Guozhao Wang

A class of Bézier-like curves

In this paper, a new basis, to be called C-Bezier basis, is constructed for the space Γn = span{1, t, t2,...,tn-2, sin t, cos t by an integral approach. Based on this basis, we define C-Bezier curves. We then show that such basis and curves share the same properties as the Bernstein basis and the Bezier curves in polynomial spaces respectively.

NUAT B-spline curves

This paper presents a new kind of splines, called non-uniform algebraic-trigonometric B-splines (NUAT B-splines), generated over the space spanned by {1, t,...,tk-3, cos t, sin t} in which k is an arbitrary integer larger than or equal to 3. We show that the NUAT B-splines share most properties of the usual polynomial B-splines. The subdivision formulae of this new kind of curves are given. The generation of tensor product surfaces by these new splines is straightforward.

Uniform hyperbolic polynomial B-spline curves

This paper presents a new kind of uniform splines, called hyperbolic polynomial B-splines, generated over the space Ω = span{sinht, cosht, tk-3, tk-4 ...,t, 1} in which k is an arbitrary integer larger than or equal to 3. Hyperbolic polynomial B-splines share most of the properties as those of the B-splines in the polynomial space. We give the subdivision formulae for this new kind of curves and then prove that they have the variation dimishing properties and the control polygons of the subdivisions converge. Hyperbolic polynomial B-splines can take care of freeform curves as well as some remarkable curves such as the hyperbola and the catenary. The generation of tensor product surfaces by these new splines is straightforward. Examples of such tensor product surfaces: the saddle surface, the catenary cylinder, and a certain kind of ruled surface are given in this paper.

Optimal multi-degree reduction of Bézier curves with G2-continuity

In this paper we present a novel approach to consider the multi-degree reduction of Bezier curves with G^2-continuity in L2-norm. The optimal approximation is obtained by minimizing the objective function based on the L2-error between the two curves. In contrast to traditional methods, which typically consider the components of the curve separately, we use geometric information on the curve to generate the degree reduction. So positions, tangents and curvatures are preserved at the two endpoints. For avoiding the singularities at the endpoints, regularization terms are added to the objective function. Finally, numerical examples demonstrate the effectiveness of our algorithms.

Curvature continuity between adjacent rational Be´zier patches

Abstract This paper discusses the curvature continuity between two adjacent rational Bezier surfaces which may be rectangular or triangular patches. The necessary and sufficient conditions are derived, and further, a series of simple sufficient conditions are developed. These conditions are either descriptive or constructive. Therefore with them one can both check the geometric continuity between two surfaces and construct a rational surface possessing curvature continuity with a given rational patch along a certain boundary. This is an important feature in CAGD applications.

The rational cubic Be´zier representation of conics

Abstract The rational cubic Bezier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bezier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bezier vertices.

Optimal properties of the uniform algebraic trigonometric B-splines

In this paper, we construct a matrix, which transforms a generalized C-Bezier basis into a generalized uniform algebraic-trigonometric B-spline (C-B-spline or UAT B-spline) basis. We also show that it is a totally positive matrix and give a normalized B-basis of the generalized UAT B-splines.

Inflection points and singularities on C-curves

We show that all so-called C-curves are affine images of trochoids or sine curves and use this relation to investigate the occurrence of inflection points, cusps, and loops. The results are summarized in a shape diagram of C-Bezier curves, which is useful when using C-Bezier curves for curve and surface modeling.

GC n continuity conditions for adjacent rational parametric surfaces

Abstract In this paper, the constraints on the homogeneous surface belonging to a certain rational surface are derived which are both necessary and sufficient to ensure that the rational surface is n th-order geometric continuous. This gives up the strong restriction that requires the homogeneous surface to be as smooth as the rational surface. Further the conditions for the rectangular rational Bezier patches are developed, and some simple and practical sufficient conditions are presented which might give a valid means for the construction of GC n connecting surfaces.

On the degree elevation of B-spline curves and corner cutting

In this paper we prove that the degree elevation of B-spline curves can be interpreted as corner cutting process in theory. We also discover the geometric meaning of the auxiliary control points during the corner cutting. Our main idea is to gradually elevate the degree of B-spline curves one knot interval by one knot interval. To this end, a new class of basis functions, to be called bi-degree B-spline basis functions, is constructed and discussed by the integral definition of spline. The transformation formulas between usual and bi-degree B-spline basis functions leads to the corner cutting for degree elevation of B-spline curves.