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No one will question that computers are revolutionizing the design industry. It is pointed out in [Bezier84] that before CAD/CAM, a surface was defined by tracing cross sections on a drawing and then carving these sections in wood, plastic or metal. The final model was determined by someone interpolating between the sections. This labor intensive art is being replaced by techniques of computer aided geometric design.

Surface fitting methods play an important role in many scientific fields as well as in computer aided geometric design. The problem treated here is that of constructing a smooth surface that interpolates data values associated with scattered nodes in the plane. The data is said to be convex if there exists a convex interpolant. The problem of convexity-preserving interpolation is to determine if the data is convex, and construct a convex interpolant if it exists.

Ferrets are amusing, flexible creatures that have been under represented in computer models. Because their bodies can assume almost any curved shape, splines are the natural tool for modelling ferrets. Surface pasting is a hierarchical method of modelling with spline surfaces, where features are added onto a base surface. Existing surface pasting techniques are limited to modelling rectilinear shapes. Using the task of modelling a ferret as a driving force, I propose a method of pasting cylinders in world space; I looked at methods for reducing distortion of pasted features; and I created a method for pasting trimmed features to allow for features that do not have the rectilinear shape of standard pasting. With my methods, modelling ferrets with surface pasting is easier, and the resulting models are closer to a real ferret.

Ferrets are amusing, flexible creatures that have been under represented in computer models. Because their bodies can assume almost any curved shape, splines are the natural tool for modelling ferrets. Surface pasting is a hierarchical method of modelling with spline surfaces, where features are added onto a base surface. Existing surface pasting techniques are limited to modelling rectilinear shapes. Using the task of modelling a ferret as a driving force, I propose a method of pasting cylinders in world space; I looked at methods for reducing distortion of pasted features; and I created a method for pasting trimmed features to allow for features that do not have the rectilinear shape of standard pasting. With my methods, modelling ferrets with surface pasting is easier, and the resulting models are closer to a real ferret.

The design of curves and surfaces in C.A.D. systems has many applications in car, plane or ship industry. Because they offer more flexibility, rational functions are often preferred to polynomial functions to modelize curves and surfaces. In this work, several methods to generate rational Bezier curves and surfaces which minimize some functionals are proposed. The functionals measure a technical smoothness of the curves and surfaces, and are related to the energy of beams and plates in the sense of the elasticity theory.

This work focuses on generating approximations of complex T-spline surfaces with similar but less complex T-splines. Two approaches to simplifying T-splines are proposed: a bottom-up approach that iteratively refines an over-simple T-spline to approximate a complex one, and a top-down approach that evaluates existing control points for removal in producing an approximations. This thesis develops and compares the two simplification methods, determining the simplification tasks to which each is best suited. In addition, this thesis documents supporting developments made to T-spline research as simplification was developed.

En esta tesis se presenta un método para analizar antenas montadas sobre estructuras arbitrarias. La Optica Geométrica (GO) y la Teoría Uniforme de la difraccion (UTD), han sidoempleadas para analizar los efectos que la estructura produce sobre el diagrama de radiación de la antena emisora. Para la descripción geométrica de la estructura, han sido utilizados parches NURBS (Non Uniform Rational B-Spline), por lo que el método presentado, es compatible con la mayoría de los programas gráficos disponibles en el mercado.EL tratamiento de geometrías arbitrarias requiere un código eficiente en el análisis de tres dimensiones.Por otro lado, para obtener resultados satisfactorios, la descripción de la superficie de la estructura, debe ser muy próxima al modelo real, sin embargo, esto complica el tratamiento computacional. Aquí la estructura es modelada mediante un conjunto de parches NURBS, que unidos entre sí, describen el modelo completo. Esta descripción permite manipular superficies arbitrarias con un bajo numero de parches, lo que significa un volumen de información reducido.La descripción inicial por NURBS del modelo, es acompañada con información complemetaria como por ejemplo: la tipología de las superficies, las curvas frontera, el tipo de material, etc. Esto es imprescindible para la aplicación de criterios de selección dedicados a la aceleración del proceso.El método tras leer la descripción del modelo, descompone los parches NURBS en superficies racionales de Bezier. Un parche de Bezier es también una superficie paramétrica definida en términos de una combinación lineal de los polinomios de Bernstein.Las antenas son modeladas usando modelos numéricos simples basados en agrupaciones de dipolos infinitesimales eléctricos y magnéticos. Esta caracterización de la antena es muyventajosa ya que con un numero reducido de datos de entrada, la fuente queda definida para cualquier dirección del espacio y el valor del campo radiado puede ser calculado fácilmente.El análisis electromagnético de los efectos que contribuyen al campo dispersado por la geometría comienza con una selección rigurosa de la geometría iluminada desde la fuente.Unicamente los parches de Bezier iluminados serán almacenados por el ordenador durante el análisis. La filosofía de este proceso es descartar aquella parte de la geometría que no contribuye a los efectos de dispersión.El campo total calculado es la superposición de los siguientes efectos pertenecientes a la GO y a la UTD: campo directo procedente de la fuente, campo reflejado por los parches de Bezier, campo difractado por las aristas del modelo definidas como curvas de Bezier, ondas de superficie, dobles reflexiones, reflexione-difraccion y difraccion-reflexión. El método ha sido diseñado para analizar campo cercano y lejano. El mayor gasto computacional se debe a la búsqueda de los puntos de dispersión, por lo que antes de emplear los algoritmos de intersección es necesario aplicar un conjunto de criterios rápidos dependientes de la dirección de observación.El principio de Fermat en combinación con el Gradiente Conjugado (CGM) es usado para obtener de manera eficiente los puntos de dispersión sobre la estructura. Para cada efecto, laposible ocultación de la trayectoria completa del rayo es examinada, por ello, si el rayo corta alguno de los parches de Bezier su contribución será descartada. Los dobles efectos son tratados como una generalización de los simples efectos.El método desarrollado es eficiente ya que precisa de un numero reducido de superficies para modelar objetos complejos lo que se traduce en bajos requerimientos de memoria y reducidos tiempos de calculo. / In this thesis a method to analyze antennas on board of complex bodies is presented. The Geometrical Optics (GO) and Uniform Theory of Diffraction (UTD) have been used to analyze the effect of the structure in the radiation pattern of the antennas. The bodies are geometrically modelled by using NURBS (Non Uniform Rational B-Spline) surfaces. In addition to be accurate and efficient, the method is compatible with most of the modern CAGD (Computer Aided Geometric Design) available programs.The treatment of arbitrary geometries requires a code which can carry out an efficient 3D analysis. To obtain accurate results the description of the surface must be close to the real model, however this complicates the computational procedure. Here the structure is modeled by a collection of individual N.U.R.B.S. surface patches joined to form a complete description of the surface model. The NURBS description is able to manipulate free form surfaces with a low number of patches, and therefore, with a low amount of information. The initial description of the model by NURBS surfaces is accompanied with other complementary data for example : the topology of the surfaces, the boundary curves, the types of material and other inputs. It is very interesting to apply criteria to make the complete analysis faster.The method reads the NURBS description of the model and transforms the NURBS into the rational BEZIER surfaces. A rational BEZIER patch is also a parametric surface defined in terms of a linear combination of Bernstein polynomials.The antennas are modelled using simple numerical models based on arrays of electric and magnetic infinitesimal dipoles. This antenna modelization is very advantageous because with a little input data, the source is defined in any direction and the field value is readily accessible.The electromagnetic analysis of the contributive effects to the scattering field by the geometry, starts with the rigorous selection of the geometry illuminated from the source. Only the Bezier patches illuminated will be in memory of the computer during the analysis. The philosophy of this previous process is to discard in the process the part of the geometry which does not contribute to the scattering effects.The total field is the superposition of the following GO and UTD field components: direct field from the source, reflected fields from the Bezier patches of the model, diffracted fields from the arbitrary edges defined as a Bezier curves, creeping waves, double reflected field and diffracted-reflected and reflected-diffracted fields. The search of specular and diffraction points are the most CPU time consuming, thus before using the intersection algorithms it is necessary to apply a set of fast selection criteria which depend on the observation direction.The Fermat principle in conjunction with the Conjugate Gradient Method (CGM) is used for obtaining efficiently the reflection points and diffraction points on the structure. For each effect the complete ray path is examined to see whether or not it is interrupted by any Bezier patch of the model, in this case the field component is not computed. The double effects are treated using a generalization of the single effects algorithms. The method has been developed to analyze the near and far field cases for different frequencies.The developed method is quite efficient because it makes use of a small number of surfaces to model complex bodies, so it requires few memory and low computing time.