An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

Algorithms for Manufacturing Paperclips and Sheet Metal Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a "carpenter's ruler") can be straightened, a problem that was open for several years and has only recently been solved in the affirmative

Animation in relational information visualization

In order to be able to navigate in the world without memorizing each detail, the human brain builds a mental map of its environment. The mental map is a distorted and abstracted representation of the real environment. Unimportant areas tend to be collapsed to a single entity while important landmarks are overemphasized. When working with visualizations of data we build a mental map of the data which is closely linked to the particular visualization. If the visualization changes significantly due to changes in the data or the way it is presented we loose the mental map and have to rebuild it from scratch. The purpose of the research underlying this thesis was to investigate and devise methods to create smooth transformations between visualizations of relational data which help users in maintaining or quickly updating their mental map

Animation in relational information visualization

In order to be able to navigate in the world without memorizing each detail, the human brain builds a mental map of its environment. The mental map is a distorted and abstracted representation of the real environment. Unimportant areas tend to be collapsed to a single entity while important landmarks are overemphasized. When working with visualizations of data we build a mental map of the data which is closely linked to the particular visualization. If the visualization changes significantly due to changes in the data or the way it is presented we loose the mental map and have to rebuild it from scratch. The purpose of the research underlying this thesis was to investigate and devise methods to create smooth transformations between visualizations of relational data which help users in maintaining or quickly updating their mental map

Articulating Space: Geometric Algebra for Parametric Design -- Symmetry, Kinematics, and Curvature

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be applied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing algorithms, these methods participate in the advancement of geometric thinking, developing an intuitive spatial articulation that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions

Realizability of Free Spaces of Curves

The free space diagram is a popular tool to compute the well-known Fr\'echet distance. As the Fr\'echet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often, the question arises whether a certain pattern in the free space diagram is "realizable", i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore, we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in $\mathbb{R}^{> 2}$, showing $\exists\mathbb{R}$-hardness. We use this to show that for curves in $\mathbb{R}^{\ge 2}$, the realizability problem is $\exists\mathbb{R}$-complete, both for continuous and for discrete Fr\'echet distance. We prove that the continuous case in $\mathbb{R}^1$ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in $\mathbb{R}^1$, we show that the problem becomes solvable in polynomial time.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms And Computations (ISAAC 2023

Towards hybrid methods for solving hard combinatorial optimization problems

Tesis doctoral leída en la Escuela Politécnica Superior de la Universidad Autónoma de Madrid el 4 de septiembre de 200

Control of objects with a high degree of freedom

In this thesis, I present novel strategies for controlling objects with high degrees of freedom for the purpose of robotic control and computer animation, including articulated objects such as human bodies or robots and deformable objects such as ropes and cloth. Such control is required for common daily movements such as folding arms, tying ropes, wrapping objects and putting on clothes. Although there is demand in computer graphics and animation for generating such scenes, little work has targeted these problems. The difficulty of solving such problems are due to the following two factors: (1) The complexity of the planning algorithms: The computational costs of the methods that are currently available increase exponentially with respect to the degrees of freedom of the objects and therefore they cannot be applied for full human body structures, ropes and clothes . (2) Lack of abstract descriptors for complex tasks. Models for quantitatively describing the progress of tasks such as wrapping and knotting are absent for animation generation. In this work, we employ the concept of a task-centric manifold to quantitatively describe complex tasks, and incorporate a bi-mapping scheme to bridge this manifold and the configuration space of the controlled objects, called an object-centric manifold. The control problem is solved by first projecting the controlled object onto the task-centric manifold, then getting the next ideal state of the scenario by local planning, and finally projecting the state back to the object-centric manifold to get the desirable state of the controlled object. Using this scheme, complex movements that previously required global path planning can be synthesised by local path planning. Under this framework, we show the applications in various fields. An interpolation algorithm for arbitrary postures of human character is first proposed. Second, a control scheme is suggested in generating Furoshiki wraps with different styles. Finally, new models and planning methods are given for quantitatively control for wrapping/ unwrapping and dressing/undressing problems