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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

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

 

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 ( 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

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 ( 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

  1. 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.