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:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M12">View MathML</a>

If o is odd:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M13">View MathML</a>

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:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M14">View MathML</a>

If o is odd:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M15">View MathML</a>

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:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M16">View MathML</a>

If o is odd:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M17">View MathML</a>

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:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M18">View MathML</a>

If o is odd:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M19">View MathML</a>

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 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.

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280   doi:10.1186/1471-2148-11-280

Open Data