形状可调的插值型球样条曲线及其双目标能量极小化

Interpolated Ball Spline Curve with Shape Adjustability and Its Bi-objective Energy Minimization

球B样条曲线(BBSC)是一种基于骨架的参数实体模型,可用于构造自然界中广泛存在的可变厚度管状物体。然而,当控制球保持不变时,球B样条曲线的形状无法调整,而且不易构造出插值球样条曲线。通过将五次Catmull-Rom样条曲线中的控制顶点替换为控制球,构造了一种带参数的插值型球样条曲线,称为α-BSC。α-BSC不仅可在控制球保持固定时通过所含的参数对其形状进行自由调整,还可直接插值于给定的控制球。进一步地,将空间参数曲线的能量极小化方法推广至α-BSC,给出了利用能量极小化确定α-BSC所含参数的最优取值方法。该方法同时考虑了α-BSC的弯曲能与扭曲能,通过求解双目标优化问题获得α-BSC所含参数的最优取值。实例表明,所提出的α-BSC及其双目标能量极小化方法简单有效,满足可变厚度管状物体造型的需要。

Ball B-spline curve(BBSC) is a skeleton-based parametric solid model, which can be used to construct tubular objects with variable thickness widely existing in nature. However, when the control balls remain unchanged, the shape of ball B-spline curve cannot be adjusted, and it is not easy to construct the interpolated ball spline curve. By replacing the control points in quintic Catmull-Rom spline curve with control balls, an interpolated ball spline curve with a parameter, named α-BSC,was constructed. α-BSC can not only adjust its shape freely by altering the value of the contained parameter when the control balls are fixed, but also interpolate directly into the given control balls. Furthermore, the energy minimization method of spatial parametric curve was extended to α-BSC to determine the optimal value of the parameter contained in α-BSC. The proposed method considered the bending energy and twisting energy of α-BSC synchronously and the optimal value of the parameter contained in α-BSC was obtained by solving a bi-objective optimization problem. Some examples show that α-BSC and its bi-objective energy minimization method are simple and effective, and meet the needs of modeling of tubular objects with variable thickness.