一类代数空间逼近样条

A kind of algebraic approximating spline in space

给定空间不共面的四个有序数据点,可以形成一个四面体。在四面体内,Bernstein-Bézie(rB-B)形式定义两类正则实多项式代数曲面片,一类是二次的,一类是三次的。此两类曲面片在四面体内的交集为一条正则曲线段。先固定二次曲面片,并得到其参数形式,然后约简三次曲面片所对应的Bernstein系数,使之为带有三个形状调整的形状因子,其中两个分别代表曲线段端点处的曲率,另外一个作为形状的调整。利用二次曲面的参数形式,由三次曲面片可得到曲线的隐参数约束形式,从而得到曲线的参数形式。对给定的空间点列,利用两个形状因子较容易的拼接出G2-连续的逼近曲线,突破了现行代数曲线生成方法,即空间连续曲线均是通过三角形仿射变换,由B-B形式生成的平面弧拼接而成。

Tetrahedron can be formed with four sequence and non-coplanar points in space,and any points in it can be represented by barycentric coordinate system with space affine transform.Both regular algebraic patches of cubic and quadratic with three variables,which is constructed in the tetrahedron,represented in algebraic Berustein -B6zier zero contour.The intersection set of both patches constructs a regular segment.The quadratic patches is fixed,and its parametric representation method is gained.The cubic patches reduce to three local factors through reducing Bernstein coefficients to control shape of curve,two of them are used to join segments into G^2 continuity approximating curves for given sequence data in space,and the other is used to adjust shape of segment.The parametric representation of segment can be gained from quadratic patch and constrained equations with cubic patches.A G^2 -continuity approximate curve for a sequence points similar to cubic Bezier splines with four control points is generated.The method is quite different from popularly algebraic method by joining planar arc into a continuous curve,and the latter is also represented by barycentric coordinate system with planar affine transform.