Figure 1.
Basic geometric forms are depicted with their sets of parametric 3D equations.
Transformations and reverse transformations are given. All forms are expressed as combinations of sines and cosines and .
Figure 2.
The torus as a model depicted with a Darboux frame.
A, A torus in x, y, z space with parameters u and v, R is the radius of a whorl and r is the radius of the aperture; B, A torus with u and v indicated with respect to major and minor radii, respectively; C, A torus with a Darboux frame consisting of a unit tangent vector (t), a unit normal vector (u), and a tangent normal vector (v).
Figure 3.
Tangent lines and planes representing Jacobians on a shell model with a gridded surface.
A Darboux frame is shown as a trihedron of orthonormal vectors at points on the surface.
Figure 4.
Illustrations of mollusk shell models created from parametric 3D equations.
A, System 1 gastropod shell 1; B, System 1 gastropod shell 2; C, cone shell; D, System 1 gastropod shell 3; E, System 2 gastropod shell; F, bubble shell; G, ammonite; H, oyster; I, scallop; J, limpet; K, scaphopod; L, clam. Pictures of gastropod shells and their models: M, Melampus coffeus; N, Scaphella junonia; O, Turritella sp.
Table 1.
Measurement of maximum whorl and aperture radii (Rmax and rmax) of gastropod shells.
Table 2.
Material examined from the University of Michigan Museum of Zoology (UMMZ).
Figure 5.
A, PCA ordination of modeled and actual mollusk shells as well as echinoid models and spiral curves in morphospace; ? = modeled clams; • = modeled scallops. Insets depicted are ▵ = Vermicularia (upper left), ? = spirals (middle left), ? = isometric turritellid series (lower center), ◊ = echinoids (upper right). Models in each inset match symbol order in the ordination from left to right; B, PC2 vs. PC1; C, PC3 vs. PC1.
Figure 6.
PCA of basic geometric forms and select mollusks to heuristically explore the properties of surface morphospaces based on Jacobians.
Figure 7.
Ammonite shell models with R and r held constant, showing the result of combinations of coefficient l (degree of coiling) in the x- and y-directions and coefficient q (degree of pleating) in the z-direction.
Coefficient q is related to change in aperture (r and v), and coefficient l is related to change in whorls (R and u).