Table 2.

Number of dimensions in the different components of shape space under object symmetry with rotation only or with rotation and reflection, for landmark data in two and three dimensions

	2D	3D
Symmetry under rotation only:
Symmetric	2k - 2	3k + c - 3
Asymmetric	2k(o - 1) + 2c - 2	3k(o - 1) + 2c - 4
Symmetry under rotation and reflection:
Completely symmetric	2p + b + m - 1 = k - 1	3p + 2b + 2m + c - 2
Reflection symmetry only	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: $3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2$
Rotational symmetry only	2p + b + m - 1 = k - 1	3p + b + m - 1
Completely asymmetric	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: $3p\left(o-1\right)+3b\frac{o-1}{2}+3m\frac{o-1}{2}+c-2$

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.