Figure 4

Sectors and types of landmarks for complex object symmetry with a rotation and reflection. To compute the dimensionalities of the different components of shape space, it is helpful to subdivide the configuration of landmarks into sectors and to distinguish different types of landmarks. The diagram shows an example of symmetry under rotation of order 4 and reflection. Therefore, the configuration can be divided into four sectors: the regions that correspond to each other when the rotation is applied (sector boundaries are indicated by solid black lines). If the symmetry also includes reflection, as in this example, the arrangement of landmark in each sector is also bilaterally symmetric about the midline or mid-plane of each sector (dashed lines). Several types of landmarks can be distinguished. There may be a landmark in the centre of rotation or, for 3D data, there may be multiple landmarks of the axis of rotation (c = 0, 1 for 2D data; c ≥ 0 for 3D data). Each sector contains k landmarks. If the order of rotation is denoted o, the total number of landmarks is therefore c + ko (in the diagram, c = 1, k = 5 and o = 4, so that there are 1 + 5 × 4 = 21 landmarks). If the symmetry group contains reflection as well as a rotation, the k landmarks of each sector can be subdivided into b landmarks on the sector boundary, m landmarks on the midline or mid-plane of the sector, and p pairs of corresponding landmarks on either side of the midline (therefore, k = b + m + 2p). We define the sector boundary as running through the axis or plane of reflection on at least one side of the centre or axis of rotation (if the order of rotation is even, two sector boundaries are in the axis or plane of reflection).