论文标题
关于欧拉的旋转定理
On Euler's rotation theorem
论文作者
论文摘要
众所周知，欧几里得平面的刚性运动最多可以写为三种反射的组成。也许并不是那么众所周知，在任何数量的维度中，相似的结果都适用于欧几里得空间。首先，本文的目的是通过明确构建合适的反射序列，在维度3中呈现这种结果的自然证明，其次，对这种结构进行仔细的分析如何为欧拉的旋转定理提供快速而令人愉悦的几何途径，以及对固定空间的完整分类，无论是定向的完整分类，是否是定向的。最后，我们提出了一个示例，其中我们使用证据的一般方案来对两个明确给定的定向保留等异构体的组成进行分类。我们认为，我们的演讲将强调结果的基本本质，并希望读者，尤其是那些更熟悉通常的线性代数方法的读者，将欣赏论点的简单性和几何风味。
It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a similar result holds for Euclidean space in any number of dimensions. The purpose of the present article is, firstly, to present a natural proof of this result in dimension 3 by explicitly constructing a suitable sequence of reflections, and, secondly, to show how a careful analysis of this construction provides a quick and pleasant geometric path to Euler's rotation theorem, and to the complete classification of rigid motions of space, whether orientation preserving or not. Finally, we present an example where we use the general scheme of our proofs to classify the composition of two explicitly given orientation preserving isometries. We believe that our presentation will highlight the elementary nature of the results and hope that readers, perhaps especially those more familiar with the usual linear algebra approach, will appreciate the simplicity and geometric flavour of the arguments.
