Triangular domain extension of algebraic trigonometricB′ezier-like basis

In computer aided geometric design （CAGD）, B′ezier-like bases receive more andmore considerations as new modeling tools in recent years. But those existing B′ezier-like basesare all defined over the rectangular domain. In this paper, we extend the algebraic trigono-metric B′ezier-like basis of order 4 to the triangular domain. The new basis functions definedover the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry,boundary representation, linear independence and so on. We also prove some properties of thecorresponding B′ezier-like surfaces. Finally, some applications of the proposed basis are shown.

New cubic rational basis with tension shape parameters

By using the blossom approach, we construct four new cubic rational Bernsteinlike basis functions with two shape parameters, which form a normalized B-basis and include the cubic Bernstein basis and the cubic Said-Ball basis as special cases. Based on the new basis, we propose a class of C2 continuous cubic rational B-spline-like basis functions with two local shape parameters, which includes the cubic non-uniform B-spline basis as a special case.Their totally positive property is proved. In addition, we extend the cubic rational Bernsteinlike basis to a triangular domain which has three shape parameters and includes the cubic triangular Bernstein-B′ezier basis and the cubic triangular Said-Ball basis as special cases. The G1 continuous conditions are deduced for the joining of two patches. The shape parameters in the bases serve as tension parameters and play a foreseeable adjusting role on generating curves and patches.

Triangular domain extension of linear Bernstein-like trigonometric polynomial basis

In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spaces(trigonometric polynomial,hyperbolic polynomial,or blended space) has also been studied.However,none of them was extended to the triangular domain.In this paper,we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis,which is linearly independent and satisfies positivity,partition of unity,symmetry,and boundary represen-tation.We prove some properties of the corresponding surfaces,including differentiation,subdivision,convex hull,and so forth.Some applications are shown.

Bézier curves with shape parameter

In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape parameter have most properties of Bernstein basis and the Bézier curves. Moreover the shape parameter can adjust the curves’ shape with the same control polygon. As the increase of the shape parameter, the Bézier curves with shape parameter approximate to the control polygon. In the last, the Bézier surface with shape parameter is also constructed and it has most properties of Bézier surface.

Representing conics by low degree rational DP curves

A DP curve is a new kind of parametric curve defined by Delgado and Pena （2003）; Jt has very good properties when used in both geometry and algebra, i.e., it is shape preserving and has a linear time complexity for evaluation. It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape, and the disadvantage of the B6zier curve that is shape preserving but slow for evaluation. It also has potential applications in computer-aided design and manufacturing （CAD/CAM） systems. As conic section is often used in shape design, this paper deduces the necessary and suffi- cient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves. The main idea is based on the transformation relationship between low degree DP basis and Bemstein basis, and the representation tbeory of conics in rational low degree B6zier form. The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form, i.e., give positions of the control points and values of the weights of rational cubic or quartic DP conics. Finally, several numerical examples are presented to validate the effectiveness of the method.