A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane.

欧几里得空间中的周期格是基向量的所有整数线性组合的无限集。任何晶格都可以由无限多个不同的基生成。这种模糊性得到了部分解决，但在模拟原子位移的扰动下，标准约简仍然不连续。本文完成了二维晶格的连续分类，直至欧几里德等距（或同余）、刚性运动（无反射）和相似性（具有均匀缩放）。新的齐次不变量允许在考虑到上述等价性的格子上轻松计算度量。直到刚性运动的度量尤其重要，并且解决了有关晶格基（不）连续性的所有剩余问题。这些度量产生了实值手性距离，可以连续测量与更高对称性邻居的晶格偏差。几何方法扩展了德龙、康威和斯隆过去的工作。