Modelling And Reshaping Spline Curves And Surfaces

In this paper, we have made a systemic theoretic research on modelling and reshaping several types of spline curves/surfaces in CAGD. And some creative productions are given as follows.At first, the uniform C-B-spline basis is proved to be a normalized totally positive basis, and can be extended to a normalized B-basis. C-B-spline and C-Bézier basis, similar to B-spline basis and Bézier basis in algebraic space, are two bases in algebraic trigonometric space. They are presented in need of special curves and surfaces in engineering applications. In order to prove that the uniform C-B-spline basis is normalized totally positive, we compute the matrix which transforms the C-Bézier basis into the uniform C-B-spline and decompose this matrix into a product of. bidiagonal and stochastic factors. Furthermore, we give a normalized B-basis by inserting knots.Secondly, certain curves calledα-paths are found by modifyingαof C-curves. These paths can be linearized, so some nonlinear problem about C-curves can be linearized. We give a geometric interpretation of theα-paths. These paths can closely be approximated by lines and have some nice geometric properties which may yield to a better understanding of the role ofαin terms of the shape of these curves.Thirdly, we present the properties of the envelope and paths of B-spline curves/surfaces when knots are modified, and propose several shape control methods of NURBS curves. The modification of a knot of a B-spline curve of orderκgenerates an one-parameter family curves. This family has an envelope which is also a B-spline curve of orderκ-a with the same control polygon, where a is the multiplicity of the modified knot. It's the same for the NURBS curves and surfaces. Furthermore, we present the effect of the symmetric alteration of some knots of the B-splines. Applying above theoretical results, several shape control methods are provided for cubic NURBS curves based on the modification of a knot. The proposed methods enable local shape modifications subject to position and/or tangent constraints that can be specified within well defined limits.At last, interval implicitization of rational curves/surfaces is presented, based on the properties of interval algebraic curves/surfaces, barycentric coordinate and Bernstein basis functions. Parametric and algebraic curves/surfaces are two common types of representations of geometric objects in CAGD and Geometric Modelling. Thus it is important to have both representations at the same time. We find an uniform interval curve/surfaces with lower degree bounding a given rational curve/surfaces and minimizing some objective function. An algorithm and some examples are provided to demonstrate the theory.