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This paper presents a new approach to computation of geometric continuity for parametric bi-cubic patches, based on a simple mathematical reformulation which leads to simple additional conditions to be applied in the patching computation. The paper presents an Hermite formulation of a bicubic parametric patch, but reformulations can be made also for B\'ezier and B-Spline patches as well. The presented approach is convenient for the cases when valencies of corners are different from the value 4, in general.

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric B&eacute, zier (GHT-B&eacute, zier) curves. The GHT-Bernstein basis functions and B&eacute, zier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-B&eacute, zier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical B&eacute, zier curve and some other applications by using the proposed GHT-B&eacute, zier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

In this paper, we propose a novel method for constructing developable surfaces using generalized C-Bezier bases with shape parameters. Based on the duality between points and planes in 3D projective space, the generalized developable C-Bezier surfaces, whose shape can be adjusted by changing multiple shape parameters, are designed using control planes with extensional C-Bezier basis functions. With the shape parameters taking different values, a family of developable surfaces can be constructed, which keeps most of characteristics of classic developable Bezier surfaces. Furthermore, some interesting properties of the new developable surfaces, as well as the geometric continuity conditions between two adjacent generalized developable C-Bezier surfaces, are investigated. Finally, we illustrate the convenience and efficiency of the proposed methods by several convictive and representative numerical examples.

In order to solve the problem of geometric design and architectural design of complex engineering surface, we introduce the parametric and geometric continuity constraints of generalized C-Bézier curves and surfaces with shape parameters. Firstly, based on C-Bézier basis with parameters, we study the constraints of the control points of the curves needed to be satisfied when connecting them. Moreover, we study the continuity conditions between two adjacent C-Bézier surfaces with parameters. By the continuity conditions and different shape parameters, the curve and surface can be changed easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve and surface still preserve its characteristics and geometrical configuration. Some graphical examples ensure that the proposed method greatly improves the ability to design complex curves and surfaces and easy to implement.

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric B&eacute, zier (GDCT-B&eacute, zier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-B&eacute, zier surfaces are made by means of GDCT-B&eacute, zier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-B&eacute, zier surfaces are designed by maintaining the classic B&eacute, zier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-B&eacute, zier surface. The conditions appropriate and suitable for G1, Farin&ndash, Boehm G2, and G2 Beta continuity in two adjacent GDCT-B&eacute, zier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

With the aim to tackle the problem that the complex composite curves can not be constructed by using a single curve, the geometric continuity conditions for H-Bezier curves of degree n with shape parameter are investigated. The H-Bezier curves of degree n not only inherit the major properties of the classical Bezier curve of degree n, but also have a prominent performance on adjusting their shapes by changing shape parameter. Based on the analysis of the basis functions and terminal properties, the necessary and sufficient conditions of G1 and G2 geometric continuity between two adjacent H-Bezier curves of degree n are proposed. Modeling examples provided show that the derived continuity conditions are effective and hence can greatly enhances problem-solving abilities in complex curves design by using H-Bezier curves.

We consider the weighted-multi-degree reduction of Bezier curves. Based on the fact that exact degree reduction is not possible, therefore approximative process to reduce a given Bezier curve of high degree n to a Bezier curve of lower degree m, \(m

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{G}^{1}$\end{document}G1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{G}^{2}$\end{document}G2 between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design.

In this paper we present a novel algorithm for the multi-degree reduction of Bezier curves with geometric constraints. Based on the given constraints, we construct an objective function which is abstracted from the approximation error in L"2-norm. Two types of geometric constraints are tackled. With the constraints of G^2-continuity at one endpoint and G^1-continuity (or C^r-continuity) at the other endpoint, we derive the optimal degree-reduced curves in explicit form. With the constraints of G^2-continuity at two endpoints, the problem of degree reduction is equivalent to minimizing a bivariate polynomial function of degree 4. Compared with the traditional methods, we derive the optimal degree-reduced curves more effectively. Finally, evaluation results demonstrate the effectiveness of our method.

A class of @[email protected]@c-Bernstein-Bezier basis functions over triangular domain, which include the cubic Ball basis functions over triangular domain and the cubic Bernstein-Bezier basis functions over triangular domain, is constructed. Based on these new basis functions, a kind of triangular Bernstein-Bezier-type patch with three exponential shape parameters is proposed. The shapes of the triangular Bernstein-Bezier-type patch can be modified intuitively and foreseeable by changing the values of the three exponential shape parameters under the same control net. The conditions for G^1 continuous smooth joining two triangular Bernstein-Bezier-type patches are given.

In this paper, linear methods to find the multi-degree reduction of Bezier curves with G^1-, G^2-, and G^3-continuity at the end points of the curves are derived. This is a significant improvement over existing geometric continuity degree reduction methods. The general equations for G^2- and G^3-multi-degree reduction schemes are non-linear; we were able to simplify these non-linear equations to linear ones by requiring C^1-continuity. Our linear solution is given in an explicit, non-iterative form, and thus has lower computational costs than existing methods which were either non-linear or iterative. Further, there are no other existing G^3-methods for multi-degree reduction. We give some examples and figures to demonstrate the efficiency, simplicity, and stability of our methods.

An accurate and efficient clothoid approximation approach is presented in this paper using Bézier curves based on the minimization of curvature profile difference. Compared with existing methods, the proposed approach is able to guarantee higher order geometric continuity with smaller approximation error in terms of position, orientation, and curvature. The approximation scheme takes place in three stages. First, a subset of clothoids with specific winding angle constraints referred to as elementary clothoids is approximated using quintic Bézier curves. Then, a basic clothoid defined in the first quadrant is formulated, which is composed of a series of transformed elementary clothoids. An adaptive sampling stra-tegy is applied to ensure that the resulting Bézier segments are computed within a specified accuracy and all the required information can be obtained offline and stored in a lookup table. Finally, a general clothoid with arbitrary parameters can be conveniently approximated based on the lookup table through appropriate geometric transformations. A comparison with the recent circular interpolation and rational Bézier curve based approximation shows that the proposed approach is able to achieve equivalent or greater computational efficiency in most scenarios. Accepted version

This paper aims to simplify the continuity conditions of Bezier curves. For this purpose, a special family of Bezier curves with three parameters, to be called adjustable Bezier curves, is constructed. They have the same structure as the quartic Bezier curves. The newly constructed curves possess some of the basic properties of Bezier curves, such as the convex hull property, symmetry, geometric invariance, etc., and they have shape adjustability. Moreover, under the geometric continuity of order 1 ( G 1 ) conditions of the usual Bezier curves, the adjustable Bezier curves can reach geometric continuity of order k ( G k ); here, k is one of the parameters of the newly constructed curves. The recursive evaluation algorithm of the new curves is provided. We also discuss how to construct the adjustable Bezier curves with a given tangent polygon. Numerical examples illustrate the correctness and validity of the proposed method.

Segments of two given circular arcs can be blended to produce a segment of a new curve. The new curve that been produced which also known as blending curve is form in a C-shape. That’s mean the two circular arcs are blend at the same endpoints. Bezier Curve refer to [1] is the main application in this construction of blending curve. As the two circular arcs are create using the Rational Bezier Curve for the shape refer to [2]. First degree of Bezier Curve is use in blending function along with functionH(t). Blending can provide a smooth transition from one curve to another and can give various degrees of smoothness at the endpoints of the blend, where the smoothness is measured analogously to parametric continuity, Cn and geometric continuity, Gn. The accuracy of the approximation to a best blending curve obtained by different blending formulas is compared via analysis. Two types of blending formula introduced, which are Blend A and B. Blend A which involve only parametric continuity, C0, C1 and C2 Blend ...

In this paper, we study the C^1G^2 continuity of surfaces by a shape-blending process. Furthermore, we study the continuity of the ruled surfaces constructed by linear interpolation between two pairs of C^1G^2 continuous curves. We give some conditions for the C^1G^2 continuity of composite surfaces in a shape-blending process. A practical approach is proposed to maintain the C^1G^2 continuity of Bezier surfaces pairs in a shape-blending process by adjusting the control points along the common boundary of the resulting surface-pair. We finish by proving and justifying the efficiency of the approaching method with some graphical examples.

Necessary and sufficient conditions for geometric continuity of arbitrary order between surface patches are derived and discussed. An analogy between the continuity conditions for curves and for patches is used to interpret these conditions. The paper presents a procedure of finding the order of continuity between two polynomial or rational patches, and describes two possible approaches to constructing patches joined smoothly via constructing their cross derivatives. One of these approaches leads to a costly numerical procedure based on the necessary and sufficient conditions, the other uses explicit formulae corresponding to stronger sufficient conditions. The results presented here are based on the algebraic results obtained in (Kiciak, 1995).

In this paper, the necessary and sufficient conditions for two adjacent Bezier surface patches of arbitrary degrees joining G^n-continuously along a common boundary curve are presented. These conditions are represented directly by the two surface patches, or via an intermediate surface patch. Configurations of the coefficient functions attached to the adjacent Bezier patches are obtained. Based on these results, an algorithm to compute the order of continuity between two given Bezier patches along a common boundary curve is presented. The G^n compatibility conditions which involve the coefficient functions and the control points of the two adjacent Bezier patches are developed. Examples for the construction of G^1- and G^2-continuous Bezier patches are given to illustrate the conditions developed.

Geometric continuity is a central topic of Computer Aided Geometric Design. In this paper we study the geometric continuity of arbitrary order between tensor product Bezier surfaces of arbitrary degree.

Rational Bezier curves are discussed from a projective-geometrical point of view. A projectively invariant Bezier representation of rational curves is presented. Geometric continuity of second and third order is characterized by special projective maps. These maps (certain perspective collineations) preserve curvature properties of a curve at a point. As an application, constructions for geometrically continuous joints of rational curves are derived.

The conversion matrix between Beta-spline basis functions and Bezier representation is analyzed in this paper. Based on it, calculation of arbitrary degree Beta-spline basis functions can be translated into a solution of linear system of equations. The construction of Beta-spline is simplified greatly, which made it more suitable for implement in pervasive computing.

Geometric continuity is a relatively new research area, and the concept causes considerable confusion for those not familiar with it. We give a brief summary of the different types of geometric continuity, and discuss three different, but equivalent, approaches for constructing Bezier curves. The continuity of composite Bezier curves are investigated, and we show that certain restrictions on the control points of the curves will ensure that they connect with a specific type and degree of continuity. We then use these results to construct C2 and G2 piecewise-cubic curves in three dimensions, since such curves are frequently used in computer aided design.

The problems of geometric continuity for rational Bezier surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bezier surfaces are given. Meanwhile, a geometric condition for smoothness between adjacent rational Bezier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.

We give a simple geometric construction for the Bezier points of two rational curves which join together with appropriate Frenet frame continuity. This is then used to give a geometric construction, from an arbitrary sequence of control points, for the Bezier points of a sequence of rational curves of degree n which join together with Frenet frame continuity of order m , for any m ⩽ n −1.

The joining vertexes between three or several adjacent surfaces are generally required to be polished smoothly to express a whole surface in multi-surfaces geometric modeling. This paper proposed a new method of polishing the vertexes with the rational triangular Bezier surface in G1 continuous conditions The method of polishing triangular Bezier surface are conducted from the share common plane judgment condition which is based on the general necessary and sufficient Gk continuous connection condition in the geometric continuity theory. By the share common plane condition, the second layer control points of triangular Bezier are calculated with a simple algorithm to keep the generated polishing triangular Bezier surface connected with adjacent surfaces in G1 continuity. The numerical experimental result shows that the method is reliable and meets the requirements of vertexes polishing in surface modeling and processing. The method can be directly applied to the higher order geometric continuous connection in polishing the joining vertexes.

The high geometric continuous blending surface blending surface reconstruction is very important to various CAD/CAM application systems. This paper proposed a new method to reconstruct G4 continuous blending surface with Bezier patches. The method and algorithm is based on the Gn necessary and sufficient conditions in the geometric continuity theory. The free shape parameters in surface blending are introduced to control the shape of the surface. By modifying these shape parameters, the shape of the blending surface can be modified interactively. The numerical experiments have demonstrated the method have perfected reconstructed G4 continuous blending surface. The method and algorithm can be directly popularized to the higher order geometric continuity for surface blending. It can be used in various CAD/CAM application systems.

The surface rendering for simultaneously connecting with several neighboring surfaces with high order geometric continuous is quite complicated in the computer graphics and CAGD. This paper proposed a new simple geometric algorithm for generating and rendering the smooth connecting surfaces for the vertexes and edges of the polyhedron. The rectangular Bezier surfaces are generated to connect the two adjacent planes, and the triangular Bezier surfaces are generated to connect its several neighboring surfaces with high order geometric continuity. The control net points of the connecting surfaces are calculated by the geometric drawing method according to the geometric continuity connection condition or the share common plane condition. All the surfaces and planes can be connected smoothly with G2 geometry continuity in the polyhedron. The algorithm can be popularized to the situation of higher order geometric continuous connection easily.

This chapter discusses geometric continuity for surfaces. The two patches with a common boundary curve are called G 1 if they have a continuously varying tangent plane along that boundary curve. The concept of G 1 continuity is a genuine generalization of C 1 continuity. One important aspect of G 1 continuity is that, it is completely independent of the domains of the two involved surface patches. For C 1 continuity, the interplay between range and domain geometry was crucial, however at present, the domains are only needed so that one can evaluate each patch. An approach to geometric continuity is developed that solves several applications oriented problems. For triangular patches, a (sufficient) condition is constructed for Bezier patches.

In this paper, the necessary and sufficient conditions and an algorithm for G4 continuity between adjacent Bezier patches are presented. The effects of shape parameters for surface connection are investigated in details. The algorithm can be generalized directly to the case of higher order geometric continuity for surface connecting. It has important applications in surface modeling, surface blending and surface connecting.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Parametric piecewise cubic polynomials are used throughout the computer graphics industry to represent geometric curved shapes. The exploration of the use of parametric curves and surfaces can be viewed as the birth of Computer Aided Geometric Design (CAG http://archive.org/details/leastsquaresppro1094539690 Lieutenant, United States Navy Approved for public release; distribution is unlimited.

Trimming of surfaces and volumes, curve and surface modeling via Bezier’s idea of destortion, segmentation, reparametrization, geometric continuity are examples of applications of functional composition. This paper shows how to compose polynomial and rational tensor product Bezier representations. The problem of composing Bezier splines and B-spline representations will also be addressed in this paper.

Publisher Summary This chapter describes the geometrically continuous cubic Bezier curves. Traditionally, curves and surfaces in computer graphics and computer-aided design have been required to be parametrically Ck-continuous. In particular, the requirement for an overall C1-curve completely determines the Bezier curve G that bridges the gap between F and H. So far, it seems that the preceding two equations provide an easy solution to an easy problem. The problem with our solution G stems from the fact that parametric continuity has been insisted instead of geometric continuity and instead of looking at the continuity of a specific parameterization, one needs to look at the continuity of the shape of the curve, which is independent under possible reparameterization. Two curves join with geometric continuity if the curves are parametrically continuous under reparameterization. It is found that it is easy to come up with a simple satisfactory solution to our original problem of filling the gap between two given cubic Bezier curves F and H by another cubic Bezier curve.

Conversion methods are required for the exchange of data. We introduce conversion methods for B-spline and Bezier surfaces. First a given integral or rational B-spline surface will be segmented by curvature oriented arguments then these patches will be converted into integral Bezier patches with help of geometric continuity conditions. If the surfaces are trimmed, the given curves on the surfaces may have rational B-spline representation of arbitrary order in the parametric domain, these curves will be converted into integral B-spline curves of arbitrary order. — The same method can be used for merging a set of surface patches to a set with less number of patches.

This paper gives a sufficient condition on the control polygon to avoid loss of geometric continuity on a spline curve of any order whose Bezier polygon can be gained by successively ‘cutting corners’ from its control polygon.

A new classification of geometric continuity is presented. It is based on the simultaneous application of the concept of ‘contact of order r” to curves (or surfaces) and to their tangent surfaces and generalized tangent surfaces; this implies projective invariance. The developments are illustrated by algorithms for geometrically continuous Bezier curves. Moreover, we obtain new properties of certain splines derived by variational formulations.

The necessary and sufficient conditions of G1 continuity between two biquartic B-spline patches with single interior knots are obtained, and the intrinsic conditions of the common boundary control points are also presented. Further, some remarkable difference on the conditions for G1 continuity of two adjacent Bézier patches and those of two adjacent B-spline patches are studied. As a result, the construction of local scheme by using biquartic B-spline surfaces with the single interior knots does not exist.

For the circular arc of angle 0 α π we present the explicit form of the best GC 3 quartic approximation and the best GC 2 quartic approximations of various types, and give the explicit form of the Hausdorff distance between the circular arc and the approximate Bezier curves for each case. We also show the existence of the GC 4 quintic approximations to the arc, and find the explicit form of the best GC 3 quintic approximation in certain constraints and their distances from the arc. All approximations we construct in this paper have the optimal order of approximation, twice of the degree of approximate Bezier curves.

This document presents a B-spline-Bezier representation of tau splines, curvature and torsion continuous quintics which have been introduced in CAGD by Hagen in 1985. Explicit formulas are given for the conversion of the B-spline-Bezier representation to the tau spline representation and vice versa, and conditions and certain ranges of tension values are derived which insure the positivity of the design parameters. Keywords: Geometric continuity, Computer aided geometric design.

The Bezier formulation for parametric curves has many qualities, among them the intuitive relationship between the shape of the control polygon and the shape of the curve, and the ease of computation and subdivision. Other formulations, however, have become more popular because they offer local control, or because they are interpolating, or even more recently because they provide the added flexibility of shape parameters.

A simple methods is given for constructing the Bezier points of curvature continuous cubic spline curves and surfaces from their B-spline control points. The method is similar to the well-known construction of Bezier points of C^2 splines from their B-spline control points. The new construction allows the use of all results of the powerful Bernstein-Bezier technique in the realm of geometric splines. A simple introduction to nu- and beta-splines is also derived, as well as some simple geometric properties of beta-splines.