This paper solves the continuous classification problem for finite clouds of unlabelled points under Euclidean isometry. The Lipschitz continuity of required invariants in a suitable metric under perturbations of points is motivated by the inevitable noise in measurements of real objects. The best solved case of this isometry classification is known as the SSS theorem in school geometry saying that any triangle up to congruence (isometry in the plane) has a continuous complete invariant of three side lengths. However, there is no easy extension of the SSS theorem even to four points in the plane partially due to a 4-parameter family of 4-point clouds that have the same six pairwise distances. The computational time of most past metrics that are invariant under isometry was exponential in the size of the input. The final obstacle was the discontinuity of previous invariants at singular configurations, for example, when a triangle degenerates to a straight line. All the challenges above are now resolved by the Simplexwise Centred Distributions that combine inter-point distances of a given cloud with the new strength of a simplex that finally guarantees the Lipschitz continuity. The computational times of new invariants and metrics are polynomial in the number of points for a fixed Euclidean dimension.

翻译：本文解决了在欧氏等距变换下有限非标记点云团的连续分类问题。在适当度量下，所需求不变量的Lipschitz连续性受真实物体测量中不可避免的噪声驱动。该等距分类问题的最佳解决案例是中学几何中的SSS定理，该定理指出任何三角形在合同（平面等距）意义下具有由三条边长构成的连续完全不变量。然而，即使对于平面中的四个点，SSS定理也难以简单推广，部分原因在于存在一个四参数族的四点点团具有相同的六对距离。过去大多数等距不变度量的计算时间与输入规模呈指数关系。最后的障碍是先前不变量在奇异构型（例如三角形退化为直线）处的不连续性。上述所有挑战现已被基于单纯形中心分布的方法解决，该方法将给定点团的点间距离与单纯形的新强度相结合，最终保证了Lipschitz连续性。对于固定欧氏维度，新不变量和度量的计算时间与点数量呈多项式关系。