in a sample of individuals from “Vale do Taquari”, in southern Brazil. In the first evaluation, 6,506 participants were assessed for demographic variables, health data, depressive symptoms (Beck Depression Inventory - BDI) and chronotype (Munich Chronotype Questionnaire - MCTQ). In the second stage, 1,127 individuals between 18 and 65 years, selected based on their chronotype, were evaluated through the Well-being 5 items Index from the World Health Organization (WHO). The midpoint of sleep on working or school days was used as an indicator of chronotype, and considered "delayed" or “late” those who have later midpoints in relation to environmental night and "advanced" or “early”, earlier. Results: in the first article, we studied the relationship between chronotype and depression in a sample of adolescent students. The regression model that best explained the difference between groups with different levels of depression symptoms (BDI<10 X BDI!10) included female sex and late chronotype. The second article demonstrated that the Well-being index has a unidimensional structure, good internal and external validities and might be usefulness as a screening tool for depression when compared to the BDI. Thus, in the third article, we evaluated the relationship between chronotype and well-being in individuals between 18 and 65 years. The regression model that included late chronotype, increased workload, earlier working routines in the day and less exposure to sunlight, as predictor variables, and worse scores on the well-being index, as the outcome was significant for females. Discussion: the present thesis content highlighted the importance of considering age and sex as factors influencing the expression of chronotype and the relationship with mental health outcomes. It corroborated the hypothesis that social demands mediate the relationship between chronotype and the studied outcomes. The expression of chronotype during the days of work or school routines was identified as the variable that established the strongest relationship with worse well-being scores and more depression symptoms. Finally, it reinforced the need to reconsider work and school routines that, by not taking into acount individual physiological differences, have been associated consistently with negative health consequences.

Au cours de la préadolescence, la durée totale de sommeil diminue et des retards importants dans l’horaire de sommeil se produisent. Cependant, la plupart des études utilisent des données transversales et/ou des mesures subjectives du sommeil, limitant la qualité méthodologique des données. Aussi, le point milieu du sommeil, le décalage horaire social (DHS) et la variabilité intra-individuelle (VII) du sommeil restent rarement abordés. Nous avons suivi 128 préadolescents (69 filles) ayant un développement typique pendant 4 ans, entre les âges de 8 et 12 ans. L’actigraphie a été utilisée pour déterminer le sommeil typique (soit la valeur moyenne), le DHS et la VII de quatre aspects du sommeil : l’heure du coucher, l’heure du lever, le point milieu du sommeil et la durée totale de sommeil. Des courbes de croissance multiniveaux ont été modélisées pour examiner à la fois les changements intra- et inter-individuels. Avec l'âge, l'horaire de sommeil devient progressivement plus tardif, la durée de sommeil diminue linéairement, et des augmentations significatives sont observés à la fois pour le DHS et la VII. Des différences individuelles autour des moyennes du groupe ont aussi été notées pour le DHS de l'heure du coucher et de la durée de sommeil. De plus, les filles présentent des différences plus importantes dans leur sommeil entre les jours comparativement aux garçons. Ces résultats décrivent des changements significatifs dans le sommeil des préadolescents au cours des années. Nous discutons des implications potentielles de ces trajectoires sur le fonctionnement physiologique, psychologique et social des jeunes. / Pre-adolescence is a period of marked changes in sleep-wake cycles characterized by a gradual decrease in adolescents' total sleep time (TST) and later sleep schedules with age. However, much of the research supporting these developmental trends has been based on cross-sectional data and/or subjective measures of sleep, limiting the methodological quality of the evidence. Additionally, important sleep parameters, like sleep midpoint, social jetlag and variability in sleep remain seldomly studied. We followed 128 typically developing preadolescents (69 female) over 4 years, between ages 8 and 12 years. Actigraphy-derived estimates of typical (i.e., mean) sleep, social jetlag and intra-individual variability were obtained for four sleep parameters: sleep onset, sleep offset, sleep midpoint, and TST. Multilevel growth curves were modelled to examine both within-person change and between-person differences in these trajectories. Across the period of pre-adolescence, sleep schedule became progressively later, mean TST decreased linearly, and significant increases were observed for both social jetlag and variability in sleep. Important individual differences around the average group estimates were noted for the social jetlag of sleep offset and TST. Sex differences were also observed, with girls experiencing greater between-day differences in their sleep than boys. These results describe significant age-related changes in the sleep of pre-adolescents. We discuss potential implications of these trajectories on youths’ physiological, psychological, and social functioning.

In this master’s thesis we focus on the basic properties of computer curves and their practical applicability. We explain how the curve can be understood in general, what are polynomial curves and their composing possibilities. Then we focus on the description of Bezier curves, especially the Bezier cubic. We discuss in more detail some of fundamental algorithms that are used for modelling these curves on computers and then we will show their practical interpretation. Then we explain non uniform rational B-spline curves and De Boor algorithm. In the end we discuss topic rasterization of segment, thick line, circle and ellipse. The aim of master’s thesis is the creation of the set of interactive applets, simulating some of the methods and algorithm we discussed in theoretical part. This applets will help facilitate understanding and will make the teaching more effective.

In this master’s thesis we focus on the basic properties of computer curves and their practical applicability. We explain how the curve can be understood in general, what are polynomial curves and their composing possibilities. Then we focus on the description of Bezier curves, especially the Bezier cubic. We discuss in more detail some of fundamental algorithms that are used for modelling these curves on computers and then we will show their practical interpretation. Then we explain non uniform rational B-spline curves and De Boor algorithm. In the end we discuss topic rasterization of segment, thick line, circle and ellipse. The aim of master’s thesis is the creation of the set of interactive applets, simulating some of the methods and algorithm we discussed in theoretical part. This applets will help facilitate understanding and will make the teaching more effective.