Core Concepts
The proposed subdivision scheme can reproduce second-degree polynomials on a variety of non-uniform grids without requiring prior knowledge of the grid specifics.
Abstract
The paper introduces a novel uniform non-linear non-stationary subdivision scheme for generating curves in Rn, n ≥ 2. The key features of this scheme are:
Polynomial Reproduction: The scheme can reproduce second-degree polynomial data on non-uniform grids without needing to know the grid details in advance. This is achieved by leveraging annihilation operators to infer the underlying grid.
Non-Stationary Formulation: The scheme is defined in a non-stationary manner, ensuring that it progressively approaches a classical linear scheme as the iteration number increases, while preserving its polynomial reproduction capability.
Convergence Analysis: The convergence of the proposed scheme is established through two distinct theoretical methods:
By combining results from the analysis of quasilinear schemes and asymptotically equivalent linear non-uniform non-stationary schemes, the scheme is shown to be C1 convergent.
By adapting conventional analytical tools for non-linear schemes to the non-stationary case, the convergence of the proposed class of schemes is again established.
Numerical Examples: The practical usefulness of the scheme is demonstrated through numerical examples, showing that the generated curves are curvature continuous.
The proposed scheme aims to be a step towards achieving the reproduction of exponential polynomials, including conic sections, on non-uniform grids without requiring the user to provide grid information.
Stats
The paper does not contain any explicit numerical data or statistics to extract.
Quotes
"In this paper, we introduce a novel non-linear uniform subdivision scheme for the generation of curves in Rn, n ≥2. 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."
"We show its practical usefulness through numerical examples, showing that the generated curves are curvature continuous."