Surface Modeling Of Scattered Data Interpolation Surface Problem

The work studies the problem of interpolation to scatter data points in surface modeling. The technique of constructing interpolant to scatter data points is used widely in many fields of scientific research and engineering design, such as CAD, the computer graphics, the meteorology and the exploration and so on.As irregularity of surface in project application, and as randomness and disorder of scatter data points, it's difficult to express the surface with a sole mathematical formalism. Therefore the piecewise method is used generally in the surface design, finally all the surface pieces are connected smoothly to form a whole surface. The most commonly used surface patches are the triangle and the quadrangle surface patch. As the triangular interpolation method has obvious geometry significance, and the surface constructed is easy for the adjustment, more and more reseachers pay their attention to the method. At present one of commonly used methods of constructing interpolated surface can be described simply as follows: the given scattered data points are triangulated into triangle network, the surface patch over each triangle region is formed according to boundary condition of continuity, and each surface patch joins together to form overall interpolated surface with C~1 continuities. There are two methods for interpolation on triangle network, one is rational interpolation, another polynomial interpolation. Because of the merit of simple structure and easy calculating, polynomial interpolation is used more widely. Now there are many methods to construct polynomial surface to the scattered data points.Different interpolation methods often have the different shortcoming, and it can be classified simply as follow: The process of constructing is too complex and the interpolation condition is limited by many factors; The method for interpolation is feasible just at the condition of that the number of the given scattered data points is not too big; The triangle grid vertex must has certain characteristics to satisfy the need of constructing; Just forms an overall surface over the whole partial region of each data point, and the overall surface sometimes is difficult to possess the surface shape suggested by the given data points, or can't be adjusted easily; Cannot obtain the onlyinterpolated surface by the operation result and so on.In view of above question, this article presents a new method to construct interpolated surface to the scattered data points. New method triangulates the given data points into triangle network, and at the adjacent region of each point a C1 piecewise quadric interpolation patch is constructed. The surface patch on each triangle is constructed by the weighted combination of the three quadric patches at the vertices of the triangle. All the triangle patches are put together to form the whole surface with C1 continuities. Because the surface patch at the adjacent region of each point is C1 piecewise quadric interpolation patch, it has the property of keeping shape and can adjust the shape easily over the local region. In addition, the solution process of equations formed by boundary condition of continuity in different situation is analyzed, and the simple and convenience solution is given. The new method can construct smooth surface interpolating the given scattered data points effectively, and the polynomial precision set of the method includes all the polynomials of degree two.Considering different background of application, for constructing the fairing surface, a new method of interpolation based on the criterion of minimum energy is presented. The solution conditions used to construct partial surface is only obtains from this region, the unknown parameter is acquired according to the criterion of minimum energy, so it enable the constructed surface to have a more ideal partial adjustment. The new surface has the shape suggested by the primitive data points, and is more fairing over the region with bigger drape. At last examples are given to make comparison with previous method.But the above two methods finally constructs the polynomial interpolations of degree seven, which is higher for C1 interpolation with polynomial. In the following research we will make the improvement to the weight function by reduces its degree in order to obtain lower degree polynomial interpolation. The fairness of surface is an extremely important research question in CAGD, the fairness of surface can be improved based on criterions of the fairness in future research.