空间有理三次Bezier曲线的射影变换和权系数的几何性质  

THE GEOMETRICL PROPERTIES OF PROJECTIVETRANSFORMATION AND WEIGHTS FOR THESPACE RATION CUBIC BEZIER CURVE

作　　者：叶正麟[1] 孟雅琴[1] 彭国华[1] 

机构地区：[1]西北工业大学应用数学系

出　　处：《应用数学学报》1999年第2期178-185,共8页Acta Mathematicae Applicatae Sinica

摘　　要：本文讨论了空间有理三次Ｂｅｚｉｅｒ曲线的射影变换和权系数的一系列几何性质。其权系数组构成了控制四顶点基下的权心的齐次坐标；权心是六个特殊平面的公共交点；含权心和曲线“肩点”的某四个共线点之比恒为常数３；权心可作为有理曲线所在射影坐标系的单位点；此有理曲线是对应整有理曲线在射影变换下的象，此变换把控制四面体的形心映为权心；权系数是此射影变换的特征值（差一常数因子）；权系数是变换前后两曲线上对应点关于控制四顶点的重心坐标对应分量之比；此有理曲线是两个二阶锥面的交线，锥面类型由权系数组成的两个形状因子所决定。任意两条空间有理三次Ｂｅｚｉｅｒ曲线之间有类似的结论。This paper discusses a series of geometrical properties of projective transforma-tion and weights for the space rational cubic Bezier curve. The weights make up homoge-neous coordinate of the weight center in four control vertex basis. The weight center is acommon intersection point of six special planes. The ratio of some four collinear points thatinclude the weight ceneter and the ishoulder point' equals 3. The weight center may be usedas the unit point of projective coordinate system that the rational curve belongs to. Thisrational curve is the map of the projective transformation corresponding to non-rationalcurve, which projects the figure center of control tetrahedron onto the weight center. Theweights are characteristic values of the projective transformation (with a constant factor).The weights respectively equal the ratios of barycentric coordinates elements of the corre-sponding points of the curves in the four control vertex base before transformation and after.This rational curve is the line of intersection of two conicoids and the type of the conicoidsis determined by two shape factors made up by the weights. The conclusions are tenablebetween arbitrary two space rational cubic Bezier curves.

关 键 词：有理BEZIER曲线 射影变换 权系数 CAD 几何性质 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术] TP391.41[自动化与计算机技术—计算机科学与技术]