Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the --exact computation paradigm--[Yap et Dube] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions with integer vertices which respectively is included and contains the true intersection. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions

Investigations of Surface-Tension Effects Due to Small-Scale Complex Boundaries

The earliest man-made irrigation systems in recorded history date back to the ancient Egypt and Mesopotamia era. After thousands of years of experience, exploration, and experimenting, mankind have learned how to construct canals and dams and use pipes and pumps to direct and control water flow, but till this day, there are still some behaviors of water and other simple fluids that surprise us. One such example is the lotus effect: a surface-tension effect which allows raindrops to roll freely on a lotus leaf as if they were drops of mercury. One of the key factors that determine how a fluid system behave is the size-scale. Fluids flow at small scales very differently than they do at large scales. The standard comparing to which small and large are defined is the capillary length. A number of surface-tension related phenomena are unfamiliar because they are only noticeable at length-scales of a few millimeters or below, and they look nothing like what we would expect fluids to behave when dominated by gravity. As fascinating as many of them may seem at first glance, surface-tension phenomena are actually not that far away from our daily lives. Surface tension is everywhere because it costs energy to create areas of surfaces and interfaces, just like it costs energy to deform a solid (resulting in elasticity) or to elevate a weight (resulting in gravity). To minimize energy, a surface or an interface has the tendency to contract, and this tendency generates surface tension. The size of a system significantly affects the relative strengths of surface-tension effects comparing to effects of body forces, most commonly gravity. By equating the estimated magnitudes of surface tension and gravitational forces of a system, a length scale, know as the capillary length, can be defined. The capillary length of water on earth is about 2.7 mm. At the length scale of everyday objects, which is usually above the capillary length, surface-tension effects are not always prominent, because at those scales the competing force, gravity, is often much stronger. That is why the surface of a glass of water is more or less flat. However, as the size-scale decreases, surface tension decreases a lot slower than gravity, so when the size of a fluid system gets down to below the capillary length, surface tension takes over. One of the defining characteristics of this moment in human history, is the tremendous efforts we are putting into the research and engineering of micro- and nano-scale materials and structures − systems where surface tension is often the predominant force. It is important to study surface-tension effects so that we can use them to our advantage. In this Ph.D. dissertation, we have investigated some important surface-tension phenomena including capillarity, wetting, and wicking. We mainly focus on the geometric aspects of these problems, and to learn about how structures affect properties. Understanding these phenomena can help develop fabrication methods (Chapter 2), study surface properties (Chapter 3), and design useful devices (Chapter 4) at scales below the capillary length. In the first project (Chapter 2), we used numerical simulations and experiments to study the meniscus of a fluid confined in capillaries with complicated cross-sectional geometries. In the simulations, we computed the three-dimensional shapes of the menisci formed in polygonal and star-shaped capillaries with sharp or rounded corners. Height variations across the menisci were used to quantify the effect of surface tension. Analytical solutions were derived for all the cases where the cross-sectional geometry was a regular polygon or a regular star-shape. Power indices that characterize the effects of corner rounding were extracted from simulation results. These findings can serve as guide for fabrications of unconventional three-dimensional structures in Capillary Force Lithography experiments [J. Feng (2011) (a)]. Experimental demonstrations of the working principle was also performed. Although quantitative matching between simulation and experimental results was not achieved due to the limitation of material properties, clear qualitative trends were observed and interesting three-dimensional nano-structures were produced. A second project (Chapter 3) focused on developing techniques to produce three-dimensional hierarchically structured superhydrophobic surfaces with high aspect ratios. We experimented with two different high-throughput electron-beam-lithography processes featuring single and dual electron-beam exposures. After a surface modification procedure with a hydrophobic silane, the structured surfaces exhibited two distinct superhydrophobic behaviors − high and low adhesion. While both types of superhydrophobic surfaces exhibited very high (approximately 160_) water advancing contact angles, the water receding contact angles on these two different types of surfaces differed by about 50_ _ 60_, with the low-adhesion surfaces at about 120_ _ 130_ and the high-adhesion surfaces at about 70_ _ 80_. Characterizations of both the microscopic structures and macroscopic wetting properties of these product surfaces allowed us to pinpoint the structural features responsible for specific wetting properties. It is found that the advancing contact angle was mainly determined by the primary structures while the receding contact angle is largely affected by the side-wall slope of the secondary features. This study established a platform for further exploration of the structure aspects of surface wettability [J. Feng (2011) (b)]. In the third and final project (Chapter 4), we demonstrated a new type of microfluidic channel that enable asymmetric wicking of wetting fluids based on structure-induced direction-dependent surface-tension effect. By decorating the side-walls of open microfluidic channels with tilted fins, we were able to experimentally demonstrate preferential wicking behaviors of various IPA-water mixtures with a range of contact angles in these channels. A simplified 2D model was established to explain the wicking asymmetry, and a complete 3D model was developed to provide more accurate quantitative predictions. The design principles developed in this study provide an additional scheme for controlling the spreading of fluids [J. Feng (2012)]. The research presented in this dissertation spreads out across a wide range of physical phenomena (wicking, wetting, and capillarity), and involves a number of computational and experimental techniques, yet all of these projects are intrinsically united under a common theme: we want to better understand how simple fluids respond to small-scale complex surface structures as manifestations of surface-tension effects. We hope our findings can serve as building blocks for a larger scale endeavor of scientific research and engineering development. After all, the pursue of knowledge is most meaningful if the results improve the well-being of the society and the advancement of humanity

The Construction of Conforming-to-shape Truss Lattice Structures via 3D Sphere Packing

Truss lattices are common in a wide variety of engineering applications, due to their high ratio of strength versus relative density. They are used both as the interior support for other structures, and as structures on their own. Using 3D sphere packing, we propose a set of methods for generating truss lattices that fill the interior of B-rep models, polygonal or (trimmed) NURBS based, of arbitrary shape. Once the packing of the spheres has been established, beams between the centers of adjacent spheres are constructed, as spline based B-rep geometry. We also demonstrate additional capabilities of our methods, including connecting the truss lattice to (a shell of) the B-rep model, as well as constructing a tensor-product trivariate volumetric representation of the truss lattice - an important step towards direct compatibility for analysis.RYC-2017-2264

Algorithms for cartographic visualization

Maps are effective tools for communicating information to the general public and help people to make decisions in, for example, navigation, spatial planning and politics. The mapmaker chooses the details to put on a map and the symbols to represent them. Not all details need to be geographic: thematic maps, which depict a single theme or attribute, such as population, income, crime rate, or migration, can very effectively communicate the spatial distribution of the visualized attribute. The vast amount of data currently available makes it infeasible to design all maps manually, and calls for automated cartography. In this thesis we presented efficient algorithms for the automated construction of various types of thematic maps. In Chapter 2 we studied the problem of drawing schematic maps. Schematic maps are a well-known cartographic tool; they visualize a set of nodes and edges (for example, highway or metro networks) in simplified form to communicate connectivity information as effectively as possible. Many schematic maps deviate substantially from the underlying geography since edges and vertices of the original network are moved in the simplification process. This can be a problem if we want to integrate the schematized network with a geographic map. In this scenario the schematized network has to be drawn with few orientations and links, while critical features (cities, lakes, etc.) of the base map are not obscured and retain their correct topological position with respect to the network. We developed an efficient algorithm to compute a collection of non-crossing paths with fixed orientations using as few links as possible. This algorithm approximates the optimal solution to within a factor that depends only on the number of allowed orientations. We can also draw the roads with different thicknesses, allowing us to visualize additional data related to the roads such as trafic volume. In Chapter 3 we studied methods to visualize quantitative data related to geographic regions. We first considered rectangular cartograms. Rectangular cartograms represent regions by rectangles; the positioning and adjacencies of these rectangles are chosen to suggest their geographic locations to the viewer, while their areas are chosen to represent the numeric values being communicated by the cartogram. One drawback of rectangular cartograms is that not every rectangular layout can be used to visualize all possible area assignments. Rectangular layouts that do have this property are called area-universal. We show that area-universal layouts are always one-sided, and we present algorithms to find one-sided layouts given a set of adjacencies. Rectangular cartograms often provide a nice visualization of quantitative data, but cartograms deform the underlying regions according to the data, which can make the map virtually unrecognizable if the data value differs greatly from the original area of a region or if data is not available at all for a particular region. A more direct method to visualize the data is to place circular symbols on the corresponding region, where the areas of the symbols correspond to the data. However, these maps, so-called symbol maps, can appear very cluttered with many overlapping symbols if large data values are associated with small regions. In Chapter 4 we proposed a novel type of quantitative thematic map, called necklace map, which overcomes these limitations. Instead of placing the symbols directly on a region, we place the symbols on a closed curve, the necklace, which surrounds the map. The location of a symbol on the necklace should be chosen in such a way that the relation between symbol and region is as clear as possible. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. We developed algorithms to compute necklace maps and demonstrated our method with experiments using various data sets and maps. In Chapter 5 and 6 we studied the automated creation of ow maps. Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of ow between a source and a target. Good ow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We developed a new algorithm for drawing ow trees, ow maps with a single source. Unlike existing methods, our method merges lines smoothly and avoids self-intersections. Our method is based on spiral trees, a new type of Steiner trees that we introduced. Spiral trees have an angle restriction which makes them appear smooth and hence suitable for drawing ow maps. We study the properties of spiral trees and give an approximation algorithm to compute them. We also show how to compute ow trees from spiral trees and we demonstrate our approach with extensive experiments

The study of surface SHG and polygonal microcavity design for nonlinear applications on LiNbO₃

A z-cut congruent lithium niobate crystal (LiNbO3) has been used in this thesis, as a platform for the surface second harmonic generation (SHG) studies and for the designs of polygonal microcavities for nonlinear applications. Reflection second harmonic generation (RSHG) experiments were performed on LiNbO3 to reveal the interfacial layer symmetry as the crystal is rotated around the z axis. RSHG was also used, unsuccessfully as a non-destructive tool to map the domain-inverted area in the poled LiNbO3 crystals. But nevertheless, the polarity of the direction of the y-axis of the crystal was determined from RSHG data and the data shows that this direction also inverts, during domain inversion. RSHG was used unsuccessfully to monitor the relaxation of the internal field within the domain inverted area of the poled LiNbO3. A general operational principle of optical microcavities was discussed, in which a detailed theory governing the operational modes of a resonating hexagonal microcavity, made from bulk LiNbO3 crystal was reviewed for nonlinear device applications. A model for a total internal reflection (TIR) technique for the QPM method in a double-resonating hexagonal microcavity was formulated. The TIR-QPM model was based on finding a suitable hexagonal dimension in which, both the fundamental and SHG signal resonate simultaneously while at the same time allowing QPM to occur via TIR. The TIR-QPM model and the FDTD simulation were used to demonstrate the potential capability of the double-resonating hexagonal microcavity for efficient SHG. The model to achieve a nonlinear microcavity by periodically poling ring/disk resonator Ti:LiNbO3 ridge waveguide was introduced.<br/

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation

Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.Comment: LaTeX, 57 pages with 13 figure

Pattern Formation and Organization of Epithelial Tissues

Developmental biology is a study of how elaborate patterns, shapes, and functions emerge as an organism grows and develops its body plan. From the physics point of view this is very much a self-organization process. The genetic blueprint contained in the DNA does not explicitly encode shapes and patterns an animal ought to make as it develops from an embryo. Instead, the DNA encodes various proteins which, among other roles, specify how different cells function and interact with each other. Epithelial tissues, from which many organs are sculpted, serve as experimentally- and analytically-tractable systems to study patterning mechanisms in animal development. Despite extensive studies in the past decade, the mechanisms that shape epithelial tissues into functioning organs remain incompletely understood. This thesis summarizes various studies we have done on epithelial organization and patterning, both in abstract theory and in close contact with experiments. A novel mechanism to establish cellular left-right asymmetry based on planar polarity instabilities is discussed. Tissue chirality is often assumed to originate from handedness of biological molecules. Here we propose an alternative where it results from spontaneous symmetry breaking of planar polarity mechanisms. We show that planar cell polarity (PCP), a class of well-studied mechanisms that allows epithelia to spontaneously break rotational symmetry, is also generically capable of spontaneously breaking reflection symmetry. Our results provide a clear interpretation of many mutant phenotypes, especially those that result in incomplete inversion. To bridge theory and experiments, we develop quantitative methods to analyze fluorescence microscopy images. Included in this thesis are algorithms to selectively project intensities from a surface in z-stack images, analysis of cells forming short chain fragments, analysis of thick fluorescent bands using steerable ridge detector, and analysis of cell recoil in laser ablation experiments. These techniques, though developed in the context of zebrafish retina mosaic, are general and can be adapted to other systems. Finally we explore correlated noise in morphogenesis of fly pupa notum. Here we report unexpected correlation of noise in cell movements between left and right halves of developing notum, suggesting that feedback or other mechanisms might be present to counteract stochastic noise and maintain left-right symmetry.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138800/1/hjeremy_1.pd