Search

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 areas of engineering and architecture, smooth curves and surfaces are crucial to produce sleek products and to prevent the structures from breaking, especially when dealing with various shapes and lengths. The designing task of smooth curves can be carried out by following certain standards of fairness. Curvatures and internal energy are two of the metrics that are commonly used to define the fairness of shapes. In this study, the generalized fractional Bézier basis functions is used to construct a fair and smooth curve using different types of continuity. The generalized fractional Bézier curve possesses shape parameters that are used to control the flexibility of the curve. It also has a notable parameter called fractional parameter that enables designers to control the length of constructed curves. The variation of the fractional parameter is directly contributed to the stretch energy. A new type of continuity called the fractional continuity that connects curve segments at different lengths of the first curve will be very conducive in satisfying the geometric conditions as well as fairness conditions. Lastly, the influence of various fractional and shape parameters of generalized fractional Bézier curve to the internal energy is demonstrated. Fractional parameter and continuity will expedite designing tasks when dealing with conflicting requirements. [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]

Trigonometric B-spline curves have gained a remarkable attention in computer-aided geometric design (CAGD). This paper presents the cubic and rational cubic trigonometric B-spline curves using new trigonometric functions and shape parameter η ∈ (1 / 2 , 2 ]. The proposed curves inherit the basic properties of classical B-spline and have been proved. For uniform knots, both curves are C 2 continuous. On non-uniform knots, cubic trigonometric curves are C 3 and C 5 continuous, whereas rational trigonometric curves are C 3 continuous and have been derived. The applicability of proposed curves has been checked by constructing open and closed curves. Different models like glass, kettle, human hand, and vase have been designed by both schemes and compared. [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]

Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. In this article we employed B-spline curves to construct the occipital bone fracture using Digital Imaging and Communications in Medicine (DICOM) format data. The construction of occipital bone fracture starts with the boundary extraction followed by corner detection, construction of fractured part inner outer curve for each DICOM data using B-spline curves and finally the construction of fractured part in DICOM format. Method used in this article is based on DICOM data only and does not require any technique such as mirror imaging, technical help, reference skull, or to take average thickness of skull bone. Using the proposed method, the constructed fractured implant is custom made for every individual patient. At the end of this article we present a real case, in which we have constructed the occipital bone fracture using B-spline. The proposed method has been validated using post-operation DICOM data. For practical application, Graphical User Interface (GUI) has been developed. [ABSTRACT FROM AUTHOR]

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves. [ABSTRACT FROM AUTHOR]

The paper presents an investigation study of the mechanical properties of metal foam. In this investigation, a numerical simulation by finite element method is used. For this purpose, the foam structure is modelled by a regular network of Kelvin cells. In these cells, the strands are modelled as a three-dimensional (3D) finite element beam. In particular, we consider four types of strands cross sections: (1) circular (2) square (3) triangular and (4) Plateau shape. Our numerical results are in agreement with the experimental results obtained on a real Nickel metallic foam. In addition, we perform a parametric analysis to study the effect of some geometrical characteristics on the elasticity of a metal foam. Among these geometrical characteristics, we consider the shape and dimensions of strands cross section as well as the inertia variation. [ABSTRACT FROM AUTHOR]

The current study presents rational cubic trigonometric Bézier curve with two shape parameters, which is a novel technique for drawing free form curves. The proposed curve retains most of the geometric properties of conventional rational cubic Bézier but is flexible due to the presence of shape parameters. The shape of the curve can be adjusted locally by altering the values of shape parameters as well as the weights. Using the given end point curvatures, conditions on shape parameters and weights have been driven so that the curve always lies in the convex hull of its control point. Later, this curve is used to generate piecewise rational trigonometric curves stitched together using parametric and geometric Hermite continuity conditions which can facilitate a designer to generate a smooth composition of curves in a situation where single curve does not work. The piecewise curves thus constructed are used to draw the layout of certain English and Arabic alphabets as an application in font designing. [ABSTRACT FROM AUTHOR]