The Construction And Application Of The Binary Quadratic Splines On Uniform Rectangular Grids

Not only computational geometry is of great theoretical significant in the geometry, but it is important and valuable in the computer aided design and manufacturing, computer graphics, image processing and other related fields. Spline is the basis and the basic theory of computational geometry. In recent years, univariate spline have nearly perfect theoretical basis. Multivariate spline function theory and achievements of curved surface design theory is limited, relative to univariate spline function and the resulting theory of curve and surface design.Applications, however, are very rare, which very likely may be attributed to the complicated nature of the grid partitions and thus to the complexity of the programming. Since these splines have very attractive properties (local support, not-negativity, optimum global smoothness, high flexibility), it is more worth to study their applicability more closely. This paper tries to generate C1surface on rectangular grid by the theory of binary quadratic Spline. This article is consisted of the following contents:(1) The first part mainly introduces the basic concepts and basic problems involved of curves and surfaces as the mainly research objects in geometry.(2) The second part introduces univariate spline, and the curve and surface design theory of the computational geometry, such as the Bezier curve and surface, the B-spline curve and surface, rational Bezier curve and surface. NURBS method.(3) The third part introduces polyhedron splines and the definition and geometrical interpretation of the multivariate simplex spline, and covers recurrence relations of the binary quadratic spline.(4) The fourth part discusses computing process of binary quadratic spline on rect-angle region.(5) In the fifth part, I design a piecewise C1rational surface based on the quadratic spline, and study their properties. (6) Finally, I give the image of the quadratic surface, which is designed by binary quadratic splines. Then, the errors are analyzed and compared.