VISUALISATION

As mentioned previously, one advantage of the geometric morphometric methods employed in this study is the preservation of the landmark geometry of each specimen throughout the analysis. This allows the shape variability on each principal component to be interpreted via visualisation. Thus a hypothetical shape along a PC axis can be reconstructed by adding the coordinates of the mean shape (ie. that at the centroid, see Section 1.2.4 and Section 2.22.6) to the product of the PC score of the hypothetical shape and the eigenvector of the PC (O’Higgins & Jones, 1998; O’Higgins, 2000b).

Z , = J + c r

Where X^ denotes the coordinates of the hypothetical shape of

interest, X the coordinates of the mean shape (located at the centroid), c

the PC score of the hypothetical shape on the principal component of interest, and y the eigenvector of the principal component of interest.

The visualisation of shapes using just points to represent the landmarks is very difficult to interpret in 3D, therefore a series of polygons is fitted to the landmark configuration to give an outline or surface to the shape. In this case the landmarks are divided up into triangles, each of which is filled to produce a flat surface and lighting and shading effects are used to give the impression of 3D. The triangles used to produce a surface in this study were selected to represent the bony anatomy of the facial skeleton as accurately and intuitively as possible. However, it should always be borne in mind when interpreting shape variation from such 3D rendered images, that the true shape information is only at the nodes of the triangles, where the landmarks are located and not in between. The triangular polygons selected for this study are illustrated in Figure 2.2, this should be viewed in conjunction with the labelled diagrams and table of the landmarks in Figure 2.1 and Table 2.2, respectively.

As described in Section 1.2.5.1, the use of Cartesian transformation grids derived from thin plate splines is a more objective means of visualising shape differences. In this study, thin plate splines are applied to shapes of interest, whether they be along a PC axis or in the shape space in between axes. One shape is designated as the start shape and the other as the target shape. The thin plate spline deformation necessary to interpolate the differences in landmark configurations between the start and the target shape is then calculated and applied to a Cartesian transformation grid, with which the deformation can be visualised. Such thin plate splines can be fitted to any desired plane within the 3D shape.

Stage 1 2 5 7 2 3 5 3 3 6 Stage 2 6 5 11 4 4 8 8 4 12 Stage 3 10 11 21 10 9 19 11 10 21 Stage 4 7 5 12 13 9 22 8 16 24 Stage 5 3 7 10 1 2 3 4 8 12 Stage 6 14 20 34 27 31 58 28 34 62 Total 42 53 95 57 58 115 62 75 137

Table 2.1 Sample sizes of each species, sex and dental stage used in this study.

page)

Number Type Landmark Definition

1 II Nasospinale - point on the anterior nasal spine which crosses a line drawn from the lowest points of the nasal aperture

2 II Alveolare - midline point on the most inferior point of the alveolar septum between the central incisors

3 III Alare - most lateral point on the nasal aperture 4 III Point of take off of the zygomatic process of the

maxilla

5 II Point on the distal margin of 12 on the external surface of the maxillary alveolus

6 II Point on the distal margin of C on the external surface of the maxillary alveolus

7 II Point on the distal margin of P4 on the external surface of the maxillary alveolus

8 II Most inferior and distal point on the maxillary alveolus

9 II Point at which median palatine suture meets the posterior margin of the incisive canals (fossa) 10 1 Point at which the transverse and median

palatine sutures intersect

11 II Staphylion - point on median palatine suture that crosses a line drawn from the deepest part of the arches of the horizontal plate of the palatine bones

12 III Infratemporal crest

13 1 Nasion - point at which the internasal and frontonasal sutures meet in the midline

14 II Rhinion - inferior free end of the internasal suture 15 II Frontomalare temporale - point where the

frontozygomatic suture crosses the outer orbital rim

16 II Frontomalare orbitale - point where the

frontozygomatic suture crosses the inner orbital rim

17 III Superior margin of the orbit

18 II Dacryon - most superior point of the

lacrimomaxillary suture (ie. where it meets the frontal bone)

19 1 Maxillofrontale - point at which the anterior lacrimal crest and frontomaxillary suture meet 20 II Inferior free end of the lateral nasal suture 21 III Infraorbital foramen

22 II Zygoorbitale - point at which the

zygomaticomaxillary suture meets the orbital rim 23 II Zygomaxillare - most inferior point of the

zygomaticomaxillary suture

24 III Jugale - point in the depth of the notch between the frontal and temporal processes of the

zygomatic bone

25 II Hormion - midpoint of the postero-superior border of the vomer

26 III Superolateral point of the posterior nasal aperture (internal choana)

27 III Inferolateral point of the posterior nasal aperture (internal choana)

28 III Anterior most point of the inferior orbital fissure (taken interiorly)

(a) •15 24 14? /2 3 4* 2 • 12 26 • 25

Figure 2.1 Landmarks used in this study; (a) lateral view, (b) inferior view (continued on next page). Numbers refer to landmarks listed in Table 2.2. (Adapted from Fenart & Deblock, 1973a)

(C) 19* 16? •15 22 24 •20 / * 2 3

Figure 2.1 (continued from previous page) Landmarks used in this study; (c) anterior view. Numbers refer to landmarks listed in Table 2.2. (Adapted from Fenart & Deblock, 1973a)

r —

(a)

Figure 2.2 Polygon surface used in this study; (a) lateral view, (b) inferior view (continued on next page). The surface is created by filling each polygon (triangles in this case). Not all potentially visable landmarks and polygons are shown in each view. For landmarks refer to Figure 2.1 and Table 2.2. (Adapted from Fenart & Deblock, 1973a)

(C)

vvFTnTfJ

Figure 2.2 (continued from previous page) Polygon surface used in this study. The surface is created by filling each polygon (triangles in this case). Not all potentially visable landmarks and polygons are shown in each view. For landmarks refer to Figure 2.1 and Table 2.2. (Adapted from Fenart & Deblock, 1973a)

CHAPTER 3

ONTOGENETIC ALLOMETRY, SEXUAL DIMORPHISM AND