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:

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If o is odd:

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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:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M10">View MathML</a>

If o is odd:

<a onClick="popup('http://www.biomedcentral.com/1471-2148/11/280/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/11/280/mathml/M11">View MathML</a>


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.

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

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