This paper discusses the relationship between the shape of the shading, the surface whose depth at each point equals the brightness in the image, and the shape of the original surface. I suggest the shading as an initial local approximation to shape, and discuss the scope of this approximation and what it may be good for. In particular, qualitative surface features, such as the sign of the Gaussian curvature, can be computed in some cases directly from the shading. Finally, a method to compute the direction of the illuminant (assuming a single point light source) from shading on occluding contours is shown.

Robustness of a system has been defined in various ways and a lot of work has been done to model the system robustness , but quantifying or measuring robustness has always been very difficult. In this research we consider a simple system of a linear estimator and then attempt to model the system performance and robustness in a geometrical manner which admits an analysis using the differential geometric concepts of slope and curvature. We try to compare two different types of curvatures, namely the curvature along the maximum slope of a surface and the square-root of the absolute value of sectional curvature of a surface, and observe the values to see if both of them can alternately be used in the process of understanding or measuring system robustness. In this process we have worked on two different examples and taken readings for many points to find if there is any consistency in the two curvatures.

We numerically investigate crystalline order on negative Gaussian curvature capillary bridges. In agreement with the experimental results in [W. Irvine et al., Nature, Pleats in crystals on curved surfaces, 2010, 468, 947] we observe for decreasing integrated Gaussian curvature, a sequence of transitions, from no defects to isolated dislocations, pleats, scars and isolated sevenfold disclinations. We especially focus on the dependency of topological charge on the integrated Gaussian curvature, for which we observe, again in agreement with the experimental results, no net disclination for an integrated curvature down to −10, and an approximately linear behavior from there on until the disclinations match the integrated curvature of −12. In contrast to previous studies in which ground states for each geometry are searched for, we here show that the experimental results, which are likely to be in a metastable state, can be best resembled by mimicking the experimental settings and continuously changing the geometry. The obtained configurations are only low energy local minima. The results are computed using a phase field crystal approach on catenoid-like surfaces and are highly sensitive to the initialization.

We numerically investigate crystalline order on negative Gaussian curvature capillary bridges. In agreement with the experimental results in [W. Irvine et al., Nature, Pleats in crystals on curved surfaces, 2010, 468, 947] we observe for decreasing integrated Gaussian curvature, a sequence of transitions, from no defects to isolated dislocations, pleats, scars and isolated sevenfold disclinations. We especially focus on the dependency of topological charge on the integrated Gaussian curvature, for which we observe, again in agreement with the experimental results, no net disclination for an integrated curvature down to −10, and an approximately linear behavior from there on until the disclinations match the integrated curvature of −12. In contrast to previous studies in which ground states for each geometry are searched for, we here show that the experimental results, which are likely to be in a metastable state, can be best resembled by mimicking the experimental settings and continuously changing the geometry. The obtained configurations are only low energy local minima. The results are computed using a phase field crystal approach on catenoid-like surfaces and are highly sensitive to the initialization.

[pt] A geometria diferencial provê um conjunto de medidas invariantes sob a ação de um grupo de transformações, em particular rígidas, afins e projetivas. Os invariantes por transformações rígidas são usados em quase todas as aplicações de computação gráfica e modelagem geométrica. O caso afim, por ser mais geral, permite estender essas ferramentas. Neste trabalho, propriedades geométricas são apresentadas no caso de superfícies paramétricas ou implícitas, em particular, a métrica afim, os vetores co-normal e normal afins e as curvaturas Gaussiana e média afins. Alguns resultados usuais de geometria Euclidiana, como a fórmula de Minkowski, são estendidos para o caso afim. Esse estudo permite definir estimadores das estruturas afins no caso de isossuperfícies. Porém, um cálculo direto dessas estruturas resulta em um grande número de operações e instabilidade numérica. Uma redução geométrica é proposta, obtendo fórmulas mais simples e mais estáveis numericamente. As propriedades geométricas incorporadas no Marching Cubes são analisadas e discutidas. / [en] Differential Geometry provides a set of measures invariant under a set of transformations, in particular rigid, affine, and projective. The invariants by rigid motions are using almost all applications of computer graphics and geometric modeling. The affine case, since it is more general, allows to extend these tools. In this work, geometric properties are presented in the case of parametric or implicit surfaces, in particular the affine metric, the conormal and normal vectors, and the affine Gaussian and mean curvatures. Some usual results of Euclidean geometry, as the Minkowski formula, are extended for the affine case. This study allows to define estimators of affines structure in the case of isosurfaces. Although, the direct calculation of these structures greatly increases the number of operations and numerical instabilities. A geometrical reduction is proposed obtaining a much simpler and numerical stabler formulae. The geometrical properties are incorporated in the Marching Cubes algorithms, then they are analyzed and discussed.

<p>By applying a segmentation procedure to two different sets of computed tomography scans, two geometrical models of the abdominal aorta, containing one inlet and two outlets have been constructed. One of these depicts a healthy blood vessel while the other displays one afflicted with a Abdominal Aortic Aneurysm. </p><p>After inputting these geometries into the computational dynamics software FLUENT, six simulations of laminar, stationary flow of a fluid that was assumed to be Newtonian were performed. The mass flow rate across the model outlet boundaries was varied for the different simulations to produce a basis for a parameter analysis study. </p><p>The segmentation data was also used as input data to a surface description procedure which produced not only the surface itself, but also the first and second directional derivatives in every one of its defining spatial data points. These sets of derivatives were followingly applied in an additional procedure that calculated values of Gaussian curvature. </p><p>A parameter variance analysis was carried out to evaluate the performance of the surface generation procedure. An array of resultant surfaces and surface directional derivatives were obtained. Values of Gaussian curvature were calculated in the defining spatial data points of a few selected surfaces. </p><p>The curvature values of a selected data set were visualized through a contour plot as well as through a surface map. Comparisons between the curvature surface map and one wall shear stress surface map were made.</p>

By applying a segmentation procedure to two different sets of computed tomography scans, two geometrical models of the abdominal aorta, containing one inlet and two outlets have been constructed. One of these depicts a healthy blood vessel while the other displays one afflicted with a Abdominal Aortic Aneurysm. After inputting these geometries into the computational dynamics software FLUENT, six simulations of laminar, stationary flow of a fluid that was assumed to be Newtonian were performed. The mass flow rate across the model outlet boundaries was varied for the different simulations to produce a basis for a parameter analysis study. The segmentation data was also used as input data to a surface description procedure which produced not only the surface itself, but also the first and second directional derivatives in every one of its defining spatial data points. These sets of derivatives were followingly applied in an additional procedure that calculated values of Gaussian curvature. A parameter variance analysis was carried out to evaluate the performance of the surface generation procedure. An array of resultant surfaces and surface directional derivatives were obtained. Values of Gaussian curvature were calculated in the defining spatial data points of a few selected surfaces. The curvature values of a selected data set were visualized through a contour plot as well as through a surface map. Comparisons between the curvature surface map and one wall shear stress surface map were made.

<p dir="ltr">In composites manufacturing, predicting the shape change in parts is vital for making sure part dimensions are properly compensated. Different factors in the manufacturing process, such as the temperature change throughout a thermoset cure cycle, can influence shape change. The compensation process becomes more difficult for geometries with double curvature, as interactions between the two radii of curvature can reduce the effectiveness of applying methodologies for single curvature geometries. Additionally, using finite element analysis (FEA) to predict shape change can be costly and time-consuming depending on part geometry.</p><p dir="ltr">This thesis studies an approach for predicting the shape change of a symmetric thermoset laminate with a double-curved L-angle section in its geometry. Specifically, the conservation of Gaussian curvature is applied to predict shape change. The geometry studied in this thesis can be broken down and analyzed as a segment of a torus, which is attached on one end by a cylinder and on the other end by a curved flange. Varying the length of the cylinder and flange sections, the effectiveness of Gauss’s theorem is determined for the different part geometries, with developed formulas compared against finite element simulations and experimental measurements.</p><p dir="ltr">By approximating torus segments with certain geometric criteria as cylinders, linear elasticity equations for a cylinder undergoing free thermal strain can be solved and the change in the larger arc length in the double-curved geometry is predicted after deformation. The integral form of Gauss’ theorem is then applied to determine the deformed angle of the larger arc, from which geometric relations can be applied to extract the deformed radius. Abaqus is used first to study the torus segment on its own, and then to see the effects of the cylinder and flange segments on the overall geometry. Experimental measurements are also used as a comparison.</p><p dir="ltr">Generally, the formula derived using Gauss’ theorem predicts shape change very well for the torus segment on its own. When cylinder and flange segments are included in the geometry, an empirical correction factor can be introduced to account for geometrically induced stiffening effects. Future developments and next steps in this research are discussed.</p>

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The objective of this research is to evaluate and introduce a new methodology regarding rural highway safety. Current practices rely on crash prediction models that utilize specific explanatory variables, whereas the depository of knowledge for past research is the Highway Safety Manual (HSM). Most of the prediction models in the HSM identify the effect of individual geometric elements on crash occurrence and consider their combination in a multiplicative manner, where each effect is multiplied with others to determine their combined influence. The concepts of 3-dimesnional (3-D) representation of the roadway surface have also been explored in the past aiming to model the highway structure and optimize the roadway alignment. The use of differential geometry on utilizing the 3-D roadway surface in order to understand how new metrics can be used to identify and express roadway geometric elements has been recently utilized and indicated that this may be a new approach in representing the combined effects of all geometry features into single variables. This research will further explore this potential and examine the possibility to utilize 3-D differential geometry in representing the roadway surface and utilize its associated metrics to consider the combined effect of roadway features on crashes. It is anticipated that a series of single metrics could be used that would combine horizontal and vertical alignment features and eventually predict roadway crashes in a more robust manner. It should be also noted that that the main purpose of this research is not to simply suggest predictive crash models, but to prove in a statistically concrete manner that 3-D metrics of differential geometry, e.g. Gaussian Curvature and Mean Curvature can assist in analyzing highway design and safety. Therefore, the value of this research is oriented towards the proof of concept of the link between 3-D geometry in highway design and safety. This thesis presents the steps and rationale of the procedure that is followed in order to complete the proposed research. Finally, the results of the suggested methodology are compared with the ones that would be derived from the, state-of-the-art, Interactive Highway Safety Design Model (IHSDM), which is essentially the software that is currently used and based on the findings of the HSM.