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Table 4 Degrees of freedom in the Procrustes ANOVA for object symmetry with rotation and reflection in two and three dimensions

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

Effect 2D 3D
Individual (n - 1) × (2p + b + m - 1)
= (n - 1) × (k - 1)
(n - 1) × (3p + 2b + 2m + c - 2)
Rotation 2p(o - 1) + b(o - 1) + m(o - 1) + c - 1
= k(o - 1) + c - 1
If o is even:
3 p ( o - 1 ) + b ( 3 2 o - 1 ) + m ( 3 2 o - 2 ) + c - 2
If o is odd:
3 p ( o - 1 ) + 3 b o - 1 2 + 3 m o - 1 2 + c - 2
Reflection 2p + b + m - 1 = k - 1 3p + b + m - 1
Rotation × reflection 2p(o - 1) + b(o - 1) + m(o - 1) + c - 1
= k(o - 1) + c - 1
If o is even:
3 p ( o - 1 ) + b ( 3 2 o - 2 ) + m ( 3 2 o - 1 ) + c - 2
If o is odd:
3p ( o - 1 ) +3b o - 1 2 +3m o - 1 2 +c-2
Rotation × individual (n - 1) × (2p(o - 1) + b(o - 1) + m(o - 1) + c - 1)
= (n - 1) × k(o - 1) + c - 1
If o is even:
n - 1 × 3 p ( o - 1 ) + b ( 3 2 o - 1 ) + m ( 3 2 o - 2 ) + c - 2
If o is odd:
n - 1 × 3 p ( o - 1 ) + 3 b o - 1 2 + 3 m o - 1 2 + c - 2
Reflection × individual (n - 1) × (2p + b + m - 1)
= (n - 1) × (k - 1)
(n - 1) × (3p + b + m - 1)
Rotation × reflection × individual (n - 1) × (2p(o - 1) + b(o - 1) + m(o - 1) + c - 1)
= (n - 1) × k(o - 1) + c - 1
If o is even:
n - 1 × 3 p ( o - 1 ) + b ( 3 2 o - 2 ) + m ( 3 2 o - 1 ) + c - 2
If o is odd:
n - 1 × 3 p ( o - 1 ) + 3 b o - 1 2 + 3 m o - 1 2 + c - 2
Measurement error (r - 1) × n × (4po + 2bo + 2mo + 2 c - 4)
= (r - 1) × n × (2ko + 2 c - 4)
(r - 1) × n × (6po + 3bo + 3mo + 3 c -7)
  1. Notation: For rotational symmetry of order o and reflection, the complete landmark configuration can be subdivided into o different sectors. Because of the reflection symmetry, each sector must be symmetric under reflection. Therefore, the k landmarks contained in each sector can be subdivided into b landmarks on the boundary between sectors, m landmarks on the midline of the sector (the mid-plane for 3D data), and p pairs of corresponding landmarks on either side of the midline of the sector, so that k = 2p + b + m. The difference between boundaries and midlines of the sectors is important if the order of the rotation is even: in this case, the plane of reflection symmetry is assumed to pass through two boundaries between sectors (e.g. the vertical axis in Figure 4). 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.