Isotopic Equivalence from Bezier Curve Subdivision

We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035

Recursive subdivision algorithms for curve and surface design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987

Synthesis of Spatially and Intrinsically Constrained Curves Using Simulated Annealing

A general technique is presented for automatic generation of B-spline curves in a spatially constrained environment, subject to specified intrinsic shape properties. Spatial constraints are characterized by a distance metric relating points on the curve to polyhedral models of obstacles which the curve should avoid. The shape of the curve is governed by constraints based on intrinsic curve properties such as parametric variation and curvature. To simultaneously address the independent goals of global obstacle avoidance and local control of intrinsic shape properties, curve synthesis is formulated as a combinatorial optimization problem and solved via simulated annealing. Several example applications are presented which demonstrate the robustness of the technique. The synthesis of both uniform and nonuniform B-spline curves is also demonstrated. An extension of the technique to general sculptured surface model synthesis is briefly described, and a preliminary example of simple surface synthesis presented

Box2Poly: Memory-Efficient Polygon Prediction of Arbitrarily Shaped and Rotated Text

Recently, Transformer-based text detection techniques have sought to predict polygons by encoding the coordinates of individual boundary vertices using distinct query features. However, this approach incurs a significant memory overhead and struggles to effectively capture the intricate relationships between vertices belonging to the same instance. Consequently, irregular text layouts often lead to the prediction of outlined vertices, diminishing the quality of results. To address these challenges, we present an innovative approach rooted in Sparse R-CNN: a cascade decoding pipeline for polygon prediction. Our method ensures precision by iteratively refining polygon predictions, considering both the scale and location of preceding results. Leveraging this stabilized regression pipeline, even employing just a single feature vector to guide polygon instance regression yields promising detection results. Simultaneously, the leverage of instance-level feature proposal substantially enhances memory efficiency (>50% less vs. the state-of-the-art method DPText-DETR) and reduces inference speed (>40% less vs. DPText-DETR) with minor performance drop on benchmarks

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

An Automated Optimal Design of a Fan Blade Using an Integrated CFD/MDO Computer Environment

The objective of the investigation is the development of more efficient design methodologies based on the applications of established design tools including Computational Fluid Dynamics (CFD) and non-linear Multidisciplinary Design Optimization (MDO) algorithms. Well known evolutionary type optimization algorithms include the Particle Swarm Optimization (PSO), Response Surface Optimization (RSO) and Genetic (GA) Algorithms. The benchmark case study is the optimal design of a low speed fan for an industrial air-conditioning application using the Response Surface Optimization (RSO) algorithm. The optimization algorithm controls the variations of parameters that describe the three-dimensional geometry of the blade while applying performance and geometrical constraints on blade shapes that are investigated. The optimal design is defined as the blade geometry which produces the maximum total efficiency subject to specified constraints on the volume flow rate (CFM) and rotational rate (RPM) of the fan

Discrete B-splines and subdivision techniques in compter-aided geometric design and computer graphics

Journal ArticleThe relevant theory of discrete 5-sphnes with associated new algorithms is extended to provide a framework for understanding and implementing general subdivision schemes for nonuniform B-splines. The new derived polygon corresponding to an arbitrary refinement of the knot vector for an existing .B-spline curve, including multiplicities, is shown to be formed by successive evaluations of the discrete B-spline defined by the original vertices, the original knot vector, and the new refined knot vector. Existing subdivision algorithms can be seen as proper special cases. General subdivision has widespread applications in computer-aided geometric design, computer graphics, and numerical analysis. The new algorithms resulting from the new theory lead to a unification of the display model, the analysis model, and other needed models into a single geometric model from which other necessary models are easily derived. New sample algorithms for interference calculation, contouring, surface rendering, and other important calculations are presented