In architectural and sculptural practice, the eventual goal is constructing the shapes that have been designed. Due to fabrication considerations, shapes with planar faces are in demand for these practices. In this thesis, a novel computational modeling approach to design constructible shapes is introduced. This method guarantees that the resulting shapes are planar meshes with 3-valence vertices, which can always be physically constructed using planar or developable materials such as glass, sheet metal or plywood. The method introduced is inspired by the traditional sculpture and is based on the idea of carving a mesh by using slicing planes. The process of determining the slicing planes can either be interactive or automated. A framework is developed which allows user to sculpt shapes by using the in- teractive and automated processes. The framework allows user to cut a source mesh based on its edges, faces or vertices. The user can sculpt various kinds of developable surfaces by cutting the parallel edges of the mesh. The user can also introduce in- teresting conical patterns by cutting dierent vertex, edge, face combinations of the mesh.

In architectural and sculptural practice, the eventual goal is constructing the shapes that have been designed. Due to fabrication considerations, shapes with planar faces are in demand for these practices. In this thesis, a novel computational modeling approach to design constructible shapes is introduced. This method guarantees that the resulting shapes are planar meshes with 3-valence vertices, which can always be physically constructed using planar or developable materials such as glass, sheet metal or plywood. The method introduced is inspired by the traditional sculpture and is based on the idea of carving a mesh by using slicing planes. The process of determining the slicing planes can either be interactive or automated. A framework is developed which allows user to sculpt shapes by using the in- teractive and automated processes. The framework allows user to cut a source mesh based on its edges, faces or vertices. The user can sculpt various kinds of developable surfaces by cutting the parallel edges of the mesh. The user can also introduce in- teresting conical patterns by cutting dierent vertex, edge, face combinations of the mesh.

The research results presented in this thesis provide tools and methods to aid in the design of developable mechanisms. This work will help engineers design compact mechanisms onto developable surfaces, making it possible for them to be used in future applications. The thesis introduces terminology and definitions to describe conical developable mechanisms. Models are developed to describe mechanism motion with respect to the apex of the conical surface, and connections are made to cylindrical developable mechanisms using projected angles. The Loop Sum Method is presented as an approach to determine the geometry of the cone to which a given spherical mechanism can be mapped. A method for position analysis is presented to determine the location of any point along the link of a mechanism with respect to the conical geometry. These methods are also applied to multiloop spherical mechanisms. This work created tools and methods to design cylindrical and conical developable mechanisms from flat, planar patterns. Equations are presented that relate the link lengths and link angles of planar and spherical mechanisms to the dimensions in a flat configuration. These flat patterns can then be formed into curved, developable mechanisms. Guidelines are established to determine if a mechanism described by a flat pattern can exhibit intramobile or extramobile behavior. A developable mechanism can only potentially exhibit intramobile or extramobile behavior if none of the links extend beyond half of the flat pattern. The behavior of a mechanism can change depending on the location of the cut of the flat pattern. Different joint designs are discussed including lamina emergent torsional (LET) joints. It is shown that developable mechanisms on regular cylindrical surfaces can be described using cyclic quadrilaterals. Mechanisms can exist in either an open or crossed configuration, and these configurations correspond to convex and crossed cyclic quadrilaterals. Using equations developed for both convex and crossed cyclic quadrilaterals, the geometry of the reference surface to which a four-bar mechanism can be mapped is found. Grashof mechanisms can be mapped to two surfaces in open or crossed configurations. The way to map a non-Grashof mechanism to a cylindrical surface is in its open configuration. Extramobile and intramobile behavior can be achieved depending on selected pairs within a cyclic quadrilateral and its position within the circumcircle. Selecting different sets of links as the ground link changes the potential behavior of the mechanism. Different cases are tabulated to represent all possibilities.

This thesis provides resources that enable the design of novel and compact mechanical devices by providing terminology, engineering models, and design methods in the fields of developable mechanisms and origami-based engineering.The first part of this work presents engineering models to aid in the design of cylindrical developable mechanisms. These models take into account the added spatial restrictions imposed by the developable surface. Equations are provided for the kinematic analysis of cylindrical developable mechanisms. A new classification for developable mechanisms is also presented (intramobile, extramobile, and transmobile) and two graphical methods are provided for determining this classification for single-DOF planar cylindrical developable mechanisms. Characteristics specific to four-bar cylindrical developable mechanisms are also discussed. The second part addresses a key challenge in origami design: how to achieve stability while maintaining the desired folding motion. The origami stability integration method (OSIM) provides an approach for graphically combining various techniques to achieve stability. This thesis presents improvements and additions to the OSIM that allow it to be applied to many different scenarios. Existing stability techniques are also categorized into four groups based on whether they are intrinsic or extrinsic to the origami pattern and whether they exhibit gradual or non-gradual energy storage behaviors. These categorizations can help designers select appropriate techniques for their application. Four case studies are presented which use the OSIM and the technique categorization to conceptualize stability in origami-based devices.

Estudaremos as singularidades genéricas de superfécies regradas em R3. O objetivo do trabalho é mostrar que as singularidades genéricas que ocorrem no conjunto das superfícies regradas são as mesmas que ocorrem no conjunto das aplicações diferenciáveis de R2 em R3, enquanto que as singularidades genéricas das superfícies desenvolvíveis, que formam um subconjunto das superfícies regradas, são mais degeneradas. / We study generic singularities of ruled surfaces in R3. In this work we show that generic singularities appearing in the set of ruled surfaces are the same that occur in the set of map germs from R2 to R3, while the generic singularities of developable surfaces are more degenerate.

Estudaremos as singularidades genéricas de superfécies regradas em R3. O objetivo do trabalho é mostrar que as singularidades genéricas que ocorrem no conjunto das superfícies regradas são as mesmas que ocorrem no conjunto das aplicações diferenciáveis de R2 em R3, enquanto que as singularidades genéricas das superfícies desenvolvíveis, que formam um subconjunto das superfícies regradas, são mais degeneradas. / We study generic singularities of ruled surfaces in R3. In this work we show that generic singularities appearing in the set of ruled surfaces are the same that occur in the set of map germs from R2 to R3, while the generic singularities of developable surfaces are more degenerate.

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.