A Shape Preserving Rational Cubic Interpolation Spline

A smooth curve interpolation scheme for the visualization of scientific data has been developed in this thesis, which can preserve the inherited features of a shaped data. Two families of parameters, in the description of the piecewise rational cubic interpolate spline, have been constrained to preserve the shape of the data. The shape constraints are restricted on shape parameters to assure the shape preservation of the data set. The rational spline scheme has a unique representation. The degree of smoothness attained is C¹.This thesis is composed of five parts.In chapter 1, the author gives a concise depict of the history of interpolation and the current situation of shape preserving study. The most important problems restricting the rational spline's usage are deficiency of automation, additional restriction on data etc.In chapter 2, the author puts forward the definition of the piecewise rational function :here θ = (x - x_i)/h_i,u_i,v_i are shape parameters and u_i,v_i > 0, which can change the shape of curve. In most applications, the derivative parameters d_i are not given and hence must be determined from the data. So two choices are mentioned in the second section. In the third section we get u_i = v_i → ∞, S_i(x) → f_i(1 - θ) + f_i+1θ, so the effectiveness for the shape control can be seen here and that is the theoretical foundation of shape-preserving analysis.The shape preservation of the interpolation spline is stressed in chapter 3. In additional to discuss the shape-preserving of monotonic data, convex data and positive data, the problems to constrain the rational interpolation curves to lie strictlyabove or below a given piecewise linear curve and between two given piecewise linear curves are solved. When asked curve to keep monotone, S'(x) > (<)0 requires shape parameters u, > Jï¿¡,Vi > ^. In the same way, S"(x) > 0(< 0) requires shape parameters u,- > ,,fr^,,iii > f/VZ^- S(x) > 0 requires shape parameters ut > max(0, -!h^Ii),vi > max(0, hidifi^<+1 ? After getting these results, we enhance restriction on U{,Vi that keeps the curve between two given piecewise linear curves. Thus we get it that putting the shape-preserving curve between two given piecewise linear curves.The approximation properties of the rational cubic spline are studied in chapter 4. Firstly, with Peano-Kernel theory we get the approximation properties of a rational spline based on function values only. With another method, an O(h2) convergence result is obtained for convex preserving spline and an O(h?) convergence result is obtained for monotonic preserving spline.The four chapters above are concluded in chapter 5. Five sets of random data and three groups of scientific data are used to demonstrate the results, and all shape-preserving curves are automatically generated.In the thesis we can get shape-preserving curve by the piecewise rational spline with cubic numerator and quadratic denominator automatically. The approximation properties of the rational spline whose derivative parameters d* weru replaced with arithmetic average difference quotient have been obtained.