Refined methods for estimating spirals and spiral deviations

Abstract

Logarithmic spirals are near-perfect fits to the outlines of many accretionary structures. Spiral fitting has recently proved to be efficient at revealing shape changes, and growth periodicities important to taxonomic and paleoclimatic studies. However, the fitting lacks guidelines for choosing the type, or even a best spiral for an outline. Described here is a family of 12 logarithmic spirals that represent many types of accretion. A simple logarithmic spiral has one constant expansion rate relative to the angle turned about an axis. Increasing complexity leads to 11 more logarithmic spirals: three are piecewise logarithmic spirals, three are log-polynomial spirals, a pair combine logarithmic and log-polynomial spirals, then lastly, three are spirals that feature a transition between logarithmic segments. The supplementary R package logspiral generates and fits all 12 spirals, and includes a comprehensive catalogue of examples. Comparing an actual accretionary outline to spirals in the catalogue highlights all nearest matches, ranked in order of a similarity ratio. Fitting nearest spirals to the outline helps choose the best spiral and its associated spiral deviations. Heuristics continue to be both quick and reliable for a final choice of spiral. The best spiral fit for an outline often produces a deviation pattern that is surprisingly consistent with all nearest spirals. The immediate value of other than a simple logarithmic spiral lies in the detection of otherwise hard to see breakpoints and other phase changes in growth that affect overall shape. Including a phase change, or changes in a spiral fit, then reveals any lurking cycles or periodicities in the growth. These periodicities as spiral deviations are considered to represent annual and seasonal growth.

Tooth whorl structure, growth and function in a helicoprionid chondrichthyan Karpinskiprion (nom. nov.) (Eugeneodontiformes) with a revision of the family composition

“…Based on a computer fit to the apices of the smaller of the two pieces of PIN 1655/1, the spiral angle in the C. ivanovi specimen PIN 1655/1 was found to equal 67.4°. In principle it is possible to determine the spiral angle of a logarithmic spiral even from a specimen preserving a small fraction of a volution and even to determine whether the spiral is of logarithmic form or of another form (Aldridge 2020). In practice, such a determination is difficult to make on an imperfect specimen comprising only a small fraction of a volution.…”