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Table 3 Degrees of freedom in the Procrustes ANOVA for object symmetry with rotation only in two and three dimensions

From: Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

Effect 2D 3D
Individual (n - 1) × (2k - 2) (n - 1) × (3k + c - 3)
Rotation 2k(o - 1) + 2c - 2 3k(o - 1) + 2c - 4
Rotation × individual (n - 1) × (2k(o - 1) + 2c - 2) (n - 1) × (3k(o - 1) + 2c - 4)
Measurement error (r - 1) × n × (2ko + 2c - 4) (r - 1) × n × (3ko + 3c - 7)
  1. Notation: For rotational symmetry of order o, the complete landmark configuration can be subdivided into o different sectors (Figure 4). Each sector contains k landmarks. In addition, there are c landmarks on the centre or axis of rotation (for 2D data, c is 0 or 1; for 3D data, c is 0 or greater). The sample consists of n individuals (specimens), and each specimen has been digitized r times.