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

(n - 1) × (2p + b + m - 1)

= (n - 1) × (k - 1)

(n - 1) × (3p + 2b + 2m + c - 2)

2p(o - 1) + b(o - 1) + m(o - 1) + c - 1

= k(o - 1) + c - 1

If o is even:

$3p\left(o-1\right)+b\left(\frac{3}{2}o-1\right)+m\left(\frac{3}{2}o-2\right)+c-2$

If o is odd:

$3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2$

2p + b + m - 1 = k - 1

3p + b + m - 1

2p(o - 1) + b(o - 1) + m(o - 1) + c - 1

= k(o - 1) + c - 1

If o is even:

$3p\left(o-1\right)+b\left(\frac{3}{2}o-2\right)+m\left(\frac{3}{2}o-1\right)+c-2$

If o is odd:

$3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2$

(n - 1) × (2p(o - 1) + b(o - 1) + m(o - 1) + c - 1)

= (n - 1) × k(o - 1) + c - 1

If o is even:

$\left(n-1\right)\times \left(3p\left(o-1\right)+b\left(\frac{3}{2}o-1\right)+m\left(\frac{3}{2}o-2\right)+c-2\right)$

If o is odd:

$\left(n-1\right)\times \left(3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2\right)$

(n - 1) × (2p + b + m - 1)

= (n - 1) × (k - 1)

(n - 1) × (3p + b + m - 1)

(n - 1) × (2p(o - 1) + b(o - 1) + m(o - 1) + c - 1)

= (n - 1) × k(o - 1) + c - 1

If o is even:

$\left(n-1\right)\times \left(3p\left(o-1\right)+b\left(\frac{3}{2}o-2\right)+m\left(\frac{3}{2}o-1\right)+c-2\right)$

If o is odd:

$\left(n-1\right)\times \left(3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2\right)$

(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)