An enhanced basis for producing Bezier-like curves

TL;DR

The paper addresses the problem of shaping Bézier-like curves without altering the given control points by constructing a Bernstein-like basis $\mathcal{T}$ from any blending family $\mathcal{F}$ using an auxiliary function $\varphi$ and a shape parameter $\sigma$, producing $\mathbf{C}_n^{\sigma}(t)=\sum_{i=0}^n T_{n,i}(t)\mathbf{P}_i$ that smoothly transitions between $\mathcal{F}$-based curves and the line segment joining $\mathbf{P}_0$ and $\mathbf{P}_n$. The main contributions include precise conditions on $\varphi$ to preserve positivity, partition of unity, symmetry, endpoint interpolation and tangency, together with monotonicity-preserving results; the authors develop $C^1$ and $C^2$ monotone interpolation schemes with feasible constructions and several concrete examples demonstrating the method's flexibility. The work offers a practical framework for adjustable curve design in CAGD, enabling controlled shaping while maintaining core geometric properties, with potential extensions to Bernstein-like operators and applications to PDEs or integral equations. Overall, the approach provides a versatile tool for shape control and monotone interpolation in geometric modeling.

Abstract

This study aims on proposing a new structure for constructing Bernstein-like bases. The structure uses an auxiliary function and a shape parameter to construct a new family of bases from any family of blending functions. The new family of bases inherit almost all algebraic and geometric properties of the initial blending functions. The corresponding curves have the freedom to travel from the curve constructed from the initial blending functions to the line segment joining the first and last control points. The new bases have the monotonicity preservation property and the shape of the curve could be adjusted by changing the parameter.

Figures (5)

• Figure 1: (a): The original p-curves with different values of parameter $\gamma$, (b) and (c): the p-Bézier curves with $\gamma=1, 0.01$ (resp.) and different values of $\sigma$.
• Figure 2: Curves constructed by basis functions (\ref{['eourzhu']}), (a) and (b) are the original $\lambda \mu$-curves, and (c) and (d) are the $\sigma \lambda \mu$-curves for various values of parameters
• Figure 3: Plot of the new curves, according to Example \ref{['ex3']}-case (i), for different values of parameters
• Figure 4: Plot of the new curves, according to Example \ref{['ex3']}-case (ii), for different values of parameters
• Figure 5: (a): The $C^{1}$ monotonicity-preserving interpolant for different values of $\sigma$, (b): Corresponding error functions.

Theorems & Definitions (32)

• Definition 2.1
• Theorem 2.2
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• Definition 3.4: Monotonicity-preserving system carnicer1994monotonicity
• Proposition 3.5: Monotonicity preservation
• proof