Fig 1.
A simplistic two-dimensional diagram describing shape quantification using the Dirichlet normal energy method.
(a) Given two surfaces i and j, normal vectors of magnitude one are derived for equal-length regions of interest. End-points of normal vectors define ni and nj, the normal maps of i and j respectively. (b) Δn represents the change in position of end-points of normal vectors, or the change in the normal map. Superimposing origin points of normal vectors corrects Δn for the change in surface position (Δi or Δj). Arc length of superimposed normal vectors reflects degree of surface bending. (c) Stated explicitly, surface bending for a region of interest can be said to be characterized by change in the normal map (Δn) relative to change in surface position (Δi or Δj). Surface i shows greater bending. For the three-dimensional polygon mesh case used here, regions of interest are individual polygon vertices (see text for more details).
Fig 2.
Diagram demonstrating edge vectors u and v of given polygon and approximated normal vectors (red) for polygon vertices.
End-points of vertex normals form a polygon with edge vectors nu and nv. Translating vertex normals to a common origin point visualizes spreading of nu and nv relative to spreading of u and v. Polygons on more curved surfaces will produce greater relative spreading of nu and nv. e(p) quantifies relative spreading to calculate degree of surface bending per polygon.
Fig 3.
Comparison of triangulated mesh and DEM grid formats of second mandibular molar tooth surfaces for species Cercocebus atys and Theropithecus gelada.
Teeth are presented in oblique perspective, with distal and buccal aspects toward bottom-right and bottom-left respectively. Color scaling reflects elevation. Triangulated mesh data is used for calculation of 3D-OPCR and DEM grid data is used for calculation of DEM-OPCR. Triangulated mesh data used here represents molar surface at a relatively finer resolution compared to DEM data.
Fig 4.
Box plot of DEM-OPCR and 3D-OPCR by species.
Fig 5.
Results of DEM-OPCR and 3D-OPCR algorithms applied to molar tooth surfaces from Fig 3.
Results from both algorithms are presented in occlusal perspective, with distal aspect at top and buccal aspect toward right. 3D-OPCR results are also shown in oblique perspective, with distal and buccal aspects toward bottom-right and bottom-left respectively. Color wheel at bottom left indicates patch aspect direction for occlusal perspective. 3D-OPCR results are presented with surface shading while DEM-OPCR results are not.
Table 1.
Species mean by OPCR treatment.
Table 2.
ANOVAs on OPCR treatments with species factor.
Table 3.
Pairwise post-hoc comparisons of 3D-OPCR between species.
Table 4.
ANOVA on ΔOPCR with species factor.
Table 5.
Pairwise post-hoc comparisons of ΔOPCR between species.