It is known that B\'{e}zier curves and surfaces may have multiple representations by different control polygons. The polygons may have different number of control points and may even be disjoint. Up to our knowledge, Pekerman et al. (2005) were the first to address the problem of testing two parametric polynomial curves for coincidence. Their approach is based on reduction of the input curves into canonical irreducible form. They claimed that their approach can be extended for testing tensor product surfaces but gave no further detail. In this paper we develop a new technique and provide a comprehensive solution to the problem of testing tensor product B\'ezier surfaces for coincidence. In (Vlachkova, 2017) an algorithm for testing B\'ezier curves was proposed based on subdivision. There a partial solution to the problem of testing tensor product B\'ezier surfaces was presented. Namely, the case where the irreducible surfaces are of same degree $(n,m)$, $n,m \in \mathbb{N}$, was resolved under certain additional condition. The other cases where one of the surfaces is of degree $(n,m)$ and the other is of degree either $(n,n+m)$, or $(n+m,m)$, or $(n+m,n+m)$ remained open. We have implemented our algorithm for testing tensor product B\'ezier surfaces for coincidence using Mathematica package. Experimental results and their analysis are presented.

翻译：已知Bézier曲线和曲面可能通过不同的控制多边形具有多种表示形式。这些多边形可能具有不同数量的控制点，甚至可能互不相交。据我们所知，Pekerman等人（2005）首次提出了测试两条参数多项式曲线是否重合的问题。他们的方法基于将输入曲线约化为标准不可约形式，并声称该方法可推广用于测试张量积曲面，但未提供进一步细节。本文提出了一种新技术，为测试张量积Bézier曲面重合性问题提供了全面解决方案。Vlachkova（2017）曾提出一种基于细分算法的Bézier曲线测试方法，并给出了张量积Bézier曲面重合性问题的部分解：即在满足特定附加条件的情况下，解决了不可约曲面具有相同度数$(n,m)$（$n,m \in \mathbb{N}$）的情形。而其他情形——当一曲面的度数为$(n,m)$，另一曲面度数分别为$(n,n+m)$、$(n+m,m)$或$(n+m,n+m)$时——仍悬而未决。我们使用Mathematica软件包实现了测试张量积Bézier曲面重合性的算法，并给出了实验结果与分析。