Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

On automatic tuning of basis functions in Bezier method

A transition from the fixed basis in Bezier's method to some class of base functions is proposed. A parameter vector of a basis function is introduced as additional information. This achieves a more universal form of presentation and analytical description of geometric objects as compared to the non-uniform rational B-splines (NURBS). This enables control of basis function parameters including control points, their weights and node vectors. This approach can be useful at the final stage of constructing and especially local modification of compound curves and surfaces with required differential and shape properties; it also simplifies solution of geometric problems. In particular, a simple elimination of discontinuities along local spline curves due to automatic tuning of basis functions is demonstrated

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

Procedurally Generating Biologically Driven Bird and Non-Avian Dinosaur Feathers

A key element in computer-graphics research is representing the world around us, and immense inspiration may be found in nature. Algorithms and procedural models may be developed that can describe the three-dimensional shape of objects and how they interact with light. This thesis focuses particularly on bird and other dinosaur feathers and their structure. More specifically, it addresses the problem of procedurally generating biologically driven geometry for modeling feathers in computer graphics. As opposed to previously published methods for generated feather geometry, data is derived from a myriad of real-world specimens of feathers and used in creating graphical models of feathers. Modeling feathers is of interest both for media production and also for various fields of research such as ornithology, paleontology, and material science. In order to create realistic, computer-graphics feathers, the anatomy of feathers is analyzed in detail with the aim of understanding their structure and variation in order to apply that understanding to modeling. Data concerning the shape of actual feathers was collected and analyzed to drive attribute parameters for modeling accurate synthetic feathers, during which methods for generating geometry informed by the data were investigated. Synthesized image results, capabilities, limitations, and extensions of the developed techniques are presented

Procedurally generated models for Isogeometric Analysis

Increasingly powerful hard- and software allows for the numerical simulation of complex physical phenomena with high levels of detail. In light of this development the definition of numerical models for the Finite Element Method (FEM) has become the bottleneck in the simulation process. Characteristic features of the model generation are large manual efforts and a de-coupling of geometric and numerical model. In the highly probable case of design revisions all steps of model preprocessing and mesh generation have to be repeated. This includes the idealization and approximation of a geometric model as well as the definition of boundary conditions and model parameters. Design variants leading to more resource-efficient structures might hence be disregarded due to limited budgets and constrained time frames. A potential solution to above problem is given with the concept of Isogeometric Analysis (IGA). Core idea of this method is to directly employ a geometric model for numerical simulations, which allows to circumvent model transformations and the accompanying data losses. Basis for this method are geometric models described in terms of Non-uniform rational B-Splines (NURBS). This class of piecewise continuous rational polynomial functions is ubiquitous in computer graphics and Computer-Aided Design (CAD). It allows the description of a wide range of geometries using a compact mathematical representation. The shape of an object thereby results from the interpolation of a set of control points by means of the NURBS functions, allowing efficient representations for curves, surfaces and solid bodies alike. Existing software applications, however, only support the modeling and manipulation of the former two. The description of three-dimensional solid bodies consequently requires significant manual effort, thus essentially forbidding the setup of complex models. This thesis proposes a procedural approach for the generation of volumetric NURBS models. That is, a model is not described in terms of its data structures but as a sequence of modeling operations applied to a simple initial shape. In a sense this describes the "evolution" of the geometric model under the sequence of operations. In order to adapt this concept to NURBS geometries, only a compact set of commands is necessary which, in turn, can be adapted from existing algorithms. A model then can be treated in terms of interpretable model parameters. This leads to an abstraction from its data structures and model variants can be set up by variation of the governing parameters. The proposed concept complements existing template modeling approaches: templates can not only be defined in terms of modeling commands but can also serve as input geometry for said operations. Such templates, arranged in a nested hierarchy, provide an elegant model representation. They offer adaptivity on each tier of the model hierarchy and allow to create complex models from only few model parameters. This is demonstrated for volumetric fluid domains used in the simulation of vertical-axis wind turbines. Starting from a template representation of airfoil cross-sections, the complete "negative space" around the rotor blades can be described by a small set of model parameters, and model variants can be set up in a fraction of a second. NURBS models offer a high geometric flexibility, allowing to represent a given shape in different ways. Different model instances can exhibit varying suitability for numerical analyses. For their assessment, Finite Element mesh quality metrics are regarded. The considered metrics are based on purely geometric criteria and allow to identify model degenerations commonly used to achieve certain geometric features. They can be used to decide upon model adaptions and provide a measure for their efficacy. Unfortunately, they do not reveal a relation between mesh distortion and ill-conditioning of the equation systems resulting from the numerical model

Vector Graphics for Real-time 3D Rendering

Algorithms are presented that enable the use of vector graphics representations of images in texture maps for 3D real time rendering. Vector graphics images are resolution independent and can be zoomed arbitrarily without losing detail or crispness. Many important types of images, including text and other symbolic information, are best represented in vector form. Vector graphics textures can also be used as transparency mattes to augment geometric detail in models via trim curves. Spline curves are used to represent boundaries around regions in standard vector graphics representations, such as PDF and SVG. Antialiased rendering of such content can be obtained by thresholding implicit representations of these curves. The distance function is an especially useful implicit representation. Accurate distance function computations would also allow the implementation of special effects such as embossing. Unfortunately, computing the true distance to higher order spline curves is too expensive for real time rendering. Therefore, normally either the distance is approximated by normalizing some other implicit representation or the spline curves are approximated with simpler primitives. In this thesis, three methods for rendering vector graphics textures in real time are introduced, based on various approximations of the distance computation. The first and simplest approach to the distance computation approximates curves with line segments. Unfortunately, approximation with line segments gives only C0 continuity. In order to improve smoothness, spline curves can also be approximated with circular arcs. This approximation has C1 continuity and computing the distance to a circular arc is only slightly more expensive than computing the distance to a line segment. Finally an iterative algorithm is discussed that has good performance in practice and can compute the distance to any parametrically differentiable curve (including polynomial splines of any order) robustly. This algorithm is demonstrated in the context of a system capable of real-time rendering of SVG content in a texture map on a GPU. Data structures and acceleration algorithms in the context of massively parallel GPU architectures are also discussed. These data structures and acceleration structures allow arbitrary vector content (with space-variant complexity, and overlapping regions) to be represented in a random-access texture

New Models for High-Quality Surface Reconstruction and Rendering

The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface