Research On The Theories And Methods Of Geometric Modeling Based On The Curves And Surfaces By The Trigonometric Polynomials

The research on the geometric modeling and processing comes of the impulse of applications in geometric design. Although, in recently years, the non-uniform rational B-spline (NURBS) curves and surfaces are standard mathematical methods, which have been adopted in many civil and external design systems, the solution to many mathematical models or methods are still desiderated. Especially, both the impulse of design approaches to the new industrial products and the desire to improve the level of design proposal offer challenges to the old methods. So it is essential for us to get rid of the stale and bring forth the fresh by the contents and means for the geometric design. And the blending of trigonometric and algebraic spline therefore becomes a new focus of the investigation. In view of these facts, the dissertation discusses the theories and methods of geometric modeling based on the trigonometric polynomials. The main results in this dissertation are outlined as follows:Free form curves and surfaces modeling(1) Bézier-type and B-spline-type curves and surfaces based on the blending of algebraic and trigonometric polynomial are constructed, which inherit all the characters that the Bézier and B-spline curves and surfaces have, such as terminal properties, symmetry, geometric invariability, convex hull property, C2 continuity, etc.(2) A shape parameterαis introduced into the base functions of the Bézier-type and B-spline-type curves and surfaces to adjust the shapes of corresponding curves and surfaces, which allow a user to reinforce the flexibility other than only by using the control points when the forms of curves and surfaces should be reshaped. The modification can be both local and global one in a sense of geometric continuity.(3) The shape parameter a that we introduced can extend the scope that the defined curve exhibits. The corresponding curves include the C-Bézier and C-B-spline curves as special cases and can approximate the Bézier and B-spline curves from the both sides. Moreover, the curves that we presented can be adjusted easily and freely by using the shape parameter a, such that dp(α,t) is linear with respect to a for the fixed t.(4) With the shape parameters and the control points chosen properly, the blended curves and surfaces by the algebraic and trigonometric polynomials can be used to represent conic section, cycloid, helix, cordioid and some other transcendental curves precisely, and the corresponding tensor product surfaces can also represent sphere and some quadric surfaces exactly.Interpolation curves and surfaces modeling(1) The methods of curves and surfaces modeling based on B-spline type interpolation spline and another one with parameters by the blended trigonometric and algebraic polynomials are presented. The introduced splines hold some elementary properties that B-spline has, such as weight property, symmetry, C~2 continuity, etc. Compared with B-spline interpolation curves and cubic polynomial interpolation spline curves, the two kinds of defined interpolation splines satisfy the interpolation conditions without solving the linear equation systems. Meanwhile, a sequence of local control parameters are proposed to afford extra freedom in order that the requisite of the curves fairing and shape preserving can be fulfilled. In addition, the interpolation curves are not sensitive about the boundary conditions, so users can easily handle the shapes of curves. Compared with theα-spline ([TL99]), the interpolating property is independent of tension parameterαby the introduced spline curves in this dissertation. The varieties of tension parameters cannot influence the high order continuity of our spline curves. But concerningα-spline, the interpolating control points only depend on the fact that the tension parameterαequals zero. In the circumstances, theα-spline curves are not high order continuous but C~0 continuous.(2) Two kinds of Hermite-type trigonometric polynomial interpolation splines are presented in the dissertation. These corresponding spline curves are of variable degrees by introducing tension parameters, which can be available in eliminating some redundant inflexions. Compared with the cubic polynomial interpolation spline curves, the defined spline curves and surfaces satisfy the interpolation conditions and C~2 continuity without solving linear equation systems when all tension parameters are greater than two. At the same time, the tension parameters provide an additional freedom, so that we can easily adjust the shapes of curves and surfaces by changing the corresponding tension parameters. In addition, the modifications are both local and global.(3) We present a class of circular trigonometric Hermite interpolation (CTHI-) spline curves and surfaces, which are defined on circular arcs. Compared with the circular Bernstein-Bézier (CBB-) polynomial curves and surfaces in [AS95], the proposed methods are not approximation but interpolation. We can get the C~2 continuous interpolation spline just by solving a tridiagonal linear equation system. But using the methods in [AS95], we must calculate the control points inversely in order to interpolate the data points. The additional difficulties in the above calculating process, including the joint process for the C2 continuity, lie in the choice of base functions in terms of trigonometric functions. These difficulties cause a lot of inconvenience when the user modifies the shape of curves and surfaces. Compared with the normal cubic polynomial interpolation spline methods, Bézier methods and B-spline methods, our approaches are established by the barycentric coordinates on the circular arcs. So it is convenient in modeling some complicated or appropriate curves and surfaces using our methods.Skinning surfaces designingSome methods for the skinning surfaces designing are presented based on the blended interpolation spline by the algebraic and trigonometric polynomials (ATBI-spline).(1) Using the ATBI-splines can slide over the procedures of solving linear equation system and calculating control points inversely as in the case by normal B-splines. The new methods both satisfy the requisite of high order continuity and make it feasible for users to directly modify the key positions and the orientations of corresponding cross-section curves.(2) The new methods, which combine the method of swept surface designing and the method of swung surface designing, can avoid both that all cross-section curves must be ascertained one by one in advance in the swept surfaces and that the profile curves must swing round the z-axis in the swung surfaces. We regard the design of skinning surfaces as the movement of a baseline along a spine. In this moving process, the baseline can be rotated and stretched in order to improve the more flexibility of skinning surfaces.(3) For the skinning surfaces constructed by some special cross-section curves, we can deal with the cross-section curves and the spine curves in the Cartesian coordinates and polar coordinates respectively at the pretreatment stage. The new methods for the skinning surfaces designing therefore are presented based on the blended coordinates, which can further enhance the design precision of the target surfaces.