Parametric Interpolation To Scattered Data [QA281. A995 2008 f rb].

Dua skema interpolasi berparameter yang mengandungi interpolasi global untuk data tersebar am dan interpolasi pengekalan-kepositifan setempat data tersebar positif dibincangkan. Two schemes of parametric interpolation consisting of a global scheme to interpolate general scattered data and a local positivity-preserving scheme to interpolate positive scattered data are described

Bernstein Polynomial-Based Method for Solving Optimal Trajectory Generation Problems

The article of record as published may be found at http://dx.doi.org/10.3390/s22051869This paper presents a method for the generation of trajectories for autonomous system operations. The proposed method is based on the use of Bernstein polynomial approximations to transcribe infinite dimensional optimization problems into nonlinear programming problems. These, in turn, can be solved using off-the-shelf optimization solvers. The main motivation for this approach is that Bernstein polynomials possess favorable geometric properties and yield computationally efficient algorithms that enable a trajectory planner to efficiently evaluate and enforce constraints along the vehicles� trajectories, including maximum speed and angular rates as well as minimum distance between trajectories and between the vehicles and obstacles. By virtue of these properties and algorithms, feasibility and safety constraints typically imposed on autonomous vehicle operations can be enforced and guaranteed independently of the order of the polynomials. To support the use of the proposed method we introduce BeBOT (Bernstein/B�zier Optimal Trajectories), an open-source toolbox that implements the operations and algorithms for Bernstein polynomials. We show that BeBOT can be used to efficiently generate feasible and collision-free trajectories for single and multiple vehicles, and can be deployed for real-time safety critical applications in complex environments.This research was supported by the Office of Naval Research, grants N000141912106, N000142112091 and N0001419WX00155. Antonio Pascoal was supported by H2020-EU.1.2.2-FET Proactive RAMONES, under Grant GA 101017808 and LARSyS-FCT under Grant UIDB/50009/2020. Isaac Kaminer was supported by the Office of Naval Research grant N0001421WX01974.This research was supported by the Office of Naval Research, grants N000141912106, N000142112091 and N0001419WX00155. Antonio Pascoal was supported by H2020-EU.1.2.2-FET Proactive RAMONES, under Grant GA 101017808 and LARSyS-FCT under Grant UIDB/50009/2020. Isaac Kaminer was supported by the Office of Naval Research grant N0001421WX01974

Efficient and High-Quality Rendering of Higher-Order Geometric Data Representations

Computer-Aided Design (CAD) bezeichnet den Entwurf industrieller Produkte mit Hilfe von virtuellen 3D Modellen. Ein CAD-Modell besteht aus parametrischen Kurven und Flächen, in den meisten Fällen non-uniform rational B-Splines (NURBS). Diese mathematische Beschreibung wird ebenfalls zur Analyse, Optimierung und Präsentation des Modells verwendet. In jeder dieser Entwicklungsphasen wird eine unterschiedliche visuelle Darstellung benötigt, um den entsprechenden Nutzern ein geeignetes Feedback zu geben. Designer bevorzugen beispielsweise illustrative oder realistische Darstellungen, Ingenieure benötigen eine verständliche Visualisierung der Simulationsergebnisse, während eine immersive 3D Darstellung bei einer Benutzbarkeitsanalyse oder der Designauswahl hilfreich sein kann. Die interaktive Darstellung von NURBS-Modellen und -Simulationsdaten ist jedoch aufgrund des hohen Rechenaufwandes und der eingeschränkten Hardwareunterstützung eine große Herausforderung. Diese Arbeit stellt vier neuartige Verfahren vor, welche sich mit der interaktiven Darstellung von NURBS-Modellen und Simulationensdaten befassen. Die vorgestellten Algorithmen nutzen neue Fähigkeiten aktueller Grafikkarten aus, um den Stand der Technik bezüglich Qualität, Effizienz und Darstellungsgeschwindigkeit zu verbessern. Zwei dieser Verfahren befassen sich mit der direkten Darstellung der parametrischen Beschreibung ohne Approximationen oder zeitaufwändige Vorberechnungen. Die dabei vorgestellten Datenstrukturen und Algorithmen ermöglichen die effiziente Unterteilung, Klassifizierung, Tessellierung und Darstellung getrimmter NURBS-Flächen und einen interaktiven Ray-Casting-Algorithmus für die Isoflächenvisualisierung von NURBSbasierten isogeometrischen Analysen. Die weiteren zwei Verfahren beschreiben zum einen das vielseitige Konzept der programmierbaren Transparenz für illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle und zum anderen eine neue hybride Methode zur Reprojektion halbtransparenter und undurchsichtiger Bildinformation für die Beschleunigung der Erzeugung von stereoskopischen Bildpaaren. Die beiden letztgenannten Ansätze basieren auf rasterisierter Geometrie und sind somit ebenfalls für normale Dreiecksmodelle anwendbar, wodurch die Arbeiten auch einen wichtigen Beitrag in den Bereichen der Computergrafik und der virtuellen Realität darstellen. Die Auswertung der Arbeit wurde mit großen, realen NURBS-Datensätzen durchgeführt. Die Resultate zeigen, dass die direkte Darstellung auf Grundlage der parametrischen Beschreibung mit interaktiven Bildwiederholraten und in subpixelgenauer Qualität möglich ist. Die Einführung programmierbarer Transparenz ermöglicht zudem die Umsetzung kollaborativer 3D Interaktionstechniken für die Exploration der Modelle in virtuellenUmgebungen sowie illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle. Die Erzeugung stereoskopischer Bildpaare für die interaktive Visualisierung auf 3D Displays konnte beschleunigt werden. Diese messbare Verbesserung wurde zudem im Rahmen einer Nutzerstudie als wahrnehmbar und vorteilhaft befunden.In computer-aided design (CAD), industrial products are designed using a virtual 3D model. A CAD model typically consists of curves and surfaces in a parametric representation, in most cases, non-uniform rational B-splines (NURBS). The same representation is also used for the analysis, optimization and presentation of the model. In each phase of this process, different visualizations are required to provide an appropriate user feedback. Designers work with illustrative and realistic renderings, engineers need a comprehensible visualization of the simulation results, and usability studies or product presentations benefit from using a 3D display. However, the interactive visualization of NURBS models and corresponding physical simulations is a challenging task because of the computational complexity and the limited graphics hardware support. This thesis proposes four novel rendering approaches that improve the interactive visualization of CAD models and their analysis. The presented algorithms exploit latest graphics hardware capabilities to advance the state-of-the-art in terms of quality, efficiency and performance. In particular, two approaches describe the direct rendering of the parametric representation without precomputed approximations and timeconsuming pre-processing steps. New data structures and algorithms are presented for the efficient partition, classification, tessellation, and rendering of trimmed NURBS surfaces as well as the first direct isosurface ray-casting approach for NURBS-based isogeometric analysis. The other two approaches introduce the versatile concept of programmable order-independent semi-transparency for the illustrative and comprehensible visualization of depth-complex CAD models, and a novel method for the hybrid reprojection of opaque and semi-transparent image information to accelerate stereoscopic rendering. Both approaches are also applicable to standard polygonal geometry which contributes to the computer graphics and virtual reality research communities. The evaluation is based on real-world NURBS-based models and simulation data. The results show that rendering can be performed directly on the underlying parametric representation with interactive frame rates and subpixel-precise image results. The computational costs of additional visualization effects, such as semi-transparency and stereoscopic rendering, are reduced to maintain interactive frame rates. The benefit of this performance gain was confirmed by quantitative measurements and a pilot user study

On derivative bounds for the rational quadratic Bézier paths

New derivative bounds for the rational quadratic Bézier paths are obtained, both for particular weight vectors and for classes of equivalent parametrisations. A comprehensive analysis of our bounds against existing bounds is made

Rational Cubic Ball Interpolants For Shape Preserving Curves And Surfaces

Interpolan pengekalan bentuk adalah satu teknik rekabentuk lengkung/ permukaan yang sangat penting dalam CAD/-CAM dan rekabentuk geometric Shape preserving interpolation is an essential curve/surface design technique in CAD/CAM and geometric desig

C1 Positivity Constrained Interpolation By Weighted Rational Cubic Triangles

The construction of positivity bivariate 1 C interpolants to scattered data using rational cubic Bézier triangular patches is considered. The interpolating surface is formed piecewise as a convex combination of three rational cubic Bézier triangles. Sufficient 1 C continuity conditions along the common boundary of two adjacent rational cubic Bézier triangles are shown. The sufficient positivity conditions on rational cubic Bézier triangle are derived. The initial values of the Bézier ordinates are determined by the data and the estimated gradient at the data sites while the weights are given the value one

On the intersection and derivative bound estimation for triangular Bézier surfaces

近年来，在CAGD和CAD中，三角Bézier曲面受到越来越广泛关注.它在表示拓扑结构复杂的几何模型时比张量积型Bézier曲面更灵活.因此，它更适合作为复杂几何体的拟合、插值离散数据点的工具.本文我们以三角Bézier曲面为研究对象，针对三角Bézier曲面在求交问题及其有理形式在低阶导矢界的改进两大方面的应用进行了研究.主要成果如下: 1.提出了implicitBézierlinearclipping(IBLC)算法来解决两平面Bézier曲线的交.在研究射线与三角Bézier曲面的求交方法之前，我们先研究了两Bézier曲线的求交，希望得到一些启发.IBLC算法利用了Bézier曲线的...In recent decades, triangular Bézier representations have attracted widely attention in the fields such as CAGD and CAD. They are a topologically more flexible modeling tool than tensor products. Hence, they are considered to be more suitable for representing models with complex geometries and fitting or interpolating scattered data points. In this dissertation, our objective is to study triangula...学位：理学博士院系专业：数学科学学院_计算数学学号：1902013015416