Towards an Iterative Algorithm for the Optimal Boundary Coverage of a 3D Environment

This paper presents a new optimal algorithm for locating a set of sensors in 3D able to see the boundaries of a polyhedral environment. Our approach is iterative and is based on a lower bound on the sensors' number and on a restriction of the original problem requiring each face to be observed in its entirety by at least one sensor. The lower bound allows evaluating the quality of the solution obtained at each step, and halting the algorithm if the solution is satisfactory. The algorithm asymptotically converges to the optimal solution of the unrestricted problem if the faces are subdivided into smaller part

Securing Pathways with Orthogonal Robots

The protection of pathways holds immense significance across various domains, including urban planning, transportation, surveillance, and security. This article introduces a groundbreaking approach to safeguarding pathways by employing orthogonal robots. The study specifically addresses the challenge of efficiently guarding orthogonal areas with the minimum number of orthogonal robots. The primary focus is on orthogonal pathways, characterized by a path-like dual graph of vertical decomposition. It is demonstrated that determining the minimum number of orthogonal robots for pathways can be achieved in linear time. However, it is essential to note that the general problem of finding the minimum number of robots for simple polygons with general visibility, even in the orthogonal case, is known to be NP-hard. Emphasis is placed on the flexibility of placing robots anywhere within the polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure

Doctor of Philosophy

dissertationRay tracing presents an efficient rendering algorithm for scientific visualization using common visualization tools and scales with increasingly large geometry counts while allowing for accurate physically-based visualization and analysis, which enables enhanced rendering and new visualization techniques. Interactivity is of great importance for data exploration and analysis in order to gain insight into large-scale data. Increasingly large data sizes are pushing the limits of brute-force rasterization algorithms present in the most widely-used visualization software. Interactive ray tracing presents an alternative rendering solution which scales well on multicore shared memory machines and multinode distributed systems while scaling with increasing geometry counts through logarithmic acceleration structure traversals. Ray tracing within existing tools also provides enhanced rendering options over current implementations, giving users additional insight from better depth cues while also enabling publication-quality rendering and new models of visualization such as replicating photographic visualization techniques

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Shared-Frustum stereo rendering

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.Includes bibliographical references (p. 52-54).by Michael Vincent Capps.S.M

Physically based mechanical metaphors in architectural space planning

Physically based space planning is a means for automating the conceptual design process by applying the physics of motion to space plan elements. This methodology provides for a responsive design process, allowing a designer to easily make decisions whose consequences propagate throughout the design. It combines the speed of automated design methods with the flexibility of manual design methods, while adding a highly interactive quality and a sense of collaboration with the design. The primary assumption is that a digital design tool based on a physics paradigm can facilitate the architectural space planning process. The hypotheses are that Newtonian dynamics can be used 1) to define mechanical metaphors to represent the elements in an architectural space plan, 2) to compute architectural space planning solutions, and 3) to interact with architectural space plans. I show that space plan elements can be represented as physical masses, that design objectives can be represented using mechanical metaphors such as springs, repulsion fields, and screw clamps, that a layout solution can be computed by using these elements in a dynamical simulation, and that the user can interact with that solution by applying forces that are also models of the same mechanical objects. I present a prototype software application that successfully implements this approach. A subjective evaluation of this prototype reveals that it demonstrates a feasible process for producing space plans, and that it can potentially improve the design process because of the quality of the manipulation and the enhanced opportunities for design exploration it provides to the designer. I found that an important characteristic of this approach is that representation, computation, and interaction are all defined using the same paradigm. This contrasts with most approaches to automated space planning, where these three characteristics are usually defined in completely different ways. Also emerging from this work is a new cognitive theory of design titled 'dynamical design imagery,' which proposes that the elements in a designer's mental imagery during the act of design are dynamic in nature and act as a dynamical system, rather than as static images that are modified in a piecewise algorithmic manner