一类新的拟Bernstein-Bézier曲线

构造了一类新的带双参数形状可调的拟Bernstein基函数,它是在三次Bernstein多项式的基础上扩展而成的一组n次拟Bernstein基.在此基础上,定义了带双形状参数的拟Bernstein-Bézier曲线,它保留了Bézier曲线的几何特征,并具有形状可调的特性.在控制点给定的情况下,可通过改变形状参数的值整体或局部地调控曲线的形状,同时给出参数控制及曲线拼接应用的实例.     

Judging or setting weight steady-state of rational Bezier curves and surfaces

Many works have investigated the problem of reparameterizing rational B~zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.     

单色仪定标曲线的非线性拟合

采用两种方案对WDF棱镜单色仪的定标曲线进行非线性拟合。通过实验找出棱镜的折射率与TT鼓轮的关系,得到利用科希公式拟合定标曲线的可行性方案;利用高阶多项式对定标曲线进行拟合,运用统计的方法探讨不同的拟合阶数对拟合结果的影响,并得到了最佳的拟合阶数和高精度的拟合结果。     

椭圆曲线y~2=px(x~2+2)有正整数点的判别条件

设p是适合p≡1(mod8)的奇素数.运用四次剩余和Pell方程的性质,给出了椭圆曲线y~2=px(x~2+2)有正整数点(x,y)的若干判别条件.     

摆线亮相高考风景这边独好

摆线即旋轮线是自选模块《坐标系与参数方程》中的内容,在数学史上,发现圆锥曲线后,受到科学家关注最多的曲线是摆线.摆线种类丰富,性质优美,应用广泛.摆线问题类属动态几何问题,解决摆线问题不仅需要平面几何、三角函数和解析几何等相关知识的支撑,而且还需要一定的运动想像能力和探究能力.2010年高考摆线首次亮相北京卷文理科第14题,2011年高考摆线第二次亮相于江西省文理科第10题,均属客观题的压轴题.这几道难得的考题视角独特、设计新颖、卓尔不群,运动中考能力,立意上求创新.     

超椭圆曲线y~k=x(x+1)(x+3)(x+4)上的有理点

针对超椭圆曲线y~k=x(x+1)(x+3)(x+4)上是否存在有理点这一问题,运用了分类讨论的方法求解出当k≥3且k≠4时,该超椭圆曲线上的有理点只有(0,0);(-1,0);(-3,0);(-4,0).     

新题征展(132)

A题组新编1.设△ABC的内角A,B,C的对边长分别为a,b,c,已知cos(A-C) +cosB=t(t是已知的正数),根据下列条件分别求出角B的大小:(1)a,b,c成等比数列;(2)a,b,c成等差数列.2.(1)求数列{2(n-1)/x(2n-1)+1}的前n项和Sn;(3n+1)+(3n+4)+(3(2)求数列(3n-2)+(3n+1)+(3n+4)+(3n+7)/(3n-2)(3n+1)(3n+4)(3n+7)的前n项和Tn.3.(1)证明:2(2n)-1 (n ∈ N*)至少有n个不同的素因数;(2)求C12n,C32n,C52n,…,C2n-12n的最大公约数.B藏题新掘4.已知曲线C:x｜x｜/a2-y｜y｜/b2=1,下列叙述中错误的是A.垂直于x轴的直线与曲线C只有一个交点B.直线y=kx +m(后,m∈R)与曲线C最多有三个交点C.曲线C关于直线y=-x对称D.若P1(x1,y1),P2(x2,y2)为曲线C上任意两点,则有(y1-y2)/(x1-x2) ＞05.(二项式定理)在(x+y)n的展开式中,若第七项系数最大,则n的值可能等于____.     

曲线积分的换元法

给出了曲线积分的换元法,丰富了曲线积分的计算方法.     

《从零学相对论》连载(27)

正第9讲虫洞和时间机器美国康奈尔大学的天体物理学家萨根(Sagen)在1985年出版的著名小说《接触》(Contact)中描写了一个通过虫洞作快速星际旅行的故事.小说初稿写的是穿越黑洞的旅行,出版前,作者曾将书稿寄给美国加州理工学院的相对论专家索恩(Thorne),请他协助把小说中涉及引力物理的部分尽可能加以准确化.索恩对黑洞非常熟悉,深知穿越黑洞的旅行几