Table 4 

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

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: If o is odd: 
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: If o is odd: 
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: If o is odd: 
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: If o is odd: 
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) 


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 midplane 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. 

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280 doi:10.1186/1471214811280 