This study is concerned with digital two-dimensional rivers for games. Specifically the aim is to create a river shape from the basis of a spline curve and make it look like a river with a water flow-rate. This is achieved by developing an artifact capable of river simulations in 2D, with the flow-rate calculated using the necessary hydraulics equations. The scientific process described in Design Science Research Methodology for Information Systems Research is presented and followed. Artifact simulations are demonstrated and evaluated, especially calculations for river mean velocity and discharge are shown to be possible by assuming that the river channel is of trapezoidal shape. Simulations show a type of river not usually seen in games, a river with more accuracy than other game simulations. The artifact river simulations are however limited by a missing smoothness when going between different channel shapes. Also there could be a way of improving the texture used, or possibly use more then one texture in one simulation. The conclusion is that the methods used for river generation by this study could be part of a design tool that targets 2D games.

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In this thesis the cubic spline guidance law and its variants are derived. A detailed analysis is carried out to find the initial conditions for successful interception. The results are applied to three dimensional guidance design and for solving waypoint following problems. The basic cubic spline guidance law is derived for intercepting a stationary target at a desired impact angle in a surface-to-surface engagement scenario. The guidance law is obtained using an inverse method, from a cubic spline curve based trajectory. For overcoming the drawbacks of the basic cubic spline guidance law, it is modified by introducing an additional parameter. This modification has an interesting feature that the guidance command can be obtained using a single cubic spline polynomial even for impact angles greater than π/2, while resulting in substantial improvement in the guidance performance in terms of lateral acceleration demand and length of the trajectory. For imparting robustness to the cubic spline guidance law, in the presence of uncertainties and acceleration saturation, an explicit guidance expression is also derived. A comprehensive capturability study of the proposed guidance law is carried out. The capturability for the cubic spline guidance law is defined in terms of the set of all feasible initial conditions for successful interception. This set is analytically derived and its dependence on various factors, such as initial engagement geometry and interceptor capability, are also established. The basic cubic spline guidance and its variants are also derived for a three dimen- sional scenario. The novelty of the present work lies in the particular representation of the three dimensional cubic spline curve and the adoption of the analytical results available for two dimensional cubic spline guidance law. This enables selection of the boundary condition at launch for given terminal boundary condition and also in avoiding the singularities associated with the inverse method based guidance laws. For establishing the feasibility of the guidance laws in the real world, the rigid body dynamics of the interceptor is presented as a 6 degrees-of-freedom model. Further, using a simplified model, elementary autopilots are also designed. The successful interception of the target in the presence of the rigid body dynamics proves practical applicability of the cubic spline based guidance laws. Finally, the theory developed in the first part of the thesis is applied to solve the waypoint following problem. A smooth path is designed for transition of vehicle velocity from incoming to outgoing direction. The approach developed is similar to Dubins’ path, as it comprises line–cubic spline–line segments. The important feature of this method is that the cubic spline segments are fitted such that the path curvature is bounded by a pre-specified constrained value and the acceleration demand for following the smooth path obtained by this method, gradually increases to the maximum value and then decreases. This property is advantageous from a practical point of view. All the results obtained are verified with the help of numerical simulations which are included in the thesis. The proposed cubic spline guidance law is conceptually simple, does not use linearised kinematic equations, is independent of time-to-go es- timates, and is also computationally inexpensive.

Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement. / Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.

In the first part I discuss ancestors of NURBS curves and surfaces, rather Ferguson, Beziere, Coons and B-spline curves and surfaces and furthermore B-spline functions. In the second part I devote to NURBS curves and surfaces, their description as a linear combination of B-spline functions in the projective space. I specify conical arcs more detailed, their submit in the projective space and NURBS surfasec given as tensor product of NURBS curves. Last part is devote to describtion programs for modeling conicals and NURBS surface.