Geometric Calculation Based On The Mixed Polynomial Interpolation And Approximation,

Curve and surface modeling is the core content of Computer Aided Geometry Design(CAGD). Bezier and B.spline method as the commonly used tools of free curve and surface modeling are still unable to express the conical curve accurately. This article constructed several interpolation curves and spline based on the mixed polynomial in order to address this deficiency of Bezier and B.spline method. In this article, the main job is as follows:Firstly, the thesis introduces the provenienee,development and application of Computer Aided Geometry Design(CAGD), and synthetically analyzes all kinds of methods of constructing interpolating free form curve and surface including the background, advantages and disadvantages. At the same time, the current use of non-polynomial solution to representing exactly circular, conic and other secondary Curve is described briefly.Secondly, the thesis constructs three types of mixed polynomial Hermite interpolational curves based on space that mixed by polynomial and trigonometric polynomial. The first and second category can achieve C1 continuity. The third category can achieve C2 continuity. The three mixed types of polynomial Hermite interpolational curves in space mixed by polynomial and hyperbolic polynomials are constructed similarly, and all of them can achieve C continuity. At the same time, these mixed polynomial interpolation curves have a better effect compared to polynomial interpolation curves on interpolating some special triangular mixed interpolation curve, exponential mixing curve,s the hyperbolic curves of the hybrid.Lastly, a class of spline curves with a shape parameter based on polynomial and hyperbolic polynomial space is constructed in this thesis. The given curve have the most of properties on cubic B-spline curve and achieve the Gz continuity. Meanwhile can accurately represents a straight line and hyperbola. Moreover, a class of hyperbolic polynomial B-spline curve is constructed with similar construction and the same continuity as the quadratic non-uniform B-spline curves and C1 continuous. The given curves can be closer to the given control polygon than the quadratic B-spline curves. Its main advantage lies in the ability to represent hyperbolic precisely. Both of two types of spline curves with a shape control parameter can adjust the shape of the curve to achieve a better approximation or smoother effect.