Shell Coiling in Gastropods: Analysis by Stereographic Projection

Abstract

Shell coiling is readily visualized as a revolution and expansion of the shell's aperture around the coiling axis. Stereographic projections provide an effective tool for plotting the orientations of the aperture through timelspace. In isometric growth, poles to the aperture (e.g., perpendiculars to the aperture plane) plot on a small circle on the stereographic projection, and the center of the small circle defines the orientation of the shell's coiling axis. The stereographic analysis requires no a priori determinations of the coiling axis position in the shell. Stereographic analyses consider shell growth in terms of rigid body motions of the entire aperture, as opposed to conventional methods that consider only the form of individual spiral trajectories (single points on the aperture). Stereographic projections offer a novel and instructive approach to the analysis of shell coiling.