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This paper discusses singularities, inflection points and convexity of FB-spline curves in terms of their control polygons, and gives the necessary and sufficient conditions for testing when FB-spline curves have one or two inflection points, or a cusp, or a loop, or none of the above points by the envelope theory and the topological mapping method.

文章利用包络理论和拓扑映射的方法，讨论了FB-样条曲线的奇拐点和曲线性质，并给出了依据控制多边形判断FB-样条曲线出现1个或2个拐点、1个尖点、1个二重点以及处处为凸的充分必要条件；最后给出了FB-样条曲线的奇点、拐点以及二重点在λ-μ平面上的分布图。

In this paper,we construct the transitive signature scheme based on bilinear pairings over elliptic curves or super elliptic curves,If one-more CDH problem is hard and standard signature scheme is unforgeable under adaptive chosen-message attack,our scheme is transitively unforgeable under adaptive chosen-message attack.

构造了基于椭圆曲线或超椭圆曲线上双线性对的可传递签名方案，并且证明了在one—more CDH问题是难的和标准签名方案是自适应选择信息攻击下不可伪造的条件下，该文的可传递签名是在自适应选择信息攻击下不可伪造的。

This paper presents a new class of piecewise trigonometric polynomial curves and a weighted trigonometric polynomial curve which has simple structure and can be used to design curves.

提出了一类新的分段三角多项式曲线，给出了加权三角多项式曲线，表示式结构简单，能用于曲线设计。

This trigonometric polynomial can be used to construct open and closed curves. Moreover, the class of piecewise trigonometric polynomial curves can be accurate and flexible to express ellipse.

利用该类分段三角多项式，给出了开曲线和闭曲线的构造方法，并提出了分段三角多项式可精确、灵活地表示椭圆。

By analyzing the characters of Bezier, we construct trigonometric polynomial curves in the space of trigonometric functions, which assume the characters of Bezier curves.

本文主要工作：(1)以Bézier曲线的特点为基础，在三角函数空间上构造了具有Bézier曲线特性的三角函数多项式曲线，称其为TC-Bézier曲线。

The trigonometric polynomial curves retain the main superiority of the quadratic Bézier curves.

给出了二次三角多项式形式的Bézier曲线，基函数由一组带形状参数的二次三角多项式组成。

In order to make up the deficiency of ordinary trigonometric polynomial curves in aspect of shape adjustment, then in view of polynomial spline cannot represent some transcendental curves, a new kind of parametric trigonometric polynomial spline with a shape parameter is constructed by linear singular blending technique.

为了弥补普通三角多项式样条曲线在形状调整方面的不足，同时又考虑到多项式样条曲线不能精确表示一些超越曲线，我们利用奇异混合的思想构造了一种新的带有形状控制参数的三角多项式样条曲线。

Interpolating trigonometric polynomial parametric curves with C~2 (or G~1) continuity can be automatically generated without having to solve any system of equations or do any iterative computation. Then, the convexity of the constructed curves can be guaranteed by the appropriate value of the shape parameter.

在无需反求方程组、迭代计算和最值求解的前提下，即可产生插值给定点列的C~2或G~1连续的一族三角多项式样条曲线；进一步研究了该类曲线的保凸性，得到插值曲线保凸时形状参数的取值范围。

The trigonometric polynomial curves retain the main superiority of cubic Bézier curves.

给出了二次三角多项式B zier曲线，基函数由一组带形状参数的二次三角多项式组成。

Experiments show that C-B spline curve, uniform B-spline with shape parameter, hyperbolic polynomial uniform B-spline with shape parameter and trigonometric polynomial uniform B-spline with shape parameter can be used to produce some frequently used free form curves in the industrial field when the shape parameter is assigned with some specific values. This process is simpler than producing these free form curves by using controlling vertex.

实验证明，C-B样条曲线、带形状参数的均匀B样条曲线、带形状参数双曲多项式的均匀B样条曲线、带形状参数三角多项式的均匀B样条曲线都可利用形状参数的特定取值表示一些工业领域常用的自由曲线，这比起用控制顶点表示同样的自由曲线更为简单。

Liapunov—Schmidt method is one of the most important method in the bifurcation theory.

Liapunov—Schmidt方法是分叉理论的最重要方法之一。

Be courteous -- even when people are most discourteous to you .

要有礼貌──即使当別人对你最不礼貌的时候。