Coaxing a planar curve to comply

AbstractA long-standing problem in computer graphics is to find a planar curve that is shaped the way you want it to be shaped. A selection of various methods for achieving this goal is presented. The focus is on mathematical conditions that we can use to control curves while still allowing the curves some freedom. We start with methods invented by Newton (1643–1727) and Lagrange (1736–1813) and proceed to recent methods that are the subject of current research. We illustrate almost all the methods discussed with diagrams. Three methods of control that are of special interest are interpolation methods, global minimization methods (such as least squares), and (Bézier) control points. We concentrate on the first of these, interpolation methods

Applying inversion to construct rational spiral curves

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including those, assuming inflection.Comment: 18 pages, 11 figure

Log-Aesthetic Magnetic Curves and Their Application for CAD Systems

Curves are the building blocks of shapes and designs in computer aided geometric design (CAGD). It is important to ensure these curves are both visually and geometrically aesthetic to meet the high aesthetic need for successful product marketing. Recently, magnetic curves that have been proposed for computer graphics purposes are a particle tracing technique that generates a wide variety of curves and spirals under the influence of a magnetic field. The contributions of this paper are threefold, where the first part reformulates magnetic curves in the form of log-aesthetic curve (LAC) denoting it as log-aesthetic magnetic curves (LMC) and log-aesthetic magnetic space curves (LMSC), the second part elucidates vital properties of LMCs, and the final part proposes G2 LMC design for CAD applications. The final section shows two examples of LMC surface generation along with its zebra maps. LMC holds great potential in overcoming the weaknesses found in current interactive LAC mechanism where matching a single segment with G2 Hermite data is still a cumbersome task

GCS approximation

The discipline of Computer Aided Geometric Design (CAGD) deals with the computational aspects of geometric objects. This thesis is concerned with the construction of one of the most primitive geometric objects, curves. More specifically, it relates to the construction of a high quality planar curve. The Generalised Cornu Spiral (GCS) is a high quality planar curve that is beginning to show value in Computer Aided Design (CAD) and Computer Aided Manufacture (CAM) applications. However in its current form it is incompatible with current CAD/CAM systems. This thesis addresses the issue with the development of a robust and efficient polynomial replacement for the GCS

Sketch-based path design

We first present a novel approach to sketching 2D curves with minimally varying curvature as piecewise clothoids. A stable and efficient algorithm fits a sketched piecewise linear curve using a number of clothoid segments with G2 continuity based on a specified error tolerance. We then present a system for conceptually sketching 3D layouts for road and other path networks. Our system makes four key contributions. First, we generate paths with piecewise linear curvature by fitting 2D clothoid curves to strokes sketched on a terrain. Second, the height of paths above the terrain is automatically determined using a new constraint optimization formulation of the occlusion relationships between sketched strokes. Third, we present the break-out lens, a novel widget inspired by break-out views used in engineering visualization, to facilitate the in-context and interactive manipulation of paths from alternate view points. Finally, our path construction is terrain sensitive. ii Acknowledgements I would like to acknowledge the efforts of my supervisor, Karan Singh, and thank him for his guidance over the duration of the Masters program. I learned much from him a

Hybrid multi-objective trajectory optimization of low-thrust space mission design

Mención Internacional en el título de doctorThe overall goal of this dissertation is to develop multi-objective optimization algorithms for computing low-thrust trajectories. The thesis is motivated by the increasing number of space projects that will benefit from low-thrust propulsion technologies to gain unprecedented scientific, economic and social return. The low-cost design of such missions and the inclusion of concurrent engineering practices during the preliminary design phase demand advanced tools to rapidly explore different solutions and to benchmark them with respect to multiple conicting criteria. However, the determination of optimal low-thrust transfers is a challenging task and remains an active research field that seeks performance improvements. This work contributes to increase the efficiency of searching wide design spaces, reduce the amount of necessary human involvement, and enhance the capabilities to include complex operational constraints. To that end, the general low-thrust trajectory optimization problem is stated as a multi-objective Hybrid Optimal Control Problem. This formulation allows to simultaneously optimize discrete decisionmaking processes, discrete dynamics, and the continuous low-thrust steering law. Within this framework, a sequential two-step solution approach is devised for two different scenarios. The first problem considers the optimization of low-thrust multi-gravity assist trajectories. The proposed solution procedure starts by assuming a planar shape-based model for the interplanetary trajectory. A multi-objective heuristic algorithm combined with a gradient-based solver optimize the parameters de_ning the shape of the trajectory, the number and sequence of the gravity assists, the departure and arrival dates, and the launch excess velocity. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation transcription scheme. In this step, the sequence of planetary gravity assists is known and provided by the heuristic search, dynamics is three-dimensional, and the steering law is not predefined. Operational constraints to comply with launch asymptote declination limits and fixed reorientation times during the transfer apply. The presented approach is tested on a rendezvous mission to Ceres, on a yby mission to Jupiter, and on a rendezvous mission to Pluto. Pareto-optimal solutions in terms of time of ight and propellant mass consumed (or alternatively delivered mass) are obtained. Results outperform those found in the literature in terms of optimality while showing the effectiveness of the proposed methodology to generate quick performance estimates. The second problem considers the simultaneous optimization of fully electric, fully chemical and combined chemical-electric orbit raising transfers between Earth's orbits is considered. In the first step of the solution approach, the control law of the electric engine is parameterized by a Lyapunov function. A multi-objective heuristic algorithm selects the optimal propulsion system, the transfer type, the low-thrust control history, as well as the number, orientation, and magnitude of the chemical firings. Earth's shadow, oblateness and Van-Allen radiation effects are included. In the second step, candidate solutions are deemed as initial guesses to solve the Nonlinear Programming Problem resulting from applying a direct collocation scheme. Operational constraints to avoid the GEO ring in combination to slew rate limits and slot phasing constraints are included. The proposed approach is applied to two transfer scenarios to GEO orbit. Pareto-optimal solutions trading of propellant mass, time of ight and solar-cell degradation are obtained. It is identified that the application of operational restrictions causes minor penalties in the objective function. Additionally, the analysis highlights the benefits that combined chemical-electric platforms may provide for future GEO satellites.El objetivo principal de esta trabajo es desarrollar algoritmos de optimización multi-objetivo para la obtención de trayectorias espaciales con motores de bajo empuje. La tesis está motivada por el creciente número de misiones que se van a beneficiar del uso de estas tecnologías para conseguir beneficios científicos, económicos y sociales sin precedentes. El diseño de bajo coste de dichas misiones ligado a los principios de ingeniería concurrente requieren herramientas computacionales avanzadas que exploren rápidamente distintas soluciones y las comparen entre sí respecto a varios criterios. Sin embargo, esta tarea permanece como un campo de investigación activo que busca continuamente mejoras de rendimiento durante el proceso. Este trabajo contribuye a aumentar la eficiencia cuando espacio de diseño es amplio, a reducir la participación humana requerida y a mejorar las capacidades para incluir restricciones operacionales complejas. Para este fin, el problema general de optimización de trayectorias de bajo empuje se presenta como un problema híbrido de control óptimo. Esta formulación permite optimizar al mismo tiempo procesos de toma de decisiones, dinámica discreta y la ley de control del motor. Dentro de este marco, se idea un algoritmo secuencial de dos pasos para dos escenarios diferentes. El primer problema considera la optimización de trayectorias de bajo empuje con múltiples y-bys. El proceso de solución propuesto comienza asumiendo un modelo plano y shape-based para la trayectoria interplanetaria. Un algoritmo de optimización heurístico y multi-objetivo combinado con un resolvedor basado en gradiente optimizan los parámetros de la espiral que definen la forma de la trayectoria, el número y la secuencia de las maniobras gravitacionales, las fechas de salida y llegada, y la velocidad de lanzamiento. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. En este paso, las secuencia de y-bys es conocida y determinada por el paso anterior, la dinámica es tridimensional, y la ley de control no está prefinida. Además, se pueden aplicar restricciones operacionales relacionadas con las declinación de la asíntota de salida e imponer tiempos de reorientación fijos. Este enfoque es probado en misiones a Ceres, a Júpiter y a Plutón. Se obtienen soluciones óptimas de Pareto en función del tiempo de vuelo y la masa de combustible consumida (o la masa entregada). Los resultados obtenidos mejoran los disponibles en la literatura en términos de optimalidad, a la vez que reflejan la efectividad de la metodología a propuesta para generar estimaciones rápidas. El segundo problema considera la optimización simultanea de transferencias entre órbitas terrestres que usan propulsión eléctrica, química o una combinación de ambas. En el primer paso del método de solución, la ley de control del motor eléctrico se parametriza por una función de Lyapunov. Un algoritmo de optimización heurístico y multi-objetivo selecciona el sistema propulsivo óptimo, el tipo de transferencia, la ley de control del motor de bajo empuje, así como el número, la orientación y la magnitud de los impulsos químicos. Se incluyen los efectos de la sombra y de la no esfericidad de la Tierra, además de la radiación de Van-Allen. En el segundo paso, las soluciones candidatas se usan como estimación inicial para resolver el problema de optimización no lineal que resulta de aplicar un método de transcripción directa. El método de solución propuesto se aplica a dos transferencias a GEO diferentes. Se obtienen soluciones óptimas de Pareto con respecto a la masa de combustible, el tiempo de vuelo y la degradación de las células solares. Se identifican que la aplicación de las restricciones operacionales penaliza mínimamente la función objetivo. Además, los análisis presentados destacan los beneficios que la propulsión química y eléctrica combinada proporcionarían a los satélites en GEO.Programa de Doctorado en Mecánica de Fluidos por la Universidad Carlos III de Madrid; la Universidad de Jaén; la Universidad de Zaragoza; la Universidad Nacional de Educación a Distancia; la Universidad Politécnica de Madrid y la Universidad Rovira i Virgili.Presidente: Rafael Vázquez Valenzuela.- Secretario: Claudio Bombardelli.- Vocal: Bruce A. Conwa

Vortex laser development employing an interferometric output coupler

This thesis focuses on the design and implementation of new laser cavity designs for vortex mode generation. The vortex generation method is integrated into both solid-state laser cavities and systems using doped fibre gain mediums. Optical vortex laser beams have attracted a lot of attention due to their proposed applications in a wide range of industries from communication to particle manipulation to microscopy to material processing. More recently, vortex generation directly from the laser has attracted research due to the potential for higher purity, higher power and a compact system. In this work a modified Sagnac interferometer, dubbed the vortex output coupler (VOC), is integrated as the output coupler of a laser cavity and used to convert the fundamental Gaussian intracavity mode into a first order Laguerre Gaussian output. The VOC was first implemented into various solid-state cavities, with a Vanadate gain medium operating at 1064 nm, where vortex generation was successfully demonstrated. A record, from-the-source generated, vortex power of 31.3 W was achieved, with a laser slope efficiency of 62.5%. The mode purity was 95.2% and the M2 = 2.25. The handedness of the generated vortex was pure and switchable during operation. First order Hermite Gaussian modes with the same power were also demonstrated. The VOC was also shown to function in a pulsed cavity without any detrimental effects. It was found that the VOC has mode filtering properties, which helped maintain the a fundamental Gaussian in the cavity, despite mode mismatch between the pump beam and the fundamental Gaussian mode in the cavity. Fibre laser systems have the advantage of being compact, alignment insensitive and maintaining a close resemblance to the fundamental Gaussian mode through the use of a single mode fibre. To demonstrate the implementation versatility of the VOC and capitalise on its power scaling potential, the VOC was integrated into a non-polarisation maintaining fibre laser system as a bulk output coupler. An Ytterbium-doped gain fibre was used, operating at 1064 nm, which allowed for the same optics as in previous work to be used for the VOC. 5.08 W of vortex output power was achieved from the fibre laser system, with a mode purity and quality of 96.1% and M2 (X/Y) = 2.03/2.22, respectively. This system was also used as a first order vortex source for higher order vortex generation, using a spiral phase plate. Vortices with orbital angular momentum values of l = +2 and +3 were generated from the first order vortex (l = +1) input using spiral phase plates, which imparted +1 and +2 orbital angular momentum (helical phase ramps of 2π and 4π respectively). The VOC is made up of a beamsplitter and three turning mirrors, which are all high power damage threshold components. By choosing appropriate optical coatings for these components, considering wavelength and polarisation, the VOC can be implemented across the output spectrum making it incredibly versatile. The VOC is shown to function in a pulsed laser system, with a vortex pulse with duration 20 ns and energy 303 μJ shown in this work. The output mode can be switched between left and right vortex handedness and also between the first order Hermite Gaussian modes, all during operation. This pulsed operation and output mode versatility make it very interesting for material surface processing, particle levitation and manipulation, free-space communication and broadband, or ultrashort pulse, vortex generation. A VOC enhanced vortex laser can also be used as a high power and purity first order Hermite-Gaussian or Laguerre-Gaussain source for further conversion to higher order modes using other methods.Open Acces