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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.

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.

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.

Quad meshes can be interpreted as tensor-product spline control meshes as long as they form a regular grid, locally. We present a new option for complementing bi-3 splines by bi-4 splines near irregularities in the mesh layout, where less or more than four quadrilaterals join. These new generalized surface and IGA (isogeometric?analysis) elements have as their degrees of freedom the vertices of the irregular quad mesh. From a geometric design point of view, the new construction distinguishes itself from earlier work by a notably better distribution of highlight lines. From the IGA point of view, increased smoothness and reproduction at the irregular point?yield fast convergence. Bi-3 tensor-product splines are complemented by bi-4 splines near irregular points.The vertices of the irregular quad mesh serve as spline-like control points.The resulting surfaces have a good distribution of highlight lines.The resulting surfaces have a increased smoothness and reproduction at irregular points.

We explore a class of polynomial tensor-product spline surfaces on 3-6 polyhedra, whose vertices have valence n=3 or n=6. This restriction makes it possible to exclusively use rational linear transition maps between the pieces: transitions between the bi-cubic tensor-product spline pieces are either C1 or they are G1 (tangent continuous) based on one single rational linear reparameterization. The simplicity of the transition functions yields simple formulas for a hierarchy of splines on subdivided 3-6 polyhedra. Graphical abstractDisplay Omitted HighlightsWe explore a class of polynomial tensor-product spline surfaces on 3-6 polyhedra.A 3-6 polyhedron has quadrilateral facets with vertices of valence either 3 or 6.This restriction allows exclusively rational linear transition maps between splines.Bi-cubic tensor-product splines are joined either C1 or G1.The simple transition supports a hierarchy of splines.

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.

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.

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.

We develop a class of rational, G^2-connected splines of degree 3 that allow modeling multiple basic shapes, such as segments of conics and circle arcs in particular, in one structure. @? This can be used, for example, to have portions of a control polygon exactly reproduce segments of the shapes while other portions blend between these primary shapes. We also show how to reparameterize the splines to obtain parametrically C^2 transitions.

Concepts and techniques for the construction of smooth surfaces over irregular meshes are developed and made concrete by defining curvature continuous splines based on 3-sided patches. These splines extend the B-spline paradigm for the construction of parametric piecewise polynomial surfaces to control meshes with nonquadrilateral cells and more or fewer than four cells meeting at a point. Mesh points serve as control points and are locally averaged to obtain a Bernstein-Bezier representation which in turn defines surface points as averages.

The promise of modeling by subdivision is to have simple rules that avoid cumbersome stitching-together of pieces. However, already in one variable, exactly reproducing a variety of basic shapes, such as conics and spirals, leads to non-stationary rules that are no longer as simple; and combining these pieces within the same curve by one set of rules is challenging. Moreover, basis functions, that allow reading off smoothness and computing curvature, are typically not available. Mimicking subdivision of splines with non-uniform knots allows us to combine the basic shapes. And to analyze non-uniform subdivision in general, the literature proposes interpolating the sequence of subdivision control points by circles. This defines a notion of discrete curvature for interpolatory subdivision. However, we show that this discrete curvature generically yields misleading information for non-interpolatory subdivision and typically does not converge, not even for non-uniform spline subdivision. Analyzing the causes yields three general approaches for solving or at least mitigating the problem: equalizing parameterizations, sampling subsequences and a new skip-interpolating subdivision approach.

We show that many fundamental algorithms and techniques for B-spline curves extend to geometrically continuous splines. The algorithms, which are all related to knot insertion, include recursive evaluation, differentiation, and change of basis. While the algorithms for geometrically continuous splines are not as computationally simple as those for B-spline curves, they share the same general structure. The techniques we investigate include knot insertion, dual functionals, and polar forms; these prove to be useful theoretical tools for studying geometrically continuous splines.

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.

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.

This brief report corrects errors in the triangular patch subdivision scheme described in [Shirman & Sequin '87]. Furthermore, an improvement of the quadrilateral subdivision scheme discussed in the same paper is presented.

We derive a natural extension of Boehm's free-form γ-spline, the G2 interpolating γ-spline. Primarily, the conditions under which singularities in the spline formulation occur are investigated. Also, the effect that these singularities have on the interpolant are studied. Comparisons are made to the behavior of the interpolating ν-spline.

A surface interpolation method with local control for meshes of cubic curves is described. This mesh is filled with Gregory patches, so that the surface is represented by a union of geometrically continous bicubic quadrilateral and/or quartic triangular Gregory patches. Along any boundarycurve the shape of the two adjoining patches can be locally controlled with shape parameters, suchas tilt, bulge and shear. The procedure is a simple extension of Chiyokura's method [Chiyokura & Kimura '83] that gives extra control to the user while preserving first order geometric continuity between the patches.

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.

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.

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.

This paper deals with the approximation of (regular) offset curves or offset surfaces (of given spline curves or spline sufaces of degree n) by spline curves or spline surfaces of arbitrarily chosen degree m. The approximating splines are determined by geometric continuity conditions and by parameter optimization for minimizing the range of the approximation error.

A theory of geometric continuity of arbitrary order is presented. Conditions of geometric continuity at a vertex where a number of patches meet are investigated. Geometric continuous patch complexes are introduced as the appropriate setting for the representation of surfaces in CAGD. Applications to the modelling of closed surfaces and the fitting or triangular patches into a geometric continuous patch complex are sketched.

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.

Connections are pointed out among various classes of parametric cubic splines. Formulas are given for representing a C1 piecewise cubic curve as a B-spline curve.

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.

Generalized C 2 conditions are used to define a class of cubic splines. A B spline-like design scheme is provide for these 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.

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. (KR) National Research Council and funded by the Naval Postgraduate School. http://archive.org/details/bsplinebezierrep00lass Direct Funding NA

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.