arXiv subjects
Publications and source records attributed to Tiago Novello.
This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia Sets are used as examples.
We introduce Proceduray, an engine for real-time ray tracing of procedural geometry. Its motivation is the current lack of mid-level abstraction tools for scenes with primitives involving intersection shaders. Those scenes impose strict engine design choices since they need flexibility in the shader table setup. Proceduray aims at providing a fair tradeoff between that flexibility and productivity. It also aims to be didactic. Shader table behavior can be confusing because parameters for indexing come from different parts of a system, involving both host and device code. This is different in essence from ray tracing triangle meshes (which must use a built-in intersection shader for all objects) or rendering with the traditional graphics or compute pipelines. Additionals goals of the project include fomenting deeper discussions about DirectX RayTracing (DXR) host code and providing a good starting point for developers trying to deal with procedural geometry using DXR.
In this expository paper, we present a survey about the history of the geometrization conjecture and the background material on the classification of Thurston's eight geometries. We also discuss recent techniques for immersive visualization of relevant three-dimensional manifolds in the context of the Geometrization Conjecture.
Local and global illumination were recently defined in Riemannian manifolds to visualize classical Non-Euclidean spaces. This work focuses on Riemannian metric construction in $\mathbb{R}^3$ to explore special effects like warping, mirages, and deformations. We investigate the possibility of using graphs of functions and diffeomorphism to produce such effects. For these, their Riemannian metrics and geodesics derivations are provided, and ways of accumulating such metrics. We visualize, in "real-time", the resulting Riemannian manifolds using a ray tracing implemented on top of Nvidia RTX GPUs.
This paper presents a path tracer algorithm to compute the global illumination of non-Euclidean manifolds. We use the 3D torus as an example.
Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincar\'e formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations of $C$ can be efficiently computed, in particular its homology or Morse-Smale decomposition. Given a function $f$ sampled on $C$, it is possible to derive a discrete gradient that mimics the dynamics of $f$. Many such constructions are based on some variant of a greedy pairing of adjacent cells, given an appropriate weighting. However, proving that the dynamics of $f$ is correctly captured by this process is usually intricate. This work introduces the notion of discrete smoothness of the pair $(f,C)$, as a minimal sampling condition to ensure that the discrete gradient is geometrically faithful to $f$. More precisely, a discrete gradient construction from a function $f$ on a polyhedron complex $C$ of any dimension is studied, leading to theoretical guarantees prior to the discrete smoothness assumption. Those results are then extended and completed for the smooth case. As an application, a purely combinatorial proof that all CAT(0) cube complexes are collapsible is given.
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a Morse-Smale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.