Curve extension is a useful tool in the computer‐aided design (CAD) community. A given B‐spline curve usually needs to be extended by another curve to reach one or more target points. In this work, we aim to enlarge the representation domain of the extending part to achieve the optimal extending result in the global solution space. Inspired by this, we have made three contributions. First, we use piecewise polynomial, that is, a nonuniform B‐spline, instead of one polynomial segment to extend the original curve. Compared to one polynomial segment, curves consisting of piecewise polynomial have stronger modeling ability and therefore expand the solution space of the problem. For extension to multiple target points, we are the first to directly give extension results based on all target points rather than extending to every target step by step. Third, we exploit the matrix representation of B‐splines to obtain an explicit solution for this extension problem. The detailed formula derivations and experimental results are provided to show the validity and effectiveness of our method.

中文翻译：

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通过分段多项式扩展 B 样条

曲线扩展是计算机辅助设计 (CAD) 社区中的有用工具。给定的 B 样条曲线通常需要通过另一条曲线进行扩展才能到达一个或多个目标点。在这项工作中，我们的目标是扩大扩展部分的表示域，以在全局解空间中实现最佳扩展结果。受此启发，我们做出了三项贡献。首先，我们使用分段多项式，即非均匀 B 样条，而不是一个多项式段来扩展原始曲线。与一个多项式段相比，由分段多项式组成的曲线具有更强的建模能力，从而扩展了问题的解空间。对于多个目标点的扩展，我们首先直接给出基于所有目标点的扩展结果，而不是一步一步扩展到每个目标。第三，我们利用 B 样条的矩阵表示来获得此扩展问题的显式解决方案。提供了详细的公式推导和实验结果，以证明我们方法的有效性和有效性。