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This paper introduces a modified method for construction of a new design-control 6-point approximating refinement scheme. The construction of the new scheme is based on translation of the points of 4-point approximating refinement scheme to the new position according to the linear combination of certain displacement vectors. The initial and terminal points of these vectors are the refinement points of two 4-point approximating binary schemes. The new scheme contains three design-control parameters. These parameters increase the efficiency and flexibility of the new scheme. The mathematical and graphical analysis of the refinement scheme show that this scheme is good choice for curve modeling. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

Physically based modeling of deformable curves and surfaces has been widely used in computer graphics to meet the needs of both geometry and physics. In this paper, the weighted sum of the first and higher order generalized thin-plate spline-surface energies is introduced for the construction of transition surface. We derive the necessary and sufficient condition that the energy functional of a transition surface reaches its minimum, by which the positions of movable control points can be fully determined within the constraints imposed by the controls and continuity constraints. Furthermore, based on the excellent properties of Bernstein basis functions, we present a simple algorithm to compute energy-minimizing transition surfaces if the given two disjoint tensor product surfaces are in Bézier form. The feasibility of the method is validated by several examples. Compared with previous modeling methods, various physical characteristics are imparted on the modeling of transition surfaces in this paper, thus various energy-minimizing transition surfaces with different characteristics can be obtained. [ABSTRACT FROM AUTHOR]

In the field of CAGD/CAD, interpolation curve plays an important role in shape representation and shape design. Based on the traditional quintic Hermite interpolation curve, this paper proposes a novel class of quintic generalized Hermite interpolation curve (QG-Hermite interpolation curve, for short) with local shape parameters, which can satisfy C2 continuity automatically. Since each segment of the constructed curve has independent shape parameters, and its shape can be adjusted locally or globally by modifying the shape parameters. First, to obtain an interpolation curve satisfying certain specific geometric requirements such as the approximate shortest arc length, the strain energy minimization or the curvature variation energy minimization, etc., three shape optimization models of the QG-Hermite interpolation curve are established. Second, an improved gray wolf optimization (I-GWO) algorithm is used to solve the shape optimization models so as to determine the optimal shape parameters. Finally, the modeling examples show that the proposed method is effective and applicable in shape designing. [ABSTRACT FROM AUTHOR]

The developable surface is always a hot issue in CAGD, CAD/CAM and used in many manufacturing planning operations, e.g., for ships, aircraft wing, automobiles and garments. In some special fields, the CAD model of developable surface is designed by interpolating a given spatial characteristic curve. In this paper, we present a class of methods to construct local controlled developable H-Bézier surfaces through a given characteristic curve. First, we introduce a class of generalized cubic H-Bézier basis functions, and utilize them to design the generalized cubic H-Bézier curves with shape parameters. Then we construct generalized cubic developable H-Bézier surfaces through a given space generalized cubic H-Bézier curve which serve as the line of curvature or geodesic. The shapes of the constructed surfaces can be adjusted and altered expediently using the shape parameters. Furthermore, the sufficient and necessary conditions for the interpolating developable H-Bézier surface to be a cylinder or a cone are deduced, respectively. Finally, we give some representative examples to illustrate the convenience and efficiency of the presented methods. [ABSTRACT FROM AUTHOR]

Ball curve plays a crucial role in modeling tubular shapes with variable thickness. In this paper, energy functionals for curve design are generalized to Ball curves and the variational design of Ball curves is investigated. Given a Ball curve with some of its control balls movable and having variable radiuses, we propose a method to determine the positions and radiuses of the movable control balls which minimize the energy functional. Based on this, we provide two efficient design tools: (i) to achieve C k continuity across linked Ball curves at their joint, while decreasing their energy as low as possible, (ii) for blending disjoint Ball curves subject to C k continuity constraints and energy minimization. The feasibility of the method is verified by several examples. By adjusting the weighted coefficients, different energy functionals are defined and thus Ball curves with different shapes and thickness can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel method for constructing developable surfaces using generalized C-Bézier bases with shape parameters. Based on the duality between points and planes in 3D projective space, the generalized developable C-Bézier surfaces, whose shape can be adjusted by changing multiple shape parameters, are designed using control planes with extensional C-Bézier 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 Bézier surfaces. Furthermore, some interesting properties of the new developable surfaces, as well as the geometric continuity conditions between two adjacent generalized developable C-Bézier surfaces, are investigated. Finally, we illustrate the convenience and efficiency of the proposed methods by several convictive and representative numerical examples. [ABSTRACT FROM AUTHOR]

Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of G1 and G2 continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design. [ABSTRACT FROM AUTHOR]

To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bézier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bézier rotation surfaces are constructed using a generalized Bézier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bézier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bézier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bézier rotation surfaces provide a valuable way for the design of rotation surfaces. [ABSTRACT FROM AUTHOR]

In this paper, two methods for approximation of offset curves of H-Bézier curves and H-B spline curves are presented, which are based on the approximation of shifting control points and norm of parametric speed. Firstly, after calculating the perturbed vectors, the offset curve can be obtained by shifting the control points of the base curve. Then, both the Chebyshev approximation and optimal trigonometric polynomials approximation of parametric speed of base curve are presented, and two approximation functions of offset curves are also obtained. Finally, some examples are used to demonstrate their practicality. [ABSTRACT FROM AUTHOR]

In order to tackle the problem of shape design and shape adjustment of complex surfaces in engineering, continuity conditions between generalized Bézier-like surfaces with multiple shape parameters are studied in this paper. Firstly, the geometric model of the generalized Bézier-like surfaces is built by blending a number of Bézier-like curves with independent shape parameters. Secondly, based on the terminal properties and linear independence of Bernstein-like basis functions, the conditions for G continuity between two adjacent generalized Bézier-like surfaces are derived, and then simplified by choosing appropriate shape parameters. Finally, some properties and applications of the smooth continuity between generalized Bézier-like surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which can greatly improve the ability to construct complex surfaces by using the generalized Bézier-like surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Refine-Smooth algorithm to design a family of a -point b -ary approximating subdivision schemes with bell-shaped mask, where a ≥ 3 and b ≥ 2. We use the combination of corner cutting b -ary subdivision scheme and weighted average of ( b + 1 ) -points to construct the proposed family. We demonstrate that the proposed family has smaller complexity and support width and higher continuity than the existing Refine-Smooth subdivision schemes. We also study the shape preserving properties of the proposed family. In addition, it is observed that the proposed family is suitable for fitting the locally noisy, oscillatory, and irregular data. [ABSTRACT FROM AUTHOR]

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 $\mathrm{G}^{1}$ and $\mathrm{G}^{2}$ 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. [ABSTRACT FROM AUTHOR]