| CAD (Computer Aided Design, CAD) technology has been used in every field, covering product design and manufacturing process and effectively promoted the production of global cooperation. The level of its development and application of science and technology have become the measure of a country's modernization and one of the hallmarks of industrial modernization. Computer Aided Geometric Design (CAGD) is the research of the theoretical issues of CAD technology.Curves and surfaces design is of importance in CAGD and Computer Graphics (CG), the main research is on the design, display and analysis of curve and surface under the environment of the computer graphics system. It is the most active in CAGD and also one of the most critical areas of science. With the development and perfection of CAD/Computer Aided Made (CAM) technology of in-depth and continuous, the non-uniform rational B-Spline (NURBS) method has become an industry product-defined standard and widely used in industrial product design. It can solve the problem of the freedom curve and surface design.In curve and surface modeling, curve and surface construction method can provide the greatest possible flexibility is very important. Design methods to provide flexibility to facilitate the designer's design, to improve the design efficiency, make the design more designers model the desired shape. However, the widely used cubic spline method with no additional degrees of freedom, therefore, in curve and surface design process they do not have the flexibility, hence are not suitable for the requirements of designers. With the improvement of the precision measuring equipment, the technology of generate graphics quickly has become the basic technology in many practical applications in CG and geometric modeling (GM). We adopt the different algorithm, its accuracy, efficiency, reliability vary considerably. The results influence on the shape quality of curve and surface directly or even decisive. How to draw the graphics are more accurate, has a certain freedom, to facilitate local changes, and how to reduce the rendering time have become CAD subjects in front of the researchers. In addition to using graphics hardware itself, through the appropriate manner to reduce the geometry processing stage of the time-consuming, or improve the theoretical approaches and realizing algorithm are good methods. So the research on design of curve and surface in geometric design and calculation of minimum distance is very important. We can improve the accuracy, to improve computational efficiency, reduce the computational time and ultimately improve the rendering speed is of great significance to the practical application. Curve and surface in geometric design and design issuesof this paper discussed an innovative point on three aspects. (1) Give a method of construction of local adjustable C2 parametic quartic interpolation curve. (2) Give a method of construction of C2 continuous quartic spline curve and surface. (3) Give an algorithm of calculating the minimum distance between two NURBS objects.In many cases, the shape of cubic curves and surfaces constructed using spline function is not ideal. For example, for the representative data points of this article, the interpolation curves constructed by cubic spline function are not acceptable. Since the two shortcomings of quartic spline function, its important applications have been ignored. And it is generally agreed that the even spline curves and surfaces are not suitable for curve and surface interpolation problem. The quartic spline function has two shortcomings:(1) The continuity equation is not the corresponding tri-diagonal matrix. (2) The breakpoint of spline function is not in the positions of data points. The former costs a lot in constructing spline function and when there are many interpolation points, the calculating is unstable. While the latter makes the constructed spline curves and surfaces be inconvenient in usage. However, our results show that the existing four spline interpolation problem was mainly due to the method of the construction of curves and surfaces, rather than due to the quartic spline curves and surfaces themselves. The quartic spline interpolation does give better results sometimes. In practice, C2 continuous curves and surfaces can meet the requirements of most applications. There is a very few requirements to construct curves and surfaces that are continuous situations of C3. If the continuity of the quartic spline function reduced to C2, we can get rid of their two shortcomings, and provide additional degrees of freedom.In the numerical calculation and in the exterior design, these degrees of freedom are very useful. For scientific computing, degrees of freedom can be used to improve the interpolation accuracy of curves and surfaces. For the exterior design, degree of freedom can be used to increase the flexibility of the design and construction to control the shape of curves and surfaces. For example, through the minimization of energy and length to determine the degrees of freedom can make the curves and surfaces have a more rational shape. Therefore, quartic spline function can be more effective than cubic spline function during the construction of curves and surfaces.This paper discusses the problem of constructing local adjustable C2 quartic spline interpolation curve firstly. Spline function with the simple structure, the easily calculation and the best mechanical background, has been widely used and become one of the most important construction methods of curves and surfaces. In the application of spline function, because of the nature of a small model, the best approximation and strong convergence, cubic spline function has become the most important method and applied to the construction interpolation curves and surfaces. We control the shape of the curve by reducing quartic spline curve to C2 continuity which can provide a degree of freedom. In shape design, the freedom degrees can be used to increase the flexibility of design and construction to control the shapes of the curve and surface, and therefore make the more desirable shape of the curve and surface. Sometimes providing an additional degree of freedom for users is a burden. To solve this problem, a general method of calculating degrees of freedom is proposed. The method is presented for determining the freedom degrees with local method. The tangent vector at every data point is identified through the localized quadratic spline function at first. The tangent vectors and data points approximately determine the shape of the four spline curve. Then the freedom degrees are determined by minimizing the change rate of the spline curve. For the imperfect part of the spline curve, the corresponding tangent vectors are modified by the following way. A desirable moving vector is defined which makes the curve have better shape if it varies along the moving vector. An objective function is defined by the integral of the squared vector product of the moving vector and the tangent vector of the curve. The imperfect part of the curve is modified by minimizing the objective function. The comparisons of the new method with other methods and the examples of the local adjusting the curve by minimizing the vector product are also included. This method has the advantages of spline function, while is providing an extra degree of freedom, and can be used for improving the accuracy of interpolation the curve and adjusting its shape, which made the curve adjustable. Some parts of the curve constructed by the general approach may be not ideal, but these parts can be modified by adjusting the corresponding degree of freedom to make the curve have the shape suggested by the given data points.This paper also discusses the problem of construction of C2 continuous quartic spline interpolation surface. Reducing the continuity of quartic spline interpolation surface to C2 can provide an additional degree of freedom, which can improve the interpolation accuracy and control the shape of the surface. We discuss the continuity equation should be satisfied by C2 continuous quartic spline curve. We also propose a new method to construct C2 continuous quartic spline surfaces. The advantage of the new method is that the continuity equation satisfied by the surface must be tri-diagonal dominant and the surface discontinuity point is in a given point. The constructed surface has the precision of quartic polynomial interpolation. Finally we compare the existing cubic or quartic spline function interpolation method using a concrete example to show the interpolation accuracy of the new method.We know that the calculation of the minimum distance of geometries is a basic problem in geometric design. It has great significance in robot planning, computer simulation, and virtual reality applications. The usual method is to surround the curves and surfaces with simple geometric objects. The minimum distance between geometric objects is can be looked as the minimum distance between the curves and surfaces approximately. However, this method requires a lot of polygons detection. In some cases, for the researchers of computer graphics and computer-aided geometric design, the calculation of the minimum distance is not precise enough.There are many algorithms to calculate the minimum distance between two free-form curves and surfaces. Johnson & Cohen proposed a method to calculate the shortest distance between two complex surfaces. Liu et al. showed a algorithm to calculate the shortest distance between biquadratic NURBS Surfaces using the necessary and sufficient conditions of biquadratic Bezier surfaces which are non-negative. Turnbull & Cameron proposed a method to calculate the shortest distance between convex hull models defined by NURBS. Ma and others gave a robust algorithm of a quick calculation of the shortest distance between two NURBS curves. There are some other related methods.In this paper, we further study the problem of the calculation of the minimum distance between NURBS objects (curve-curve, curve-surface and surface-surface). And we propose a new algorithm to calculate the minimum distance between two NURBS objects. It firstly decomposes two NURBS curves and surfaces into two sets composed of their piecewise Bezier forms. A simple and fast approach is proposed to compute the bounding sphere of each set. Then we select candidate pair called Bezier pair which has the shortest distance for each set. We use bound balls and 'Four-points-condition'to improve calculation efficiency. We also use an iterative multidimensional Newton-Raphson method to calculate the minimum distance of all candidate pairs in order to find the global minimum distance. The new method is high-performance, accurate and robust and it can calculate the distance of two NURBS objects in real-time.NURBS curves and surfaces in the shape definition and design have great flexibility. They also have a unified, general and efficient standard algorithm and strong supporting technology. These advantages make them be widely used in industry. In CAGD, we often come across the problem of the calculation of the minimum distance between two NURBS objects. Our experimental results show that the algorithm has high stability, especially in dealing with complex NURBS curves and surfaces and it is easy in parallel implementation. The algorithm can be applied directly to all the geometric shape represented by Bezier or B-spline. It can also be applied to the further expansion of subdivision curves and surfaces, as subdivision curves and NURBS curves have much in common nature, such as convex hull. Therefore, this algorithm, which is a fast general-purpose algorithm, can solve the minimum distance between the two free-form curves and surfaces in a real-time. It is a kind of fast and general algorithm.In this paper we have made a research of the construction of local adjustable C2 parametic quartic interpolation curve, the construction of C2 continuous quartic spline surfaces and the calculation of the minimum distance between two NURBS objects. We have provided a series of new theories and methods for computer-aided geometric design, computer graphics and scientific computing and other areas to meet the needs of practical application.
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