Nonlinear stationary subdivision schemes that reproduce trigonometric functions

TL;DR

This work tackles reproducing conic sections with a single nonlinear, interpolatory, stationary subdivision scheme. By leveraging orthogonal rules for exponential-polynomial spaces and introducing a bounded cut-off function $\Gamma_\epsilon$, the authors construct $S_\epsilon$ to reproduce $\Pi_2$ and $W_{0,\gamma}$ for all admissible $\gamma$ without gamma-dependent rules. They prove convergence, monotonicity preservation, and, under a mild condition, $C^1$ regularity, while demonstrating strong numerical reproduction of circles, ellipses, and hyperbolas and a fourth-order one-step approximation in monotone regions. The results offer a simple, robust approach to conic reproduction in geometric modeling with potential extensions to higher dimensions and bivariate settings.

Abstract

In this paper we define a family of nonlinear, stationary, interpolatory subdivision schemes with the capability of reproducing conic shapes including polynomials upto second order. Linear, non-stationary, subdivision schemes do also achieve this goal, but different conic sections require different refinement rules to guarantee exact reproduction. On the other hand, with our construction, exact reproduction of different conic shapes can be achieved using exactly the same nonlinear scheme. Convergence, stability, approximation and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain $\mathcal{C}^1$ limit functions are also studied.

Figures (5)

• Figure 1: Curves generated by $S_\epsilon$ ($\epsilon=1$) and $T_{2,2}$ from an initial sequence $f^0 = (G_u(i))_{i\in\mathds{Z}}$, where $G_u(t) = (\cos(\gamma t+u), \sin(\gamma t+u))$, for $u=0$ and $u=u=10^{-5}$. Observe that $S^\infty_\epsilon f^0$ strongly depends on the value of $u$.
• Figure 2: Left plot: Anthropomorphic shape composed of one ellipse, two hyperbolas and one parabola. The marked points refer to Table \ref{['tab:repro']}. Center plot: $f^0$, black dots. $S^\infty_\epsilon f^0$, solid line ($\epsilon=1$). Right plot: $f^0$, black dots. $T_{2,2}^\infty f^0$, solid line. The 'exact' conic sections are represented with a dashed line in the center and right plots.
• Figure 3: Left plot: $T_{2,2}^\infty f^0$ (dotted line) and $S_\epsilon^\infty f^0$ (solid line) with $f^0$ in \ref{['eq:monotone_data1']}. Right plot: zoom of the area marked with a dashed rectangle on the left plot.
• Figure 4: Left plot: $T_{2,2}^\infty f^0$ (dotted line) and $S_\epsilon^\infty f^0$ (solid line) with $f^0$ in \ref{['eq:monotone_data2']}. Right plot: zoom of the area marked with a dashed rectangle on the left plot.
• Figure 5: Left: The curve generated (line) from the initial sequence (dots) using $S_\epsilon$. Right: A zoom of one edge of the left figure.

Theorems & Definitions (46)

• Definition 1
• Definition 2
• Theorem 3
• Remark 4
• Corollary 5
• proof
• Definition 6
• Theorem 7
• proof
• Example 8