We prove that, under generic conditions, the covariance matrix of a 3D point cloud with rotational symmetry has a simple eigenvalue, whose associated eigenvector provides the direction of the axis of rotation, and a double eigenvalue. The direction of the axis of rotation can also be computed from higher order tensors related to the point cloud, which is useful in pathological cases. This leads to a very simple algorithm for detecting rotational symmetry and computing the axis of rotation.

中文翻译：

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使用协方差矩阵和高阶张量的 3D 点云的旋转对称性


我们证明，在一般条件下，具有旋转对称性的 3D 点云的协方差矩阵具有一个简单的特征值，其关联的特征向量提供旋转轴的方向，以及一个双特征值。旋转轴的方向也可以从与点云相关的高阶张量计算出来，这在病态情况下很有用。这导致了一种非常简单的算法，用于检测旋转对称性和计算旋转轴。