Representation of NURBS surfaces by Controlled Iterated Functions System automata

The main geometric features of the nonuniform rational B-splines (NURBS) curve and surface representations are described. It is shown that most of these features are already exhibited by conics, which are a special case of NURBS. The properties typical of NURBS are discussed without dwelling on properties already present in polynomial curves. Conic sections and their representations using rational Bezier curves are reviewed. Cubic NURB curves, geometrical rational splines, rational and B-spline surfaces, and rational Bezier triangles are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Subdivision schemes generate self-similar curves and surfaces. Therefore there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by Iterated Function Systems (IFS). We demonstrate that this connection between subdivision schemes and fractals is even deeper by showing that curves and surfaces generated by subdivision are also attractors, fixed points of IFS's. To illustrate this fractal nature of subdivision, we derive the associated IFS for many different subdivision curves and surfaces without extraordinary vertices, including B-splines, piecewise Bezier, interpolatory four-point subdivision, bicubic subdivision, three-direction quartic box-spline subdivision and Kobbelt's √3-subdivision surfaces. Conversely, we shall show how to build subdivision schemes to generate traditional fractals such as the Sierpinski gasket and the Koch curve, and we demonstrate as well how to control the shape of these fractals by adjusting their control points.

DN-Edit is a virtual environment developed to allow the manipulation of non-uniform rational b-spline (NURBS) surfaces using virtual shaping tools. NURBS have become the industry standard for representation of free-form curves and surfaces. The contribution of the work presented here is in the development of shaping tools which are used to operate directly on the NURBS data and change the shape of the surfaces in a virtual environment. These shaping tools allow surface manipulations to be made using methods that match the shaping of real malleable objects. The virtual shaping tools are three-dimensional shapes that are controlled through a six degree-of-freedom tracking system that converts user hand motions into computer input. The NURBS surface updates itself in real time due to the effect of the tools on the surface. The new shape of the surface is dependent on the position and orientation of the shaping tool relative to the surface. Constraint-based surface manipulation is used to obtain multiple point direct manipulation of NURBS surfaces. In addition, computational methods to allow the user to have direct control of the first derivative of the surface over an area are implemented. This application has been developed using the C2 software libraries and Iowa State University’s C2 surround screen virtual environment.

Matrix representation for NURB curves and surfaces

Continuous surface representations, such as B-spline and Non-Uniform Rational B-spline (NURBS) surfaces are the de facto standard for modeling 3D objects - thin shells and solid objects alike - in the field of Computer-Aided Design (CAD). For performing physically based simulation, Finite Element Analysis (FEA) has been the industry standard for many years. In order to analyze physical properties such as stability, aerodynamics, or heat dissipation, the continuous models are discretized into finite element (FE) meshes. A tight integration of and a smooth transition between geometric design and physically based simulation are key factors for an efficient design and engineering workflow. Converting a CAD model from its continuous boundary representation (B-Rep) into a discrete volumetric representation for simulation is a time-consuming process that introduces approximation errors and often requires manual interaction by the engineer. Deriving design changes directly from the simulation results is especially difficult as the meshing process is irreversible. Isogeometric Analysis (IGA) tries to overcome this meshing hurdle by using the same representation for describing the geometry and for performing the simulation. Most commonly, IGA is performed on bivariate and trivariate spline representations (B-spline or NURBS surfaces and volumes). While existing CAD B-Rep models can be used directly for simulating thin-shell objects, simulating solid objects requires a conversion from spline surfaces to spline volumes. As spline volumes need a trivariate tensor-product topology, complex 3D objects must be represented via trimming or by connecting multiple spline volumes, limiting the continuity to C^0. As an alternative to NURBS or B-splines, subdivision models allow for representing complex topologies with as a single entity, removing the need for trimming or tiling and potentially providing higher continuity. While subdivision surfaces have shown promising results for designing and simulating shells, IGA on subdivision volumes remained mostly unexplored apart from the work of Burkhart et al. In this dissertation, I investigate how volumetric subdivision representations are beneficial for a tighter integration of geometric modeling and physically based simulation. Focusing on Catmull-Clark (CC) solids, I present novel techniques in the areas of efficient limit evaluation, volumetric modeling, numerical integration, and mesh quality analysis. I present an efficient link to FEA, as well as my IGA approach on CC solids that improves upon Burkhart et al.'s proof of concept with constant-time limit evaluation, more accurate integration, and higher mesh quality. Efficient limit evaluation is a key requirement when working with subdivision models in geometric design, visualization, simulation, and 3D printing. In this dissertation, I present the first method for constant-time volumetric limit evaluation of CC solids. It is faster than the subdivision-based approach by Burkhart et al. for every topological constellation and parameter point that would require more than two local subdivision steps. Adapting the concepts of well-known surface modeling tools, I present a volumetric modeling environment for CC-solid control meshes. Consistent volumetric modeling operations built from a set of novel volumetric Euler operators allow for creating and modifying topologically consistent volumetric meshes. Furthermore, I show how to manipulate groups of control points via parameters, how to avoid intersections with inner control points while modeling the outer surface, and how to use CC solids in the context of multi-material additive manufacturing. For coupling of volumetric subdivision models with established FE frameworks, I present an efficient and consistent tetrahedral mesh generation technique for CC solids. The technique exploits the inherent volumetric structure of CC-solid models and is at least 26 times faster than the tetrahedral meshing algorithm provided by CGAL. This allows to re-create or update the tetrahedral mesh almost instantly when changing the CC-solid model. However, the mesh quality strongly depends on the quality of the control mesh. In the context of structural analysis, I present my IGA approach on CC solids. The IGA approach yields converging stimulation results for models with fewer elements and fewer degrees of freedom than FE simulations on tetrahedral meshes with linear and higher-order basis functions. The solver also requires fewer iterations to solve the linear system due to the higher continuity throughout the simulation model provided by the subdivision basis functions. Extending Burkhart et al.'s method, my hierarchical quadrature scheme for irregular CC-solid cells increases the accuracy of the integrals for computing surface areas and element stiffnesses. Furthermore, I introduce a quality metric that quantifies the parametrization quality of the limit volume, revealing distortions, inversions, and singularities. The metric shows that cells with multiple adjacent boundary faces induce singularities in the limit, even for geometrically well-shaped control meshes. Finally, I present a set of topological operations for splitting such boundary cells - resolving the singularities. These improvements further reduce the amount of elements required to obtain converging results as well as the time required for solving the linear system.

A method on how to generate traditional quadrilateral NURBS surfaces, non-quadrilateral NURBS surfaces, such as triangle NURBS (Non-uniform Rational B-spline) surfaces and pentagon NURBS surfaces and their equidistant surfaces is presented. A NURBS surface is a bi-parameter surface, and the surface can be generated in accordance with its parameters. With the method given by this paper, if a series of values of the two parameters can be obtained in sequence, Surface Points and the Equidistant Surfaces Points based on these values of the two parameters can be obtained. These points can comprise a NURBS surface and its equidistant surface. Traditionally a NURBS surface is a quadrilateral surface. It is difficult to obtain a non-quadrilateral NURBS surface. A strategy is to generate non-quadrilateral surfaces by adjusting the NURBS surfaces' control points. The results show that the NURBS surface generation method is effective and the strategy to generate non-quadrilateral surfaces by adjusting the control points is feasible in some areas. What discussed in the paper can be used at computer graphics and numerical control areas.

Because of the complexity of Non-Uniform Rational B-splines (NURBS) surface, the conventional NURBS surface generation method usually first constructs a Coons surface and then changes it into NURBS surface, which adds the operation complexity. To address this problem, an NURBS boundary surface direct generation method was proposed, which can directly construct a NURBS boundary surface by using four NURBS boundary curves. Compared with the traditional method,it need not convert Coons surface to NURBS surface and have a low computing cost. The experiments show the proposed method is simple and convenient, but the generated surface still has a high quality and good continuity.

Chapter 10 - Subdivision Surfaces

Rapid advancements have been made in the field of surface reconstruction over the last two decades. Nonetheless, traditional approaches to reconstructing surface representations, although robust for a plethora of objects, fail to scale well for 3D point cloud datasets available today that contain a diverse class of shapes. Due to the widely acknowledged success of deep learning-based methods on 2D images, there has been a growing interest in using deep learning for obtaining surface representations from 3D point clouds. While several such methods have been proposed, many are not easy to adapt for applications in fields like computer-aided design and agriculture. This thesis aims to develop robust deep learning frameworks for explicit and implicit surface reconstruction that can be seamlessly integrated into research pipelines of such fields. Boundary representations (B-reps) using Non-Uniform Rational B-splines (NURBS) are the de facto standard used in CAD, but their utility in deep learning-based approaches is not well researched. In our first work, we propose a differentiable NURBS module to integrate the NURBS representation of CAD models with deep learning methods. We mathematically define the derivatives of the NURBS curves or surfaces with respect to the input parameters, which are then used to perform the “backward” evaluation performed while training deep learning models. This allows NURBS to be incorporated with the modern differentiable programming paradigm used in deep learning, making it more easily integrated with modern deep learning frameworks. Reconstructing the geometry of crops from 3D point cloud data is useful for various plant phenotyping applications. Due to very thin and slender segments, obtaining accurate surface geometry representations from the 3D point cloud data of plants is challenging. Further, defects in the point cloud data might produce errors in the reconstructed plant structures. In our second work, we leverage deep learning frameworks that learn neural implicit representations to reconstruct the surfaces of fully developed maize plants using data acquired from Terrestrial Laser Scanners (TLS).

Novel freeform optical design methods can be classified in two categories, depending on whether they focus on the generation of a starting point or the development of new optimization tools. In this paper, we design a freeform three-mirror anastigmat (TMA) and compare different surface representations using either a differential ray tracer as a new optimization tool or a commercial ray tracer (ANSYS-ZEMAX OpticStudio). For differential ray tracing, we used FORMIDABLE (Freeform Optics Raytracer with Manufacturable Imaging Design cApaBiLitiEs), an optical design library with differential ray tracing and Non-Uniform Rational B-Splines (NURBS) optimization capabilities, available under the European Software Community License (ESCL). NURBS allow a freeform surface to be represented without needing any prior knowledge of the surface, such as the polynomial degree in polynomial descriptions. OpticStudio and other commercial optical design software are designed to optimize polynomial surfaces but are not well-suited to optimize NURBS surfaces, requiring a custom optical design library. In order to demonstrate the interest in using NURBS representation, we designed and independently optimized two freeform telescopes over different iteration cycles; with NURBS using FORMIDABLE or with XY polynomials using OpticStudio. We then compared the resulting systems using their root mean square field maps to assess the optimization quality of each surface representation. We also provided a full-system comparison, including mirror freeform departures. This study shows that NURBS can be a relevant alternative to XY polynomials for the freeform optimization of reflective three-mirror telescopes as it achieves more a uniform imaging quality in the field of view.

As optical systems continue to advance, non-uniform rational B-spline (NURBS) surfaces increasingly being considered in asymmetric optical systems due to their localized control characteristics. However, the representation of NURBS surfaces has complicated the analysis of these systems, leading to a significant computational burden. To address this challenge, we propose an optimizing algorithm for imaging optical systems based on high-precision ray tracing of NURBS surfaces. This method initiates with getting a knot grid as prior information, in conjunction with the Newton-Raphson algorithm, to obtain high-precision numerical solutions for the intersection of rays with a NURBS surface. Building upon this methodology, we introduce an optimization technique that includes shape evaluation to generate an evaluation function specific to NURBS surfaces. This approach is then applied within a rapid optimization process that accounted for the region of ray influence. Under consistent control point grids and sampling ray conditions, we present an off-axis four-mirror system to showcase that our algorithm has achieved a computational efficiency improvement of approximately 14 times compared to the previous method. This high-precision imaging design based on spline surfaces fulfills the need for efficient and accurate algorithms for NURBS surface applications in various imaging systems, providing guidance for practical applications.

On-machine measurement (OMM), which is used to measure the machined surfaces of a workpiece during or after the machining of the workpiece, enables direct measurement within the workspace without moving the workpiece. However, although the three-dimensional geometric shape created by computer-aided design systems include various features, because such features are converted into non-uniform rational B-spline (NURBS) surfaces during OMM, measurement deviation occurs between a feature and its corresponding NURBS surface. In this paper, we suggest a method to generate the measurement path for 5-axis OMM in a way that recognizes the features necessary for measurement from NURBS surfaces whose feature data set was removed during the conversion processes. To verify the reliability of the measurement path generation system named OMV+, which was developed using the above method, we carried out an experiment to compare the measurement path that defines the measuring point on the feature and the path that defines the measuring point on the NURBS surface with the measured result using the touch probe on the machined workpiece.

This paper focuses on the design of the extended branch structure for rapidly and simultaneously interpolating non-uniform rational B-spline (NURBS) surfaces and their partial derivatives on CNC machines. NURBS surfaces are concerned in the applications of molding process. However, the copious and complicated operations usually limit the applications of NURBS surfaces in CNC machining systems. In this paper, by considering the derivatives of basis functions, an extended branch structure is derived such that the computation time can be significantly reduced in computing NURBS surfaces and their partial derivatives. According to the machining results on a 3-axis vertical machining center, the extended branch structure is feasible to implement on CNC machines and is more efficient than existing approaches in computation.

Computer aided geometric design (CAGD) tools use non-uniform rational B-spline (NURBS) surfaces to describe the geometrical shape of an object. A NURBS-based description is particularly useful when treating problems of EM scattering from large conducting surfaces in the framework of high-frequency techniques like physical optics (PO). When dealing with large and complex bodies, PO requires the computation of double integrals with fast oscillating kernels. The use of basic building blocks (e.g. NURBS) for describing complex surfaces can alleviate the numerical problems. The presented approach provides a field representation in terms of line-integrals for the near-field PO radiation from a general NURBS surface. The surface is illuminated by a point source located in arbitrary position. The numerical computation of geometrical parameters is more conveniently performed if each NURBS surface is transformed into a set of rational Be/spl acute/zier patches. The final physically meaningful outcome provides a representation of the PO field from the Be/spl acute/zier surface in terms of line integrals. The results are compared with those obtained with classical PO surface integration, by discussing the computational improvements and asymptotic limitations.

Chapter 5 - Rational Techniques