Infinate Interpolation On Tirangles

The work studies the problem of interpolation to scatter data points in surface modeling. The technique of constructing interpolate 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.Constructing triangular patches through boundary interpolation was first introduced by Barnhill, Birkhoff, Gordon ,which construct triangular patches using Boolean sum operator . Gregory constructed triangular patches through convex combination, which was further developed in paper The triangular patches they constructed were composed of the convex combination of three interpolating operators ,each of which satisfy the interpolation conditions on both edges of the triangle. Nielson proposed the edge-point method which also be fulfilled through three interpolation operators, each of which satisfies the interpolation condition of one point and the opposite edge. Hegen further developed the edge-point method, and used it to construct geometric triangular patches. In paper the author proposed interpolation through scattered data, which also be used in triangular patches construction. Recently in paper which propose construction triangular patches through four interpolation operators, including an inner interpolation operator and three edge-point operators. In paper they proposed the method of basic approach operator and additional operator for constructing triangular patches. Nielson proposed nine-parameter constructing patches which evaluate the unit tangent vector using the given vertex normal and function value, then compute the boundary curve and boundary unit normal vector using the edge-point method, at last constructing the triangular patches using edge-point method.There are many methods to construct polynomial interpolate on triangles ,which have different shortcoming. For example,Nielson proposed constructing surface patch with night- parameter ,which computes unit vector of the given vertex using the normal and function value ,then gets boundary curve and unit normal using side-vertex method ,finally constructs triangular surface patch using side-vertex method ,but there exist problems such that the request of parameter is too much ,and the error caused by the unconfirmed of α and β.Aiming at the above problems, A new method for constructing surface patch on triangles with six-parameter is presented in this paper. The new method defines a patch of quadric polynomial on two triangles which have a common line boundary .With the given function and outward surface normal of four vertices of the two triangles, we can get the boundary conic and the cross-boundary slope, in the same way, we can get the others. So we get the interpolation conditions on the boundary of the triangle, then we can construct the triangular patches by side-vertex method. For the boundary only belonging to one triangle, we do it specially.Comparing with exited methods, the advantage of new method is: constructing surface patch with six-parameter requests less parameter and avoids the error causedby the unconfirmed of α and β ;this method is simple to use, and avoids highcompute complexity and great error of special instance at a certain extent.Through experiment, we show the results in the end of this paper, and compared with the results of Nielson's.