Interpolation Algorithm Research Of Fair Triangular Gregory Patches

Surface modeling is one of the core techniques in the CAD/CAM. Its target is to design various complex surfaces which satisfy the industrial demand rapidly and flexibly. At present, surface modeling is still dominated by rectangular patches, but we always need to interpolate out smooth surface from irregular grid in engineering application. Under the construction of complicated surfaces using traditional quadrilateral meshes, lots of quadrilateral patches degenerate into triangular patches, which affect the smoothness of surface greatly. Therefore, there are two trends in the research of surface modeling. One is using rational splines, and the other is using triangular patches during the process of surface construction. The reason is that the polynomial surfaces and conical surfaces can be represented by the tensor product of rational spline surfaces uniformly and the triangular patches are more suitable for irregular meshes.In engineering applications, it’s difficult or even impossible to construct a complicated surface with only one patch. We usually construct a surface piecewise. First, we divide the complex surfaces into several patches and then we interpolate each one. Therefore, in order to obtain a smooth surface, the research on the smoothness between patches is a crucial issue. This subject takes the quartic Gregory patch as the object to study the smooth combination between them.The normal curvature of corner points is singular, so the second derivatives can’t be decided uniquely there. Once the 1 continuity is established, the normal curvature of corner points is decided subsequently without removal of the singularity. The solution to this problem is that establishing the 1 continuity under the constraint of normal curvature of corner points.We modify the boundary curves to reduce the error of normal curvature. During this process, we take the geometry information of two endpoints into consideration to establish constraint conditions. We calculate the control points based on 1 continuity, and then we can obtain the parametric form of controlling point. Taking the elimination of singularity into consideration, the smoothness is greatly improved especially in the corner.