The basis of T-splines are the point-based splines (PB splines) that are unstructured meshless splines. In this paper, we study associated PB splines with local knot vectors that are arbitrarily distributed in \(([0,1]\cap {\mathbb {Q}})^d, d=1,2\), where \({\mathbb {Q}}\) is the set of rational numbers. We prove the linear independence of linear PB splines under a mild assumption that their central knots are all distinct. The linearly independent property is one of important prerequisites for isogeometric analysis. Moreover, we illustrate that the same assumption can not be extended to two-dimensional case, by giving a set of linearly dependent bilinear PB splines.

T样条曲线的基础是非结构化无网格样​​条曲线的基于点的样条曲线（PB样条曲线）。在本文中，我们研究了带有局部结向量的关联PB样条，该局部结向量在\（（[0,1] \ cap {\ mathbb {Q}}）^ d，d = 1,2 \）中任意分布，其中\（ {\ mathbb {Q}} \）是有理数的集合。我们在温和的假设下证明了线性PB样条的线性独立性，即它们的中心结都是不同的。线性独立属性是等几何分析的重要先决条件之一。此外，我们通过给出一组线性相关的双线性PB样条，说明了相同的假设不能扩展到二维情况。