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High Quality Content by WIKIPEDIA articles! In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. It is a subgroup of the orthogonal group O(2), the group of all isometries which leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(2) itself is a subgroup of the Euclidean group E(2) of all isometries.