Research On Design Of Blended Surfaces And Curves And Analysis Of Orthogonal Splines

In this paper, we have made a systemic research on the design of blended surfaces and curves and the analysis of spline orthogonality, that is the design of algebraic trigonometric blended surfaces over triangular domain, the identification of the characterization diagrams of planar cubic blended hyperbolic polynomial curves, and the construction of orthogonal splines in spline space. The main creative results presented as follows.1. In respect of the design of blended surfaces over triangular domain. The existing blended surfaces are all defined over the rectangular domain. We extend the algebraic trigonometric Bezier-like basis of order4to the triangular domain and define a new basis. The new basis functions are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. So we call it Bezier-like basis over triangular domain. The corresponding Bezier-like surfaces have some properties and can accurately represent those surfaces whose boundary lines are arcs or elliptical arcs.2. In respect of the shape analysis of blended curves. We propose the geometric characterization diagrams of planar cubic H-curves based on the conditions leading to singularities of the curves. When three control points are fixed and the fourth point moves, the planar cubic H-curve may produce a cusp,-a-loop, or zero to two inflection points. The result is determined entirely by the moving control point. In this process, the plane can be divided into srveral regions according to the characterization of the curve when the fourth point lies in each region. This partitioned plane is called a characterization diagram. Since inflection points and singularities(loops or cusps) of curves are affinely invariant, we find out the geometrically intuitive relationship of these different geometric characterization diagrams in a common3-dimension characterization space. This approach completes the theory of detecting singularities of H-curves in the point of elevating geometric characterization diagrams dimension.3. In respect of the polynomial orthogonal splines. In order to resolve the theoretical problem that there is not a well-expressed orthogonal basis in polynomial spline space by now, we construct an orthogonal basis for the n-degree spline space with arbitrary knot sequence. We extend the traditional Legendre method to spline space and obtain a unified and explicit expression for orthogonal basis. We first define a set of auxiliary functions based on the B-splines of degree (2n+1). Then the proposed orthogonal splines are given as the (n+1)st-order derivatives of these auxiliary functions. We also provide some examples of cubic orthogonal splines to demonstrate our process.4. In respect of the blended polynomial orthogonal splines. Non-uniform algebraic trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. In order to develop the theory of orthogonal basis for algebraic trigonometric spline space, a novel approach is presented to construct an orthogonal basis for the n-order spline space with arbitrary knot sequence. Based on the NUAT B spline functions of order (2n-2), a set of auxiliary functions is constructed. And the proposed orthogonal splines are given as the (n-2)nd-order derivatives of these auxiliary functions. We extend this method to algebraic hyperbolic spline space.