Research On Fitting,Parameterization And Shape Optimization Problems Of Curve Modeling

Computer Aided Design and Computer Aided Manufacture technology originated in the aviation industry, since the aircraft shape is very complex, which possesses a large number of freeform surface, CAD/CAM technology closely linked with the freeform curve and surface modeling technology from the outset. So far, with the development of Computer Graphics and Computer Aided Design and Manufacture, the application of curve and surface modeling technology in the modern industry product design and manufacture becomes more and more widely, and has increasingly been adopted by all sectors of society, these fields include aerospace technology, digital city, media design, medical visualization and industrial product design, and so on, which all generate huge economic benefits. In the curve modeling technology, curve fitting is a basic work and plays an important role in the fields of pattern recognition, computer vision, reverse engineering, computer graphics, statistics and computer aided geometric design. Next, in order to construct the curve with well shape, the curve parameterizaiton is also a key problem, the selection of a suitable curve parameterizaiton method can not only bring convenience to other operations, but also can get satisfactory curve shape. Besides, the selection and construction of the shape optimizaiton model also affects the curve modeling effect directly. In recent years, there are considerable of research achievements for curve modeling technology, however, there are some difficult problems to be resolved, such as how to fitting the curve for the plane scattered data points with high precision, and so on. Currently, how to construct a new shape optimization model to ensure the curve has satisfactory shape is also a hot problem, and there is no efficient method for this to be solved problem.Based on above problems, we will discuss three core problems in this article:1) Research of the conic fitting method for plane scattered data points;2) Construction of cubic rational curve satisfy the endpoint conatraint and chord-lengh parameterization;3) Discussion on the relationship between curve shape and optimization modeling. New theories and algorithms are presented in this article, the detail research works include:1. Conic fitting method for plane scattered data points For the conic fitting problem of plane scattered data points, we chose the minimal algebraic distance between the data points and the fitting curve as the objective function, and presented two fitting method based on the coordinate transformation and coefficents weighted combination. In the fitting method based on coordinate transformation, considered the fact that if using the geometric variant as the constraint, different solutions will be obtained in different coordinate systems. So we studied the problem that how to get the new coordinat system by tanslating and rotating the original coordinate system, then fitted the conic, and shifted the results back to the original coordinat system at last, which ensured the conic fitting effect better. In the weighted combination fitting method, the specific ellipse, hyperbola and parabola were fitted to the data points recpectively at first, then the final fitting conic was produced by combining the above three specific conics and adding certain weights to the coefficients of them, by this method, the fitting conic not only preserved the original curve shapes if the conic data comes from the basic quadratic curve, but also improved the fitting effects for the general data points. In addition, for the high curvature bias problem of ellipse fitting, we also presented a correction method.2. Construction of cubic rational curves satisfy the endpoint constraint and chord-length parameterizationArc-length parameterization is the best choise for the curve modeling, but for the nonlinear polynomial curves or rational curves, we can't adopt the arc-length parameterization. Therefore, the term "chord-length" is commonly used for simulating it in discrete point data interpolation and approximation. The existing research has given the contruction method of cubic rational Bezier curve satisfy the chord-length parameterization. Based on this research, by analyzing the three characteristics of the constructed curve, we presented the construction method of cubic rational Bezier curve satisfy the endpoint constraint and chord-length parameterization. For the entire tangent angle area (α0,α1)∈[-π-,π]×[-π,π], the method adopted the piecewise curve construction idea, and proved the curve segments is3at most, the curve is G1continuous, and in some case is C1and G2continuous.3. Relationship beween curve shape and optimization modelFor the curve modeling problem, minimal strain energy or minimal curvature variation are widely used as the shape optimization models. In order to discuss the relationship between the curve shape and optimization models, we took the cubic Hermite curve and cubic spline curve as the research objective respectively. For the construction problem of cubic Hermite curve with given endpoint positions and tangent directions, we took the minimal curve strain energy and minimal curvature variation as the objective function, and used the numerical computation technology to solve the tangent lengths of the cubic Hermite curve. The experiment results indicated that, for the tangent angle area (θ,φ)∈[-π,π]×[-π,π],(1) most of cubic Hermite curves can't be constructed by solely minimizing the strain energy or curvature variation;(2) by using a local minimum value of the stain energy or curvature variation, the shapes of cubic Hermite curves could be determined for about60percens fo all case. However, lots of the cubic Hermite curves with the local minimum strain energy or curvature variation have unsatisfactory shapes. The construction of cubic spline curves by minimizing the curve strain energy or curvature variation also indicated that, in many cases, the curve have unsatisfactory shapes.As the classic shape optimization models, however, we can't construct the curve with satisfactory shapes by minimizing curve strain energy or curvature variation in many cases. Therefore, it's urgent need us to construct a new shape optimization model, which can satisfy the curve modeling problem with different interplation situations.