Abstract: To construct a G1 continuity spatial quintic Pythagorean-hodograph fitted curve for the reconstruction of spatial curve, through adding some intermediate conditions into the space sampling points to determine endpoints’ dates, it interpolates these spatial discrete data with G1 Hermite to construct spatial PH fitted curve. According to the sufficient and necessary conditions of PH space curve, the quartic derived vector composed of four quadratic polynomials is given. Its coefficients in Bernstein basis are compared with coefficients of Bézier curve’s derived vector to compose a system of equations. Following it compares Bézier curve’s derived vector coefficients with the control points of Bézier curve, by adding some free parameters to build their equal relationships. Then it composes a system of equations with aforementioned equations and equal relationships. By solving them, a G1 continuity spatial quintic Pythagorean-hodograph fitted curve is producted with G1 Hermite interpolation, which satisfies?the data of endpoints determined by intermediate conditions, and some numerical examples are presented. This construction method is intuitive, it includes multiple free parameters to control the shape of curve fitting effect.

Key words: Pythagorean-Hodograph（PH） spatial curve, curve fitting, G1 Hermite interpolation, Bernstein basis function

摘要： 为了构造一种空间五次Pythagorean-hodograph G1连续拟合曲线以重建空间曲线，对已知空间采样点数据加入中间条件确定首末端点数据，对其进行G1 Hermite插值构造拟合PH曲线。根据空间PH曲线的充分必要条件，给出由四个二次多项式组成的四次导函数，比对其与空间五次Bézier曲线的导函数在Bernstein基下分别对应的向量型系数，形成向量等式，再根据Bézier曲线导函数的系数与其控制多边形顶点的关系，引入自由参数建立五次Bézier曲线导函数的系数与首末端点的等量关系，并与前述向量等式组成方程组。通过求解方程组可得一段由G1 Hermite插值构造出的满足由中间条件给出的首末端点数据且G1连续的PH拟合曲线，并给出了数值实例。此构造方法直观，有多个自由参数可对曲线进行拟合效果的形状控制，且通过数值实验拟合效果较好。

关键词: PH空间曲线, 曲线拟合, G1 Hermite插值, Bernstein基函数