Computer Aided Geometric Design

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C 2 piecewise quintic feedrates. The more conservative formulation ensures chord tolerance, as well as prescribed bounds on velocity, acceleration and jerk Cartesian components. Since the less restrictive formulations of our strategy can usually still ensure all the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is desired but also when full kinematic control has to be guaranteed. Our approach can be applied to any planar path with a piecewise sufficiently smooth parametric representation. When Pythagoreanhodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited.

The purpose of this research is the analysis using meta-analysis of studies in the field of Educational Technology in Turkey and in the field is to demonstrate how to get to that trend. For this purpose, a total of 263 studies were analyzed including 98 theses and 165 articles published between 2010-2018. Purpose sampling method was used when selecting publications. In the research, while selecting articles and Technology used in journals. Publications have been reviewed under 11 criteria. Index, year of publication, research scope, method, education level, sample, number of samples, data collection methods, analysis techniques, and research tendency, research topics in Educational Technology Research in Turkey has revealed. The data is interpreted based on percentage and frequency and the results are shown using the table.

A class of single-valued curves in polar coordinates, which we refer to as p-B6zier cur\e, ha> been recently presented by' St]nchez-Reyes and independently discovered by R de Casteljau. From their definition and expression in terms of the Fourier basis it is obvious that every curxe of degree ;; can be expressed as a curve of degree /or;, for any natural value k. In this paper, we provide a formula for degree elevation and we describe a simple and efliciem implementation of it. ,v~ 1998 Elsevier Science B.V.

Human Interaction in meetings is one of the famous fields of social dynamics. Meeting is integral part of every organization. In this, meeting outcome is extracted using tree based approach. Meetings contents or conversation are available in forms such as audio, video and text. In this, pattern of meeting is extracted from text document. An interaction is represented in the form of tree. Meeting Output is generated using data mining technique. Firstly the contents are filtered, extracted and steamed. Secondly classification is done into six categories propose, comment, acknowledgement, request Info, ask Opinion, pos Opinion, and neg Opinion. Next the interaction tree is constructed which represent the interaction flow of meeting. Finally the meeting output is generated from interaction tree using frequent pattern mining algorithm. The behavior of person is determined which includes a person who proposed a lot of ideas, a person with positive or negative attitude.

The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the "ordinary" cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how "close" the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the "ordinary" cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.

In this paper we prove a necessary and sufficient condition ensuring that the following problem possesses a solution: construct a twice-continuous cubic parametric spline, which interpolates a given set of planar points with a given parametrization and satisfies curvature-sign-type boundary conditions, i.e., the sign of the curvature is prescribed at the boundary points of the data. Based on this solvability condition, we also derive and numerically test a simple interactive algorithm that solves the above problem.

RapidMiner is a software for machine learning, data mining, predictive analytics, and business analytics. The server will record large web log files when user visits the website. Extracting knowledge from such huge data demands for new methods. In this paper, we propose a web usage mining method with RapidMiner. At first, the redundant files in log file are deleted by Matlab and then we mines web log which has been pretreated with RapidMiner, it obtains the custom of different user to visit the website by processing and analyzing log file, and mines unusual rules, and provides the reference for the policy decision and construction of website. Experimental result analysis show that, applying RapidMiner in web usage mining, will obtain frequent model which user visits the website, manage to optimize the website structure and recommends for users.

This paper examines a special type of rational curves called rational Frenet-Serret (RF) curves distinguished by the property that the motion of their Frenet-Serret frame is rational. It is shown that a rational curve is an RF curve if and only if it has rational speed and rational curvature. The paper derives a general representation formula for RF curves suitable for geometric design and provides a geometric survey of special RF curves. The special case of a cubic helix is examined thoroughly. Additionally the paper discusses several applications including examples for the design of rational sweeping surfaces, rational pipe surfaces and rational transition surfaces joining sweeps with G t continuity.

A Cpoint interpolatory subdivision scheme with a tension parameter is analysed. It is shown that for a certain range of the tension parameter the resulting curve is C'. The role of the tension parameter is demonstrated by a few examples. The application to surfaces and some further potential generalizations are discussed.

We provide a surprisingly simple cubic BCzier curve which gives a very accurate approximation to a segment of a circle. Joining the Bkier segments we obtain an approximation to the circle with continuous tangent and curvature. For 45 ' segments the error is approximately 2.10-6, and in general the approximation is sixth order accurate.

Current trends in free form editing suggest the development of a new novel editing paradigm for CAD models beyond traditional CAD editing of mechanical parts. To this end we wish to develop accurate, robust and efficient 3D mesh deformation techniques such as 3D structural morphing.

Given a closed triangular mesh, we construct a smooth free-form surface which is described as a collection of rational tensor-product and triangular surface patches. The surface is obtained by a special manifold surface construction, which proceeds by blending together geometry functions for each vertex. The transition functions between the charts, which are associated with the vertices of the mesh, are obtained via subchart parameterization.

General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case of rational general offsets, namely, Minkowski isoperimetric-hodograph curves. Differential geometric topics in the Minkowski plane, including the notion of normality, Frenet frame, Serret-Frenet equations, involutes and evolutes are introduced. These lead to an elegant process from which an explicit parametric representation of the general offset curves is derived. Using the duality between indicatrix and isoperimetrix and between involutes and evolutes, rational curves with rational general offsets are characterized. The dual Bézier notion is invoked to characterize the control structure of Minkowski isoperimetric-hodograph curves. This characterization empowers the constructive process of freeform curve design involving offsetting techniques.

Spirals are curves with single-signed, monotone increasing or decreasing curvature. A spiral can only interpolate certain G 2 Hermite data that is referred to as admissible G 2 Hermite data. In this paper we propose a biarcbased subdivision scheme that can generate a planar spiral matching an arbitrary set of given admissible G 2 Hermite data, including the case that the curvature at one end is zero. An attractive property of the proposed scheme is that the resulting subdivision spirals are also offset curves if the given input data are offsets of admissible G 2 Hermite data. A detailed proof of the convergence and smoothness analysis of the scheme is also provided. Several examples are given to demonstrate some excellent properties and practical applications of the proposed scheme.

Pythagorean-hodograph curves and related topics Pythagorean-hodograph (PH) curves were introduced (Farouki and to provide exact solutions to a number of basic computational problems that arise in computer-aided design and manufacturing, computer graphics and animation, robot path planning, and related fields. These include analytic reduction of the arc length and bending energy integrals; construction of rational offset (parallel) curves and surfaces; formulation of real-time interpolators for motion control applications; determination of rotation-minimizing frames for specifying orientations of rigid bodies along spatial paths; and many other basic computations concerned with "free-form" curves and surfaces.

A tutorial 2D MATLAB code for solving elliptic diffusion-type problems, including Poisson's equation on single patch geometries, is presented. The basic steps of Isogeometric Analysis are explained and two examples are given. The code has a very lean structure and has been kept as simple as possible, such that the analogy but also the differences to traditional finite element analysis become apparent. It is not intended for large-scale problems.

Bézier subdivision and degree elevation algorithms generate piecewise linear approximations of Bézier curves that converge to the original Bézier curve. Discrete derivatives of arbitrary order can be associated with these piecewise linear functions via divided differences. Here we establish the convergence of these discrete derivatives to the corresponding continuous derivatives of the initial Bézier curve. Thus, we show that the control polygons generated by subdivision and degree elevation provide not only an approximation to a Bézier curve, but also approximations of its derivatives of arbitrary order. 

Chapter ?? describes the fundamental geometric setting for 3D modeling and addresses Euclidean, affine and projective geometry, as well as differential geometry. In the present chapter, the discussions will be continued with a focus on geometric concepts which are less widely known. These are projective differential geometric methods, sphere geometries, line geometry, and non-Euclidean geometries. In all cases, we outline and illustrate applications of the respective geometries in geometric modeling.

This paper aims at presenting a parametric shape grammar of traditional Suakin houses (Red Sea state, Sudan). This work systematically attempts to generate appropriate plans arrangement that allows required functional relationships between spaces to be satisfied. The topological and geometrical properties of old Suakin houses were analyzed. These properties were originated and incorporated into the traditional Suakin buildings for the past ten centuries. The shape rules, dimensional, geometric and topological patterns of houses in the corpus are used as the generative model for the language of traditional suakin style This paper concludes with a discussion of the creative and generative value of the parametric shape grammars. Moreover, it facilitates the understanding of the formal composition of Suakin old style and the revival of a contemporary Suakin building style.

This paper is the third in a sequence of papers in which a knot removal strategy for splines, based on certain discrete norms, is developed. In the first paper, approximation methods defined as best approximations in these norms were discussed, while in the second paper a knot removal strategy for spline functions was developed. In this paper the knot removal strategy is extended to parametric spline curves and tensor product surfaces. The method has been implemented and thoroughly tested on a computer. We illustrate with several examples and applications.

Paper, sheet metal, and many other materials are approximately unstretchable. The surfaces obtained by bending these materials can be flattened onto a plane without stretching or tearing. More precisely, there exists a transformation that maps the surface onto the plane, after which the length of any curve drawn on the surface remains the same. Such surfaces, when sufficiently regular, are well known to mathematicians as developable surfaces. While developable surfaces have been widely used in engineering, design and manufacture, they have been less popular in computer graphics, despite the fact that their isometric properties make them ideal primitives for texture mapping, some kinds of surface modelling, and computer animation. Unfortunately, their constrained isometric behaviour cuts across common surface formulations. We formulate a new developable surface representation technique suitable for use in interactive computer graphics. The feasibility of our model is demonstrated by applying it to the modelling of a hanging scarf and ribbons and bows. Possible extensions and interesting areas of further research are discussed.

We discus several alternatives to the rational Bézier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding Bbases in general spaces. We also revisit the C-Bézier curves presented by , which coincide with the helix spline segments developed by , and are nothing else than curves expressed in the normalized B-basis of the space P 1 = span{1, t, cos t, sin t}. Such curves provide a valuable alternative to the rational Bézier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2 = span{1, t, cos t, sin t, cos 2t, sin 2t}, Q = span{1, t, t 2 , cos t, sin t} and I = span{1, t, cos t, sin t, t cos t, t sin t} are also suitable for shape preserving design, and we find their normalized B-basis. 

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R 3 . The four dimensional cyclographic model of the set of spheres in R 3 is an appropriate framework to show that a quadratic triangular Bézier patch in R 4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties.

Computer-Aided Geometric Design modelers are now based on powerful mathematical curve and surface models, but there is still a considerable need for efficient tools to handle, analyze and modify these objects. Designing product shapes using geometric operations on free-form curves and surfaces is still a tedious task. Moreover, designers would prefer to use meaningful tools to concentrate on design objectives expressed in terms of functionalities and constraints related to engineering topics and technical matters.

A class of single-valued curves in polar coordinates, which we refer to as p-B6zier cur\e, ha> been recently presented by' St]nchez-Reyes and independently discovered by R de Casteljau. From their definition and expression in terms of the Fourier basis it is obvious that every curxe of degree ;; can be expressed as a curve of degree /or;, for any natural value k. In this paper, we provide a formula for degree elevation and we describe a simple and efliciem implementation of it. ,v~ 1998 Elsevier Science B.V.

In the last century, the expression possibilities of the art of sculpture have reached unlimited possibilities conceptually and formally. These possibilities have  completely changed the forms of practice and expression in urban and public spaces within the historical process of sculpture. The close relationship of art with technique and technology can be followed closely today. Computer-aided design modeling programs and technologies are used in sculpture as a branch of art, as a reflection of this relationship in current conditions, and the boundaries of sculpture's expression and application possibilities continue to expand. In the content of the study; The possibilities provided by computer-aided design programs to the sculpture area, the realization of sculptures in digital environment with these programs and applications in open spaces are presented. By comparing the digital
representation images and applications of sculptures produced with computer aided programs, experiences regarding these processes were shared and evaluated.

The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the "ordinary" cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how "close" the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the "ordinary" cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.

We propose an iterative algorithm to generate a sequence of a prescribed number of points on a parametric curve with control of their distribution. Our algorithm depends on a free parameter which controls the achievement of a final distribution of points lying between two extreme cases: uniform arc length distribution and bending energy dependent distribution. This is obtained by computing a reparametrization function that interpolates the inverse of the arc length function at equidistant arguments. The proposed reparametrization function is a C 1 rational linear spline, therefore if the original curve is a NURBS then the reparametrized curve is also a NURBS of the same degree, but with an enlarged set of knots.

Computer aided geometric design is an area where the betterment of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. The traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling and computer graphics since they offer a number of features from which these areas can benefit. PDEs have been utilised under different approaches that vary from the development of surface generation techniques to applications addressing concrete problems such as noise reduction. This work summarises the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.

In this paper we introduce an unified framework for topological manipulation on triangulated 2-manifolds with or without boundary. We show that there are two kinds of primitive operators on the underlying meshes: operators that change the topological characteristic of the mesh and operators that just modify its combinatorial structure. We present such operators and demonstrate that they provide a complete and coherent set of elementary operations for mesh construction and editing.

An active contour model to surface approximation is presented. It adapts to the model shape to be approximated with help of local quadratic approximants of the squared distance function. The approach completely avoids the parametrization problem. The concept is open for inclusion of smoothing operators and shape constraints.

This paper examines a special type of rational curves called rational Frenet-Serret (RF) curves distinguished by the property that the motion of their Frenet-Serret frame is rational. It is shown that a rational curve is an RF curve if and only if it has rational speed and rational curvature. The paper derives a general representation formula for RF curves suitable for geometric design and provides a geometric survey of special RF curves. The special case of a cubic helix is examined thoroughly. Additionally the paper discusses several applications including examples for the design of rational sweeping surfaces, rational pipe surfaces and rational transition surfaces joining sweeps with G t continuity. © 1997 Elsevier Science B.V.

Invention of digital technology has lead to increase in the number of images that can be stored in digital format. So searching and retrieving images in large image databases has become more challenging. From the last few years, Content Based Image Retrieval (CBIR) gained increasing attention from researcher. CBIR is a system which uses visual features of image to search user required image from large image database and user's requests in the form of a query image. Important features of images are colour, texture and shape which give detailed information about the image. CBIR techniques using different feature extraction techniques are discussed in this paper.

We present an algorithm to approximate a set of unorganized points with a simple curve without self-intersections. The moving least-squares method has a good ability to reduce a point cloud to a thin curve-like shape which is a near-best approximation of the point set. In this paper, an improved moving least-squares technique is suggested using Euclidean minimum spanning tree, region expansion and refining iteration. After thinning a given point cloud using the improved moving least-squares technique we can easily reconstruct a smooth curve. As an application, a pipe surface reconstruction algorithm is presented.

We investigate the properties of a special kind of frame, which we call the Euler-Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a characterization of PH curves whose ERF achieves rotationminimizing property. This computation leads into a new fact that this ERF is equivalent to the Frenet frame on cubic PH curves. Furthermore, we prove that the minimum degree of non-planar PH curves whose ERF is an rotation-minimizing frame is seven, and provide a parameterization of the coefficients of those curves.