Figure 1.
Simplified example of the convex hulling process.
Black ellipse represents the initial extent of a rubber band stretched to encompass all coloured points. Green polygon represents the convex hull defined by the rubber band ‘snapping to’ the green boundary points. The internal (blue) points lie within the convex hull and do not contribute to defining its maximum extent.
Figure 2.
Convex hulling applied to a human skull.
A, point cloud representing both the inner and outer surface contours of the skull; B, illustrates fit of the convex hull around the maximum extent of the skull; C, convex manifold (water-tight) polytope fitted by the hulling operation.
Table 1.
Convex hull specimen list and sources of body mass.
Figure 3.
Effect of subdividing neck of C. dromedaries on convex hull volume.
A, illustrates extent of neck convex hull without subdivision due to curvature of cervical series; B, tighter fit of convex hulls when divided into two parts.
Figure 4.
Consensus trees used in phylogenetic analysis.
Tick marks represent increments of 10 million years. A, non-primate mammal tree topology and branch lengths derived from [37]–[44]; B, primate tree derived from [45]; C, bird tree derived from [46]–[48].
Figure 5.
A, Body mass (kg) against convex hull volume (m3). For slope equations, see Table 2 (labelled in bold). B, apparent density of convex hull (kg/m3) against body mass (kg). Density did not scale with body mass in non-primate mammals. Density increases with body mass in primates (a = 1042, b = 221, r2 = 0.46, p = 0.025) yet decreases with body mass in birds (a = 1977, b = −619, r2 = 0.36, p<0.05). Black circles, non-primate mammals; green squares, primates; blue triangles, birds.
Table 2.
Ordinary least squares (OLS) and phylogenetically based regression (PGLS, phylogenetic generalised least squares; OU, Ornstein-Uhlenbeck process) of body mass (kg) against convex hull volume (m3).
Figure 6.
Scaling of density against mass in birds.
Feathered carcass density (open circles) scales negatively against body mass (a = 707, b = −50.4, r2 = 0.16, p = 0.04). Plucked carcass density (closed circles) scales negatively against body mass (a = 955. b = −72.4, r2 = 0.61, p<0.001). Note the extremely weak correlation between feathered body density and mass, compared to the much stronger correlation between plucked carcass density and mass (see text for further discussion). Feathered and plucked carcass data from Budgey (2000). Convex hull density in birds also given for reference (closed triangles).