Paul Fractal Interpolation Of Theoretical Research And Applications

Shape preserving interpolation of parametric curves in computational geometry has always been an important topic. To study the problems of shape preserving interpolation, the application of trigonometric polynomial has many advantages,such as good continuity property, high approximation and so on. In this dissertation, we construct quartic trigonometric polynomial basis function firstly. Basing on it, we construct quartic trigonometric shape preserving interpolation parametric spline curves and quartic trigonometric shape preserving GHI curves.The main results in this dissertation are outlined as follows:(l)We construct quartic trigonometric polynomial basis function over the trigonometry function spaceΦ7=span{1,sint,cost,cos2t,sin3t,oos3t,sin4t,cos4t}, which possess characteristics similar to quintic Bernstein basis function.The explicit expressions of quartic trigonometric Bezier curves are presented,which has characteristics similar to quintic Bezier curves,such as properties of end point,symmetry,convex hull property,geometrical invariability,affine invariance,variation diminishing properties and convexity preserving property. Application graphics illustrate that quartic trigonometric Bezier curves posses better approximation property and shape preserving property.(2) Based on quartic trigonometric polynomial basis function,with geometrical information of four adjacency control points,quartic trigonometric control points are directly generated by simple computation.Thereby,quartic trigonometric shape preserving interpolation parametric spline curves are constructed.There is no need to solve the system of equations,and the obtained quartic trigonometric shape preserving interpolation parametric spline curves possess GC2-continiuous.Not altering the control points, the shape of the curves can be locally adjusted by Shape parameter. Application graphics illustrate that in shape design,quartic trigonometric shape preserving interpolation parametric spline curves can satisfy users'shape preserving demands on curves interpolation conveniently and rapidly.(3) Basing on quartic trigonometric polynomial basis function,we construct quartic trigonometric shape preserving GHI curves,which interpolate the given control points and the corresponding curvatures.All of its quartic trigonometric control points are directly generated by simple computation with the given control points and the corresponding curvatures.There is no need to solve the system of equations.The calculation is simple and the obtained quartic trigonometric shape preserving GHI curves possess GC2-continiuous.Not altering the control points and the corresponding curvatures,the shape of the curves can be locally adjusted by Shape parameter.Algorithms flow chart is given.Application graphics illustrate the validity and practicality of the quartic trigonometric shape preserving GHI curves.