n-框架的宇称性及其在非Procrustean非线性化中的应用
Parity of n-Frames with Application to Non-Procrustean Orthogonalization
摘要：
欧几里得空间中的完备标准正交标架空间不是路连通的。实际上它正好有两个路径分量，分别包含n个标准坐标的坐标系和两个坐标颠倒的坐标系。对应于这些框架的矩阵分别具有行列式1和负1。我们希望避免在许多变量中实现行列式函数，而是通过纤维化和覆盖映射来处理这些紧空间。重要的纤维化是从框架中删除最终载体，并选择第一个载体。这个纤维化的长同伦序列意味着循环的等价类形成两个元素的群。这个群的一个生成元是吉文斯旋转的圆。三维空间对于证明是至关重要的。它是由罗德里格斯公式处理，连同域的不变性。另一个明确的证明，从单位四元数的映射是满射的主要路径分量是由于Itzhack。因此，使用覆盖的同伦提升定理，正交群不是路径连通的。但是Givens循环执行身份提升，所以它不可能是组生成器。作为一个应用程序，我们回顾了引用算法从数学文献中，有关的Wahba指导问题，产生的正交化的任何矩阵，是度量接近一些旋转。这种方法在Frobenius范数中有偏差，但在现实条件下与Procrustean方法一样好。
英文摘要：
The space of complete orthonormal frames in Euclidean space is not path connected. In fact it has exactly two path components, containing respectively the coordinate frame of n standard coordinates and the frame with two coordinates reversed. The matrices corresponding to these frames have determinant one and minus one respectively. We wish to avoid the implement of the determinant function in many variables, and rather work with these compact spaces by means of fibrations and covering maps. The important fibrations are deletion of the final vector from the frame, and selection of the first vector. The long homotopy sequence for this fibration implies that the equivalence classes of loops form the group of two elements. One generator of this group is a circle of Givens rotations. Dimension three is critical for the proof. It is handled by the Rodrigues formula, together with Invariance of Domain. Another explicit proof that the mapping from unit quaternions is surjective to the main path-component is due to Itzhack. Thus the Orthogonal Group is not path connected, using the Homotopy Lifting Theorem for coverings. But the Givens loop performs an identity lifting, so it cannot be the group generator. As an application we review how quoted algorithms from the astronautical literature, related to the Wahba guidance problem, give rise to an orthogonalization of any matrix that is metrically close to some rotation. This method has a bias in the Frobenius norm, but under realistic conditions serves as well as a Procrustean method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
