Extending a Ball B-spline Curve (BBSC) is a useful function in the shape modelling of freeform tubular objects. In this paper, we aim to obtain a cubic BBSC $\bar{\mathbf{B}}(\bar{t})$ that can smoothly and fairly extend a given cubic BBSC $\mathbf{B}(t)$ to a target ball R. BBSCs with one endpoint satisfying $G^2$ continuity with $\mathbf{B}(t)$ and the other endpoint passing through R are optional extending results. We choose the fairest of these BBSCs as the extension result. Our contributions are threefold. First, using one polynomial segment such as Bézier to represent extending parts is often inadequate due to its limited representation ability. We use piecewise polynomials, namely, B-spline, to expand the solution space of this problem. Second, we define a strain energy function for BBSCs to describe their fairness. Third, we exploit the matrix representation of B-splines to obtain an explicit solution of the functional optimization problem in the BBSC extension algorithm. Experimental results are provided to prove the effectiveness of our method.

扩展球B样条曲线（BBSC）在自由形状的管状对象的形状建模中非常有用。在本文中，我们旨在获得立方BBSC$\stackrel{〜}{\mathbf{乙}}（\stackrel{〜}{Ť}）$ 可以平稳合理地扩展给定的立方BBSC $\mathbf{乙}（Ť）$到目标球- [R 。一个端点满足的BBSC$G^2$ 连续性 $\mathbf{乙}（Ť）$通过R的另一个端点是可选的扩展结果。我们选择这些BBSC中最公平的作为扩展结果。我们的贡献是三倍。首先，由于有限的表示能力，使用一个多项式段（例如Bézier）来表示扩展部分通常是不够的。我们使用分段多项式，即B样条，来扩展此问题的解空间。其次，我们定义了BBSC的应变能函数来描述其公平性。第三，我们利用B样条的矩阵表示来获得BBSC扩展算法中功能优化问题的显式解决方案。实验结果证明了该方法的有效性。