# Efficient modified Jacobi-Bernstein basis transformations

## TL;DR

This paper introduces an efficient $O(n^2)$ algorithm for transforming between modified Jacobi and Bernstein bases, significantly improving the degree reduction process of Bézier curves over previous $O(n^3)$ methods.

## Contribution

The paper presents a novel $O(n^2)$ transformation algorithm between modified Jacobi and Bernstein bases, enhancing the efficiency of polynomial degree reduction in Bézier curves.

## Key findings

- Transformations performed with $O(n^2)$ complexity
- Algorithm significantly faster in practice
- Improved degree reduction of Bézier curves

## Abstract

In the paper, we show that the transformations between modified Jacobi and Bernstein bases of the constrained space of polynomials of degree at most $n$ can be performed with the complexity $O(n^2)$. As a result, the algorithm of degree reduction of B\'ezier curves that was first presented in (Bhrawy et al., J. Comput. Appl. Math. 302 (2016), 369--384), and then corrected in (Lu and Xiang, J. Comput. Appl. Math. 315 (2017), 65--69), can be significantly improved, since the necessary transformations are done in those papers with the complexity $O(n^3)$. The comparison of running times shows that our transformations are also faster in practice.