The Applications Of Spiral And Orthogonal Polynomial In CAGD

In this paper, the theme is everal kind of curve's expressions, splices and approximations, in particular, we have made an in-depth research on spiral and orthogonal polynomials in the CAGD and have obtained new creative production on the following five aspects:(1) Based on the requirement of highway design, a planar cubic C-Bezier spiral with monotone curvature of constant sign and a starting point of zero curvature is constructed, then an algorithm of the transition curve between straight lines and/or circular arcs is derived in detail which can be applied in highway design. As was done with clothoids in engineering, a single spiral is used for straight line to circular arc, two spirals suiting C-shaped or S-shaped transition are used for circular arc to circular arc, two spirals are used for straight line to straight line, and a single spiral is used for circular to circular arc when the latter is contained within the former. The concrete expressions for the first four cases are given; in the fifth case the solution cannot always be obtained. Because straight line segments and circular arcs can be represented precisely by C-Bezier curves, the issues such as highway design can be handled in the system by C-Bezier model, avoiding the difficult situation for computer-aided design system to use the clothoids defined in terms of the Fresnel integral.(2) A single C-Bezier curve with a shape parameter for G2 joining two circular arcs is constructed. It is shown that a S-shaped transition curve which is able to better manage the broader scope about two circle radii than the Bezier curves has no curvature extrema, and a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms is presented in detail, strict mathematical proof was given, and the effectiveness of the method is showed by examples. This method has the following three advantages:the pattern is unified; the parameters able to adjust the shape of the transition curves are available; the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. Therefore it is simple and easy to implement.(3) To fit the form of curve in the current Computer Aided Design system and aesthetic needs in industrial design, two approximation algorithms for logarithmic spiral segments is proposed. In the first method, the calculation formula for s-Power series is firstly derived and a fast polynomial approximation algorithm is presented, and then the calculation formula of the offset curves of the logarithmic spiral and corresponding approximation algorithm by s-Power series is presented. In the second method, The G2 Hermite interpolation formula of the two end points by the C-Bezier form is firstly derived, and then the G2 Hermite interpolation approximation algorithm by the C-Bezier form is presented. The results of example show that the algorithms are correct and effective, suitable for the use of the CAD system.(4)To solve least squares approximation problem effectively in CAGD, the transformation matrices between the weighted orthogonal basis which possesses boundary constraints characteristic and Bernstein basis are derived. Then using the matrices, two specific applications are given.(i) the optimal algorithm based on Jacobi weighted L2 norm for constrained multi-degree reducing Bezier curve, including its matrix representation, is presented. Also the degree reduction error that can be forecasted is given. the Jacobi weighted function adapting to optimal degree reduction is selected with respect to L2, L1, L∞norm, respectively. (ⅱ) To solve the inverse function of polynomial is a basic problem in CAGD, an algorithm about approximating the inverse function with Ck constrains by using the constrained Jacobi basis is proposed. The approximation method is easy and steady. Moreover, the defect that the corresponding coefficients must be recalculated when approximating every inverse function one by one is overcame. As an application, how to generate quasi arc-length parameterization of PH curves is shown. (5) For solve least squares approximation problem simply and effectively on triangular domains in CAGD, The matrices of transformation of the bivariate Bernstein basis form into the Jacobi basis of the same degree and vice versa are derived. A method for constructing bivariate Jacobi-weighted orthogonal polynomials in the Bernstein form on triangular domains was formulated firstly. And then, by using connection coefficients between the univariate Bernstein and Jacobi basis, the transformation matrices between bivariate Jacobi and Bernstein basis were presented. Finally, Then using the matrices, an explicit form of the multi-degree reduction matrix for Bezierr surface on triangular domains with respect to Jacobi weighted L2 norm was presented, and the error of the degree reduction was given.