Curve and surface framing for scientific visualization and domain dependent navigation

Thesis (Ph.D.) - Indiana University, Computer Science, 1996Curves and surfaces are two of the most fundamental types of objects in computer graphics. Most existing systems use only the 3D positions of the curves and surfaces, and the 3D normal directions of the surfaces, in the visualization process. In this dissertation, we attach moving coordinate frames to curves and surfaces, and explore several applications of these frames in computer graphics and scientific visualization. Curves in space are difficult to perceive and analyze, especially when they are densely clustered, as is typical in computational fluid dynamics and volume deformation applications. Coordinate frames are useful for exposing the similarities and differences between curves. They are also useful for constructing ribbons, tubes and smooth camera orientations along curves. In many 3D systems, users interactively move the camera around the objects with a mouse or other device. But all the camera control is done independently of the properties of the objects being viewed, as if the user is flying freely in space. This type of domain-independent navigation is frequently inappropriate in visualization applications and is sometimes quite difficult for the user to control. Another productive approach is to look at domain-specific constraints and thus to create a new class of navigation strategies. Based on attached frames on surfaces, we can constrain the camera gaze direction to be always parallel (or at a fixed angle) to the surface normal. Then users will get a feeling of driving on the object instead of flying through the space. The user's mental model of the environment being visualized can be greatly enhanced by the use of these constraints in the interactive interface. Many of our research ideas have been implemented in Mesh View, an interactive system for viewing and manipulating geometric objects. It contains a general purpose C++ library for nD geometry and supports a winged-edge based data structure. Dozens of examples of scientifically interesting surfaces have been constructed and included with the system

Network Geometry

Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity

Quantum Gravity from Causal Dynamical Triangulations: A Review

This topical review gives a comprehensive overview and assessment of recent results in Causal Dynamical Triangulations (CDT), a modern formulation of lattice gravity, whose aim is to obtain a theory of quantum gravity nonperturbatively from a scaling limit of the lattice-regularized theory. In this manifestly diffeomorphism-invariant approach one has direct, computational access to a Planckian spacetime regime, which is explored with the help of invariant quantum observables. During the last few years, there have been numerous new and important developments and insights concerning the theory's phase structure, the roles of time, causality, diffeomorphisms and global topology, the application of renormalization group methods and new observables. We will focus on these new results, primarily in four spacetime dimensions, and discuss some of their geometric and physical implications.Comment: 64 pages, 28 figure