In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.

In this thesis the main object of our concern is a Pythagorean hodograph B- spline curve. We recall notions of both Pythagorean hodograph (PH) curves and B-spline functions separately first. Then we put these fields together to generalize PH curves to their B-spline instances. We encapsulate these curves in various spaces under one algebraic structure using the formalism of Clifford algebras. We consider both Euclidean and Minkowski spaces of lower dimensions which give room for real applications and use of these curves. We support our results by giving numerous examples. 1

Ongoing developments in the field of medical imaging modalities have pushed the frontiers of modern medicine and biomedical engineering, prompting the need for new applications to improve diagnosis, treatment and prevention of diseases. Biomedical data visualization and modeling rely predominately on manual processing and utilization of voxel and facet based homogeneous models. Biological structures are naturally heterogeneous and in order to accurately design and biomimic biological structures, properties such as chemical composition, size and shape of biological constituents need to be incorporated in the computational biological models. Our proposed approach involves generating a density point cloud based on the intensity variations in a medical image slice, to capture tissue density variations through point cloud densities. The density point cloud is ordered and approximated with a set of cross-sectional least-squares B-Spline curves, based on which a skinned B-Spline surface is generated. The aim of this method is to capture and accurately represent density variations within the medical image data with a lofted surface function. The fitted B-Spline surface is sampled at uniformly distributed parameters, and our preliminary results indicate that the bio-CAD model preserves the density variations of the original image based point cloud. The resultant surface can thus be visualized by mapping the density in the parametric domain into color in pixel domain. The B-Spline function produced from each image slice can be used for medical visualization and heterogeneous tissue modeling. The process can be repeated for each slice in the medical dataset to produce heterogeneous B-Spline volumes. The emphasis of this research is placed on accuracy and shape fidelity needed for medical operations.

Molts sistemes mecànics existents tenen un comportament vibratori funcionalment perceptible, que es posa de manifest enfront d'excitacions transitòries. Normalment, les vibracions generades segueixen presents després del transitori (vibracions residuals), i poden provocar efectes negatius en la funció de disseny del mecanisme.El mètode que es proposa en aquesta tesi té com a objectiu principal la síntesi de lleis de moviment per reduir les vibracions residuals. Addicionalment, els senyals generats permeten complir dues condicions definides per l'usuari (anomenats requeriments funcionals). El mètode es fonamenta en la relació existent entre el contingut freqüencial d'un senyal transitori, i la vibració residual generada, segons sigui l'esmorteïment del sistema. Basat en aquesta relació, i aprofitant les propietats de la transformada de Fourier, es proposa la generació de lleis de moviment per convolució temporal de polsos. Aquestes resulten formades per trams concatenats de polinomis algebraics, cosa que facilita la seva implementació en entorns numèrics per mitjà de corbes B-spline. / Many mechanical systems have a vibratory behavior, functionally noticeable, which can be raised when transient excitations happen. Usually, such generated vibrations remain in the system after this transient (residual vibrations), and may imply negative effects in the mechanism's intended function.The method proposed in this thesis has as main objective, the synthesis of excitation signals for mechanical systems in order to reduce residual vibrations. In addition, generated signals are able to achieve two conditions defined by the user related to function of the mechanism (called functional requirements). This method is based on the relationship between the frequency contents of the transient signal and the residual vibration generated, depending on the system damping. From this relation, and taking advantage of the Fourier transform properties, motion laws are generated through the pulses' time convolution. Resultant laws are made up of algebraic polynomial pieces linked together, which makes them very suitable for implementation with numerical calculus through B-spline curves.