Searcharxiv⌕ Search

arXiv subjects

Mao Shi

Publications and source records attributed to Mao Shi.

7 recordsLinked to original sources
arXiv2026

On the First Derivative Bounds for Rational B\'ezier Curves

In this paper we investigate sharp upper bounds for the first derivative of rational B\'ezier curves. A long-standing conjecture posited that the linear bound $\|\mathbf{R}'(t)\| \le n\Omega \max\|\Delta_i\|$ holds for all degrees. We prove that the bound is indeed valid for $n \leq 6$, thus resolving the last open low-degree case. The problem is reformulated as maximizing a variance-like function over a compact box. Using a block argument we show that optima can only appear on one-dimensional faces, reducing the task to a finite family of polynomial inequalities, which are verified exactly via real quantifier elimination. A notable practical feature is that the bound can be evaluated in linear time with respect to the degree, making it attractive for real-time geometric processing. The same structural analysis illustrates the failure for $n=7$ and outlines how the true worst-case constant can be computed.

math.NA↗
arXiv2025

Counterexamples to a Conjecture on First Derivative Bounds of Rational B\'ezier Curves

In this paper we present an explicit counterexample of degree $n=7$, which shows that the conjecture proposed by Li et al. \cite{Li2013} regarding the first derivative bounds for rational B\'ezier curves is generally false. We further derive an explicit rational B\'ezier representation of the first derivative and propose a degree-elevation based computable upper bound for $\sup_{t\in[0,1]}\|\mathbf r'(t)\|$. The bound is valid for any finite elevation order and converges to the true supremum as the elevation degree tends to infinity. An \emph{a priori} tolerance-driven rule is provided to determine a sufficient elevation degree, and the computational complexity of the proposed procedure is analyzed. Numerical experiments validate the counterexample and demonstrate the accuracy and efficiency of the new upper bound across a range of degrees and weight patterns.

math.NA↗
arXiv2023

On the derivatives of rational B\'{e}zier curves

By studying the existing higher order derivation formulas of rational B\'{e}zier curves, we find that they fail when the order of the derivative exceeds the degree of the curves. In this paper, we present a new derivation formula for rational B\'{e}zier curves that overcomes this drawback and show that the $k$th degree derivative of a $n$th degree rational B\'{e}zier curve can be written in terms of a $(2^kn)$th degree rational B\'{e}zier curve.we also consider the properties of the endpoints and the bounds of the derivatives.

cs.GR↗
arXiv2021

Degree reduction of disk rational Bézier curves

How to quickly and stably realize the degree reduction of the rational Bezier curve is an open problem in CAGD. Based on the weighted least squares method and weighted sum method of multi-objective optimization, this paper transforms the degree reduction problem of the rational Bézier curve into a convex optimization problem and then uses quadratic programming to solve it. Prove that the solution is the minimum. Numerical experiments show that the method is fast and stable.

math.NA↗
arXiv2019

Rational Bézier Curves Approximated by Bernstein-Jacobi Hybrid Polynomial Curves

In this paper, we propose a linear method for $C^{(r,s)}$ approximation of rational Bézier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for Bézier curve is used. A error bound between rational Bézier curve and Bézier curve is presented. Finally, some examples and figures were offered to demonstrate the efficiency, simplicity, and stability of our methods.

cs.CG↗
arXiv2012

Approximating rational Bezier curves by constrained Bezier curves of arbitrary degree

In this paper, we propose a method to obtain a constrained approximation of a rational B\'{e}zier curve by a polynomial B\'{e}zier curve. This problem is reformulated as an approximation problem between two polynomial B\'{e}zier curves based on weighted least-squares method, where weight functions $\rho(t)=\omega(t)$ and $\rho(t)=\omega(t)^{2}$ are studied respectively. The efficiency of the proposed method is tested using some examples.

math.NA↗