Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric $\varrho$. For example, we will see that an expanding Thurston map is topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We also obtain existence and uniqueness results for $f$-invariant Jordan curves $\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we obtain several characterizations of Latt\`{e}s maps.Comment: 492 pages, 51 figure

Power Deposition on Tokamak Plasma-Facing Components

The SMARDDA software library is used to model plasma interaction with complex engineered surfaces. A simple flux-tube model of power deposition necessitates the following of magnetic fieldlines until they meet geometry taken from a CAD (Computer Aided Design) database. Application is made to 1) models of ITER tokamak limiter geometry and 2) MASTU tokamak divertor designs, illustrating the accuracy and effectiveness of SMARDDA, even in the presence of significant nonaxisymmetric ripple field. SMARDDA's ability to exchange data with CAD databases and its speed of execution also give it the potential for use directly in the design of tokamak plasma facing components.Comment: 13 pages, 20 figure

An Overview of Rendering from Volume Data --- including Surface and Volume Rendering

Volume rendering is a title often ambiguously used in science. One meaning often quoted is: `to render any three volume dimensional data set'; however, within this categorisation `surface rendering'' is contained. Surface rendering is a technique for visualising a geometric representation of a surface from a three dimensional volume data set. A more correct definition of Volume Rendering would only incorporate the direct visualisation of volumes, without the use of intermediate surface geometry representations. Hence we state: `Volume Rendering is the Direct Visualisation of any three dimensional Volume data set; without the use of an intermediate geometric representation for isosurfaces'; `Surface Rendering is the Visualisation of a surface, from a geometric approximation of an isosurface, within a Volume data set'; where an isosurface is a surface formed from a cross connection of data points, within a volume, of equal value or density. This paper is an overview of both Surface Rendering and Volume Rendering techniques. Surface Rendering mainly consists of contouring lines over data points and triangulations between contours. Volume rendering methods consist of ray casting techniques that allow the ray to be cast from the viewing plane into the object and the transparency, opacity and colour calculated for each cell; the rays are often cast until an opaque object is `hit' or the ray exits the volume

Digital technologies for virtual recomposition : the case study of Serpotta stuccoes

The matter that lies beneath the smooth and shining surface of stuccoes of the Serpotta family, who used to work in Sicily from 1670 to 1730, has been thoroughly studied in previous papers, disclosing the deep, even if empirical, knowledge of materials science that guided the artists in creating their master- works. In this work the attention is focused on the solid perspective and on the scenographic sculpture by Giacomo Serpotta, who is acknowledged as the leading exponent of the School. The study deals with some particular works of the artist, the so-called "teatrini" (Toy Theater), made by him for the San Lorenzo Oratory in Palermo. On the basis of archive documents and previous analogical photogrammetric plotting, integrated with digital solutions and methodologies of computer- based technologies, the study investigates and interprets the geometric-formal genesis of the examined works of art, until the prototyping of the whole scenic apparatus.peer-reviewe

Implicit Functionally Graded Conforming Microstructures

The tensor product parametric representations are the most commonly used representation in geometric modeling. Yet, other representations have advantages in certain aspects, and in this work, we focus on employing implicit representations in the construction of microstructures. An implicit function, either functionally precise, or spline trivariate-based, is used to populate a macro-shape trivariate parametric form, and construct a conforming microstructure. Either the implicit tile or the macro-shape can be functionally graded or be heterogeneous, carrying graded properties such as material, translucency, or color alongside the geometry. Further, the implicit tiles can be parametrized and hence their geometry can vary across the macro-shape. The representation is locally precise and we demonstrate that in a slicing process that employs no (piecewise-linear) approximation. Finally, we demonstrate this framework on several 3D printed heterogeneous models

Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary materials, please visit at http://www.Eyemaginary.com/Portfolio/Publications.htm

Setting intelligent city tiling strategies for urban shading simulations

Assessing accurately the solar potential of all building surfaces in cities, including shading and multiple reflections between buildings, is essential for urban energy modelling. However, since the number of surface interactions and radiation exchanges increase exponentially with the scale of the district, innovative computational strategies are needed, some of which will be introduced in the present work. They should hold the best compromise between result accuracy and computational efficiency, i.e. computational time and memory requirements. In this study, different approaches that may be used for the computation of urban solar irradiance in large areas are presented. Two concrete urban case studies of different densities have been used to compare and evaluate three different methods: the Perez Sky model, the Simplified Radiosity Algorithm and a new scene tiling method implemented in our urban simulation platform SimStadt, used for feasible estimations on a large scale. To quantify the influence of shading, the new concept of Urban Shading Ratio has been introduced and used for this evaluation process. In high density urban areas, this index may reach 60% for facades and 25% for roofs. Tiles of 500 m width and 200 m overlap are a minimum requirement in this case to compute solar irradiance with an acceptable accuracy. In medium density areas, tiles of 300 m width and 100 m overlap meet perfectly the accuracy requirements. In addition, the solar potential for various solar energy thresholds as well as the monthly variation of the Urban Shading Ratio have been quantified for both case studies, distinguishing between roofs and facades of different orientations

PARAMETRIZATION AND SHAPE RECONSTRUCTION TECHNIQUES FOR DOO-SABIN SUBDIVISION SURFACES

This thesis presents a new technique for the reconstruction of a smooth surface from a set of 3D data points. The reconstructed surface is represented by an everywhere -continuous subdivision surface which interpolates all the given data points. And the topological structure of the reconstructed surface is exactly the same as that of the data points. The new technique consists of two major steps. First, use an efficient surface reconstruction method to produce a polyhedral approximation to the given data points. Second, construct a Doo-Sabin subdivision surface that smoothly passes through all the data points in the given data set. A new technique is presented for the second step in this thesis. The new technique iteratively modifies the vertices of the polyhedral approximation 1CM until a new control meshM, whose Doo-Sabin subdivision surface interpolatesM, is reached. It is proved that, for any mesh M with any size and any topology, the iterative process is always convergent with Doo-Sabin subdivision scheme. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully