In this paper, we introduce a novel non-linear uniform subdivision scheme for the generation of curves in $\mathbb{R}^n$, $n\geq2$. This scheme is distinguished by its capacity to reproduce second-degree polynomial data on non-uniform grids without necessitating prior knowledge of the grid specificities. Our approach exploits the potential of annihilation operators to infer the underlying grid, thereby obviating the need for end-users to specify such information. We define the scheme in a non-stationary manner, ensuring that it progressively approaches a classical linear scheme as the iteration number increases, all while preserving its polynomial reproduction capability. The convergence is established through two distinct theoretical methods. Firstly, we propose a new class of schemes, including ours, for which we establish $\mathcal{C}^1$ convergence by combining results from the analysis of quasilinear schemes and asymptotically equivalent linear non-uniform non-stationary schemes. Secondly, we adapt conventional analytical tools for non-linear schemes to the non-stationary case, allowing us to again conclude the convergence of the proposed class of schemes. We show its practical usefulness through numerical examples, showing that the generated curves are curvature continuous.

翻译：本文提出了一种新颖的均匀非线性细分格式，用于生成$\mathbb{R}^n$（$n\geq2$）空间中的曲线。该格式的显著特征在于，无需预先知晓网格特性即可在非均匀网格上再现二次多项式数据。我们的方法利用湮灭算子的潜力来推断潜在网格，从而免除了最终用户指定此类信息的必要性。我们以非平稳方式定义该格式，确保随着迭代次数的增加，它逐渐逼近经典线性格式，同时保持其多项式再现能力。通过两种不同的理论方法建立了收敛性。首先，我们提出了一类新格式（包括本文格式），通过结合拟线性格式分析结果与渐近等价的线性非均匀非平稳格式结论，证明了其$\mathcal{C}^1$收敛性。其次，我们将非线性格式的常规分析工具推广至非平稳情形，从而再次论证了所提出格式类别的收敛性。通过数值示例展示了其实用价值，表明生成的曲线具有曲率连续性。