Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Coonsovo povezivanje klase C1 trokutnih dijelova

A Gordon-Coons-type surface construction starts from three differentiable triangular surface patches, which are defined on the same triangular parameter domain. If one boundary curve of each fits a curvilinear triangle, then the defined surface interpolates to these curves. The connection between the resulting surface and the constituents is C1 continuous along the common boundary curves with the exception of the corner points. This surface definition is an extension of the Gordon--Coons definition of a triangular surface patch constructed from three boundary curves.Gordon-Coonsova konstrukcija plohe kreće od tri diferencijabilna trokutna plošna dijela koji su definirani na istom trokutnom parametarskom području. Ako po jedna rubna krivulja svakog od njih odgovara krivuljnom trokutu, tada definirana ploha interpolira te krivulje. Veza između dobivene plohe i sastavnih dijelova je klase C1 duž zajedničkih rubnih krivulja, s izuzetkom vrhova. Ova definicija plohe je proširenje Gordon-Coonsove definicije trokutnog plošnog dijela konstruiranog iz tri granične točke

A survey of parametric modelling methods for designing the head of a high-speed train

With the continuous increase of the running speed, the head shape of the high-speed train (HST) turns out to be a critical factor for further speed boost. In order to cut down the time used in the HST head design and improve the modelling efficiency, various parametric modelling methods have been widely applied in the optimization design of the HST head to obtain an optimal head shape so that the aerodynamic effect acting on the head of HSTs can be reduced and more energy can be saved. This paper reviews these parametric modelling methods and classifies them into four categories: 2D, 3D, CATIA-based, and mesh deformation-based parametric modelling methods. Each of the methods is introduced, and the advantages and disadvantages of these methods are identified. The simulation results are presented to demonstrate that the aerodynamic performance of the optimal models constructed by these parametric modelling methods has been improved when compared with numerical calculation results of the original models or the prototype models of running trains. Since different parametric modelling methods used different original models and optimization methods, few publications could be found which compare the simulation results of the aerodynamic performance among different parametric modelling methods. In spite of this, these parametric modelling methods indicate more local shape details will lead to more accurate simulation results, and fewer design variables will result in higher computational efficiency. Therefore, the ability of describing more local shape details with fewer design variables could serve as a main specification to assess the performance of various parametric modelling methods. The future research directions may concentrate on how to improve such ability

Visible surface algorithms for quadric patches

technical reportThis paper describes two algorithms which find the visible portions of surfaces in a picture of a cluster of three-dimensional quadric patches. A quadric patch is a portion of quadric surface defined by a quadratic equation and by zero, one or several quadratic inequalities. The picture is cut by parallel planes called scan planes; the visibility problem is solved in one scan plane at a time by making "a good guess" as t o what is visible according to the visible portions found in the previous can plane. The algorithm for intersecting patches works in a time roughly proportional to the number of patches involved (and not to the square of this number as with some previous algorithms)

Controlling the interpolation of NURBS curves and surfaces

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements. A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$^1$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$^2$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods. Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry. For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini. Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $N$ patches to form an $N$-sided region is considered. It is shown that regions with odd $N$ can be joined with G$^1$ continuity, but those with even $N$ or requiring G$^2$ continuity can only be obtained for specific geometries

Blending techniques in Curve and Surface constructions

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An intelligent robotic inspection system for airframe structures

Robot trajectory control is currently performed in an open loop fashion with the trajectory being specified as either a series of end point positions or as a series of joint angle values which are passed through in sequence. In order to perform any complex tasks in an unstructured or semi-structured environment an 'intelligent' robot is required which can sense its environment and alter its trajectory accordingly. This thesis describes the development of an 'intelligent' robotic inspection system which is capable of automatically deploying test probes for the inspection of the structures commonly found in modern airframes with no prior knowledge of the structure being inspected. It utilises a Puma 560 (articulated arm) industrial robot which is under supervisory control from an IBM ps2 personal computer and a wrist-mounted CCD camera with a low power industrial laser to acquire information about the robot's environment. In order to improve the positional accuracy of the Puma it has been calibrated using a computerised surveying system. Extensive use is made of image processing, pattern recognition and mathematical surface modelling to build up a model of the structure being examined and this is used to define the trajectory of the end point of the Puma