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非线性均匀细分格式，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，无需预先获知网格特性即能在非均匀网格上再生二次多项式数据。我们的方法利用湮灭算子的潜力推断潜在网格，从而免除终端用户指定此类信息的必要。我们以非平稳方式定义该格式，确保其随迭代次数增加逐步趋近经典线性格式，同时保持多项式再生能力。通过两种截然不同的理论方法建立了收敛性：首先，提出包含本方案在内的一类新格式，通过结合拟线性格式分析与渐进等价线性非均匀非平稳格式的研究成果，证明了其$\mathcal{C}^1$收敛性；其次，将非线性格式的传统分析工具适配至非平稳情形，再次论证了所提格式类别的收敛性。通过数值实例展示其实际效用，证明生成的曲线具有曲率连续性。