Research On Infinite Interpolation Surface On Triangle

In the field of CAGD (computer aided geometry design) and CG (computer graphics), how to construct surface of high precision and smoothness is an important problem.Because the surface for engineering industry is irregular and the scattered data does not have regularity and order, it is hard for us to express surface using only one mathematic expression. Hence, designing the surface piecewise is usually adopted. Finally, the patches will be combined to form a whole surface. At present, quadrangle patches and triangular patches are the ones used widely. Because triangular interpolation method has obvious geometric meaning and is easy to adjust, it becomes an important method of constructing surface. One of the commonest methods of constructing interpolation surface is to triangulate the given scatter data points, and then construct triangular patch on each triangular domain, and finally combine all the patches to build the whole surface. The interpolation methods on triangle include rational interpolation method and polynomial interpolation method. The polynomial interpolation method is used widely in the applications.In the process of shape modeling, it is a common and important problem that a surface is constructed to connect other three surfaces smoothly. Because the three surfaces may be the ones of arbitrary form; hence, the surface which is used to connect the other three ones is constructed with the boundary of arbitrary form. The interpolation to arbitrary boundary conditions is called infinite interpolation. In order to get the smoothness of the whole surface, the adjacent patches should have the same function values and cross-boundary slopes on the boundary. So, constructing triangular patch which interpolates the given boundary curves and cross-boundary slopes on a triangle is the basic problem in computer aided geometry design and computer graphics.A new method of constructing C1 triangular patch of degree six by the Boolean sum of an approximation operator and an interpolation operator is presented in this paper. The triangular patch satisfies the given boundary curves and cross-boundary slopes. The approximation operator makes the triangular patch approximate the given boundary conditions, while the interpolation one makes the triangular patch interpolate the triangular patch using the vertex-side method. Since the precision of triangular patch is mainly dependent on the approximation operator, increasing the precision of the approximation operator is very crucial for building a triangular patch with better precision. The existing methods merely make use of the given boundary conditions directly, and their maximum error is usually generated in the interior of the triangle, especially at and near the center of the triangle. The new method constructs the approximation operator as a polynomial of degree six, which not only approximates the given interpolation conditions on the boundary with a better precision, but also has a better shape in the interior of the triangle, especially in the central area of big error.It is shown that the C1 triangular patch constructed using the new method has better precision and smoothness. In the construction of triangular patch, interpolation operator exerts a great affect on the quality of the triangular patch. So, how to increase the interpolation precision of the interpolation operator to build triangular patch of better precision and smoothness is the key point for the future research work.